Diophantine applications of geometric invariant theory 2856298656, 9782856298657
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Marco MACULAN
DIOPHANINE APPLICAIONS OF GEOMERIC INVARIAN HEORY
MÉMOIRES DE LA SMF 152
Société Mathématique de France 2017
Comité de rédaction Chrisine BACHOC Emmanuel BREUILLARD Yann BUGEAUD Jean-François DAT Marc HERZLICH O’ Grady KIERAN
Raphaël KRIKORIAN Julien MARCHÉ Lauren MANIVEL Emmanuel RUSS Chrisophe SABOT Wilhelm SCHLAG
Pascal HUBERT (dir.) Difusion AMS P.O. Box 6248 Providence RI 02940 USA www.ams.org
Maison de la SMF B.P. 67 13274 Marseille Cedex 9 France [email protected] Taris 2017
Vene au numéro : 35 € ($ 52) Abonnemen élecronique : 113 € ($ 170) Abonnemen avec supplémen papier : 162 €, hors Europe : 186 € ($ 279) Des condiions spéciales son accordées aux membres de la SMF. Secrétariat : Nathalie Christiaën Mémoires de la SMF Sociéé Mahémaique de France Insiu Henri Poincaré, 11, rue Pierre e Marie Curie 75231 Paris Cedex 05, France Tél : (33) 01 44 27 67 99 Fax : (33) 01 40 46 90 96 [email protected] http://smf.emath.fr/ © Sociéé Mahémaique de France 2017 Tous drois réservés (aricle L 122–4 du Code de la propriéé inellecuelle). Toue représenaion ou reproducion inégrale ou parielle aie sans le consenemen de l’édieur es illicie. Cete représenaion ou reproducion par quelque procédé que ce soi consiuerai une conreaçon sancionnée par les aricles L 335–2 e suivans du CPI.
ISSN 0249-633X (prin) 2275-3230 (elecronic) ISBN 978-2-85629-865-7
Séphane SEURET Direceur de la publicaion
MÉMOIRES DE LA SMF 152
DIOPHANINE APPLICAIONS OF GEOMERIC INVARIAN HEORY
Marco MACULAN
Société Mathématique de France 2017 Publié avec le concours du Centre National de la Recherche Scientifque
M. MACULAN Institut Mathématique de Jussieu, Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris. E-mail : [email protected] Url : http://webusers.imj-prg.fr/~marco.maculan/
2010 Mathematics Subject Classification. --- Primary 14L24, 14G40, 11K60, 14G22. Key words and phrases. --- Geometric Invariant Theory, diophantine approximation, Roth’s Theorem, Berkovich spaces, height o semi-stable points. DOI. --- 10.24033/msm.460.
Partially supported by ANR research project “Positive” ANR-2010-BLAN-0119-01.
DIOPHANTINE APPLICATIONS OF GEOMETRIC INVARIANT THEORY Marco MACULAN
Abstract. --- This text consists o two parts. In the frst one we present a proo o Thue-Siegel-Roth’s Theorem (and its more recent variants, such as those o Lang or number felds and that “with moving targets” o Vojta) as an application o Geometric Invariant Theory (GIT). Roth’s Theorem is deduced rom a general ormula comparing the height o a semi-stable point and the height o its projection on the GIT quotient. In this setting, the role o the zero estimates appearing in the classical proo is played by the geometric semi-stability o the point to which we apply the ormula. In the second part we study heights on GIT quotients. We generalise Burnol’s construction o the height and refne diverse lower bounds o the height o semi-stable points established to Bost, Zhang, Gasbarri and Chen. The proo o Burnol’s ormula is based on a non-archimedean version o Kemp-Ness theory (in the ramework o Berkovich analytic spaces) which completes the ormer work o Burnol. Résumé (Applications diophantiennes de la théorie géométrique des invariants) Ce texte est constitué de deux parties. Dans la première nous présentons une preuve du théorème de Thue-Siegel-Roth (et des variantes plus récentes, comme celle de Lang pour le corps de nombres et celle with moving targets de Vojta) basée sur la théorie géométrique des invariants (GIT). Le théorème de Roth est déduit d’une ormule reliant la hauteur d’un point semi-stable et la hauteur de sa projection dans le quotient GIT. Dans ce cadre, le rôle du « lemme des zéros » présent dans la preuve classique est joué par la semi-stabilité géométrique du point auquel on applique la ormule. Dans la deuxième partie nous étudions la hauteur sur les quotients GIT. Nous généralisons la construction de Burnol de cette hauteur et nous améliorons plusieurs minorations de la hauteur de point semi-stables précédemment établies par Bost, Zhang Gasbarri et Chen. La preuve de la ormule de Burnol porte sur une version non-archimédienne de la théorie de Kemp-Ness (dans le langage de la géométrie analytique de Berkovich), qui complète le travail antérieur de Burnol.
© Mémoires de la Société Mathématique de France 152, SMF 2017
CONENS
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1. Geometric Invariant Teory and Arakelov Geometry: basic results . . . . . . . . 13 1. T F F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2. L q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2. Diophantine approximation on P via Geometric Invariant Teory . . . . . . . . 1. S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. F F F M T . . . . . . . . . . . . . . . . . . . . . . . . . 3. U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. E : . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. S P P W . . . . . . . . . . . . . . . . . . . .
27 27 36 44 48 54 60
3. Kemp-Ness Teory in Non-Archimedean Geometry . . . . . . . . . . . . . . . . . . . . . . . 1. S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. KN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. M GI q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69 70 76 90 102
4. Heights on GI quotients: urther results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. E q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. T B, G Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. L q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113 114 120 129 136
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
INRODUCION
I , R’ T R , pq Q p q q
p q . T R’ T T. T “” f : “” . T R ( T, S G’) . O , D — R’ T — ( “D’ L”) : , . T D’ L E V [28]; N [50] F’ P T “” . T D’ L B [7]: B. U P T, F W [30] S’ S T, . T Z L, R S, . T ( [29]). T : D’ L ( P T); P T. I O [53] S [65] V [69] R’ T — “ ” — . I S’ S T. H B G [8, T 6.5.2 §6.6] q R’ T f “ ” S’ S T.
CONENS
7
T G I T ( A ) . B [20] GI q . B [12], [13], Z [75] S [64] ( ) (e.g. , …). G [32] B Z GLn . C [22] — RR [56] [67] — . I , () x x n a an P Kn P K ′n v ( K K ′ ). T , , n B [7, T 2], R’ T . L . F . I C 1 G I T R’ T. T C 4, R’ T C 2. I C 3 4 . I C 2 R’ T ( ) q F F C 2 ( “ ”). I C 3 KN [46] . I C 4 q ( ) C 1. A . I C 2 C 1 2.3, C 4 C 3 1. W .
SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
8
CONENS
Acknowledgements T J.B. B. I . D I : I A. CL, A. D, C. G, M. N A. T. F, I q .
MÉMOIRES DE LA SMF 152
CONVENIONS
H . 1. — F A n, An HA An A T PA A A PA PS A
R , M A , M HA M A PM PS M 2. — L A , M A n . S M n M n HA M A n 3. Hermitian Norms. — L E, F q ∥∥E , ∥∥F ⟨ ⟩E , ⟨ ⟩F . L r . ▷ O E C F ∥∥E F ⟨v w v ′ w ′⟩E F ⟨v v ′⟩E ⟨w w ′⟩F
v v ′ E w w ′ F . ▷ O r Sr E ∥∥Sr E q E r Sr E. I e en E, n C E, n r r n r rn r : r ∥e r enr n ∥Sr E r r r n r rn T : f Sr E д Ss E, ∥ f д∥Sr s E ∥ f ∥Sr E ∥ д∥Ss E L ∥∥Sr E : f Sr E, f x r ∥ f ∥ r ∥ f ∥S E ,x E ∥x ∥E ▷ O r r E ∥∥r E ⟨v vr w w r ⟩r E ⟨vi w j ⟩E i j r v vr w w r E.
CONENS
10
W H q 1 : (1)
∥v vr ∥r E
r ∏ i
∥vi ∥E
T ∥∥r E no q E r r E, r q ( [22, L 4.1]). ▷ F ϕ E F ϕ ( ∥∥E ∥∥F ). O HC E F ∥∥HE F ⟨ϕ ⟩HE F ϕ ◦
ϕ HC E F . I e en E, ∥ϕ ∥HE F ∥ϕe ∥F ∥ϕen ∥F
W E C F HC E F .
4. Non-archimedean norms. — L K o . I o K . M , o E E E o K : v E ∥v ∥ E K v E
T ∥∥ E o : , ϕ E F (. ) o ∥∥ E E Eo K (. ∥∥ F F Fo K) ∥∥ F F (. q ∥∥ E ϕ, , w
ϕ v w
∥v ∥ E
w F .) F r , Sr E (. r E) r Er Er Sr E (. Er r E). I , (. H q ). 5. Normalisation o places. — F K oK VK . I v K, Kv K v Cv Kv ( q v). A v p p v p Kv Qp 1. T q H’ E H H’ q .
MÉMOIRES DE LA SMF 152
ACKNOWLEDGEMENS
11
6. Hermitian vector bundles and Arakelov degree. — L K , oK VK . A E oK E , K C, ∥∥ E E E C. T . F v VK ∥∥ Ev Kv Ev E oK Kv . I E, F oK , homomorphism o hermiian vecor bundles ϕ E F oK , K C, : , v E C, ∥ϕv∥ F ∥v ∥ E
I L , , degree ∑ L #Ls L ∥s ∥ L
∑
v VK
∑
K C
∥s ∥ Lv
v VK
Kv R ∥s ∥ Lv
s L . I , P F, s. F E E E E ▷ degree: E ▷ slope: E E ▷ maximal slope: E F , F E
F E E.
Proposition (S q, [14]). — Le E F be oK -hermiian vecor bundles and le ϕ E oK K F oK K be an injecive homomorphism o K-vecor spaces. Ten, ∑ F ∥ϕ ∥ v E where, or every place v VK ,
v VK
∥ϕ ∥ v
,s Ev
∥ϕs∥ Fv ∥s ∥ Ev
SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
CONENS
12
7. Height unction with respect to a hermitian line bundle. — L K , oK X oK . ▷ A hermiian line bundle
L L ∥∥ L K C
L X , K C, ∥∥ L L X C . T ∥∥ L K C . L K ′ K, oK ′ P K ′ X. B , P oK P S oK ′ X
T oK ′ L: oK ′ P L. P L
▷ T heigh o P wih respec o L h L P
L P K ′ K
M , s H X L P, ∑ K ′ Kh L P # P LsP P L ∥s ∥ L P K C K ′,
T h LP h L XQ R heigh uncion wih respec o L.
MÉMOIRES DE LA SMF 152
CHAPER 1 GEOMERIC INVARIAN HEORY AND ARAKELOV GEOMERY: BASIC RESULS
I G I T ( F F, T 1.6) C 2 “ ” R’ T: GI q. I q ( T 2.1). E C 4 q, : . W R’ T C 2. 1. Te Fundamental Formula L K oK . 1.1. — L X oK oK 1 G L very ample G X. T E X L G. T G P E OE G. T j X P E j OE ≃ L Gq. 1. O k G — i.e. f k — reducive . O S G reducive ( G S -reducive group) : 1) G f, S ; 2) s S , s G s G S s ( s s). E S GLn S , SLn S . I oK SLn oK . T [9, C IV] [34], [24] .
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CHAPER 1. GEOMERIC INVARIAN HEORY AND ARAKELOV GEOMERY
1.2. — A x X semi-sable , f d , G s X Ld x. d C oK A d X L . A S [63, II.4, T 4] AG X Ld G d
G A oK Y P AG q X X ( G L). F
X G ( X L G ). L X Y q . S Y , f D , MD Y Gq ϕ D MD LD X 1.3. — F K C ∥∥ E E C G C. S ∥∥ E K C .
L ∥∥ O FS OE ∥∥ E ∥∥ L L. D L X L ∥∥ L K C . F y Y C t y MD ∥t ∥ MD y ϕ D t LD x x X C x y
Lemma 1.1. — Le f Y MD be a global secion.
Tere exiss a unique G-invarian global secion f X LD which vanishes idenically on X X and such ha ϕ D f f X . For every complex embedding K C,
y Y C
∥ f ∥ MD y
x X C
∥ f ∥ LD x
I , ∥t ∥ MD y t y MD , ∥∥ MD MD . Proo. — 1) I Y MD . 2) I 1).
MÉMOIRES DE LA SMF 152
□
1. HE FUNDAMENAL FORMULA
15
1.4. — L M D Y h M D M D ( [8, 2.7.17]). D h M D Q h X L G Q Y Q D ( D).
Lemma 1.2. — h X L G .
Proo. — L D MD t t N Y MD . L K ′ K, Q K ′ Y Q oK ′ Y . T i N ti Q. B , ∑ ∥ti ∥ MD Q K ′ Qh M D Q # Q MD Q ti Q MD ′
K K
∑
K ′ C
∥ti ∥ MD y
K C y Y C
I f i N K C ∥ti ∥ MD Y C. T L 1.1 (2) . □ Remark 1.3. — P L ∥∥ MD . T . A C 3 ( T 1.1). I KN ( K [48, C 8, §2], B [20] S [62, C 5]). A ( ) Z [75, T 4.10]. 1.5. Instability measure. — L v K. I v ∥∥ Lv (. ∥∥ MD v ) L (. MD ) 2 . F Cv x XCv v-adic insabiliy measure v x
д GCv
∥д s ∥ Lv д x ∥s ∥ Lv x
. T s. I x s jx P E Cv , x L
v x
д GCv
∥д x ∥ Ev ∥ x ∥ Ev
2. L x Cv X. S X , Cv x ov x X, ov Cv . T x L ov : s . E s x L q s s Cv . S ∥s ∥ Lv x v
T s x L. S [8, E 2.7.20].
SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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CHAPER 1. GEOMERIC INVARIAN HEORY AND ARAKELOV GEOMERY
Proposition 1.4. — Le v be a place o K. For every Cv -poin x X Cv and every non-zero secion t x MD , ∥t ∥ M v x v x D D ∥ t ∥ LD v x
Proo. — I ∥∥ MD G . S v . U MD MD . L y x y Yov q ov Y y ( ov Cv ). U t t ov y MD ∥t ∥ MD v y . S MD , f Y MD ov y f t. A P 1.1, f q G f X LD ov X . F д GCv . S f , ∥ f ∥ LD v д x ∥ f ∥ LD v д x
∥t ∥ MD v y ∥ t ∥ LD v д x ∥t ∥ MD v y. д GCv , v x
∥ t ∥ LD v д x ∥t ∥ M v y D D D ∥ t ∥ LD v x д GCv ∥ t ∥ LD v x
□
Remark 1.5. — F v, P q x x v , i.e. F v XoK F v GF v ( F v Cv ). I C 3 : , q x ( T 1.16). 1.6. Fundamental ormula. — S : heorem 1.6 (F F). — Le P XK be a semi-sable poin. Ten or almos all places v VK he insabiliy measure v P is zero and ∑ h L P v P h M D P D K Q v V K
I T 1.6 C:
Corollary 1.7. — For every semi-sable poin P X K, ∑ v P h X L G h L P K Q v V K
Remark 1.8. — O C 3 4 q F F ( “F F”). F , T 1.5 C 4.
MÉMOIRES DE LA SMF 152
2. LOWER BOUND OF HE HEIGH ON HE UOIEN
17
I , B [20, P 5] ( C 1.6 C 4). 2. Lower bound o the height on the quotient 2.1. Statement o the lower bound. — L E E En n oK . L F oK GL E GL E oK oK GL En GL F , , oK , uniary, i.e., K C, U E U∥∥ E U∥∥ En GLd C
∥∥ F .
heorem 2.1. — Wih he noaion inroduced above, le b b bn be n-uple o inegers and Eb Enbn F a homomorphism o hermiian vecor bundles, generically surjecive and GL E-equivarian. Ten, n n ∑ ∑ bi Ei bi Ei h P F OF SL E i i
where OF is equipped wih he Fubini-Sudy meric given by F and SL E is he oK reducive group SL E oK oK SL En .
H, i n bi , Eibi oK Eibi Eibi bi . A C 4 ( T 1.11). T q Ei ( R’ T). Remark 2.2. — T [22, T 4.2] : C K P P F , n n ∑ ∑ h OF P bi Ei bi Ei i i C’ T 2.1 q C 1.7 h OF P h P F OF SL E
Remark 2.3. — I T 2.1 bi . I, bi , , i n, Ei bi ′ Ei Ei SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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CHAPER 1. GEOMERIC INVARIAN HEORY AND ARAKELOV GEOMERY
S GL E′ GL E′ GL En′ . I
Eb Enbn F
T 2.1, GL E′ ′ E′
b
En′
bn
SL E′
F
SL E SL E′ SL En′ T q MD . I , h P F OF SL E h P F OF SL E′ P F
T T 2.1 b b N . Invariant theory or a product o linear groups 2.2. — L k . L n E E En n k . D GLE GLE k k GLEn SLE SLE k k SLEn
Deinition 2.4. — L F k . A , i.e. k , GLE GLF homogeneous o weigh b b bn Zn , k S S t tn Gm S, t E tn En t b tnbn F
Proposition 2.5. — Le GLE GLF be a homogeneous represenaion o weigh b b bn and suppose ha he subspace o SLE-invarian elemens o F is non-rivial. Ten: or every i n he dimension ei o Ei divides he ineger bi ; or every k-scheme S, any S-poin д дn o GLE and any SLE-invarian elemen w o F : (2.1)
д дn w д b e дn bn en w
Proo. — T . □ 2.3. — F N SN N ( N , S ∅ ). I E k SN N E N . E, SN x x N E, x x N x x N
Deinition 2.6. — F N Z k S N Ek E N E N . MÉMOIRES DE LA SMF 152
2. LOWER BOUND OF HE HEIGH ON HE UOIEN
19
2.4. — L b b bn n . T Sb Sb Sbn
k E b E b Enbn . T k GLE k Ek E b Ek E b k k Ek Enbn
T GLE GLEE b . P 2.52) EE Db GLE SLE . Deinition 2.7. — T Sb E b n b k i kSbi EE Eb .
T Eb EE b . T F M T I T f (c. [71, C III], [22, T 3.1, C] [2, A 1]):
heorem 2.8 (F M T I T). — Suppose ha he characerisic o k is zero. Te subspace o SLE-invarian elemens o he k-vecor space EE b is he image o he homomorphism Eb . Deinition 2.9. — W ei bi i n. C k bi Ei bi ei Eibi Ei bi E Ei : bi Ei bi ei E Ei Eibi Eiei bi ei Ei bi ei
Eibi
bi ei Eibi Ei Ei
. S Eb Eb En bn .
Corollary 2.10. — Suppose ha he characerisic o k is zero. Le F be a non-zero k-vecor space o nie dimension and GLE GLF be a represenaion. Le b b bn be a n-uple o non-negaive inegers and n ⊗ ϕ Eibi F i
be a surjecive and GLE-equivarian homomorphism o k-vecor spaces. Te represenaion is homogeneous o weigh b b bn and i he subspace o SLE-invarian elemens o F is non-zero: 1) For every i n he dimension ei o Ei divides he ineger bi .
SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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CHAPER 1. GEOMERIC INVARIAN HEORY AND ARAKELOV GEOMERY
2) Te subspace o SLE-invarians o F is he image o he homomorphism ϕ ◦ Eb ◦ Eb
n ⊗ i
kSbi
n ⊗ i
Ei bi ei F
T F M T I T : Remark 2.11. — I E G unique Gq R E E E G ( R ). T q : Gq E F G R F ◦ ◦ R E . I , ϕ E G F G . F , [49, . 182] [48, C1, §1]. Non-hermiian norms and ensor produc. — I . T [36] [33, N , . 33] . L N i N Vi ∥∥Vi . L ∥∥Vi Vi . D V V VN . Deinition 2.12. — T V , v V , ∥v ∥V
ϕ i Vi i N
∥v ∥V
R {∑
ϕ ϕ N v , ∥ϕ ∥V ∥ϕ N ∥VN
∥v ∥V ∥v N ∥VN v
R ∑
} v v N
T V q ∥∥V V VN . A, V q ∥∥V V VN . I ∥∥Vi V VN V . Proposition 2.13. — Wih he noaion inroduced above: Te -norm ∥∥ resp. he -norm ∥∥ is he smalles resp. he bigges among he norms ∥∥ on V such ha, or i N and vi Vi , ∥v v N ∥ ∥v ∥V ∥v N ∥VN . ∥v v N ∥ ∥v ∥V ∥v N ∥VN and, or i n and ϕ i Vi ,
∥ϕ ϕ N ∥ ∥ϕ ∥V ∥ϕ N ∥VN
. ∥ϕ ϕ N ∥
where ∥∥ denoes he operaor norm induced by ∥∥ on V . MÉMOIRES DE LA SMF 152
∥ϕ ∥V ∥ϕ N ∥VN
2. LOWER BOUND OF HE HEIGH ON HE UOIEN
21
∥v ∥V ∥v ∥V or all v V .
(D) Te isomorphism V ≃ V C C VN induces he ollowing isomeries: V VN V VN
V VN V VN
(F) For every i N le Wi be a nie-dimensional complex vecor space equipped wih a norm ∥∥Wi and le ϕ i Vi Wi be a linear map decreasing he norms. Ten, he induced maps ϕ ϕ N V VN W WN ϕ ϕ N V VN W WN decrease he norms. Le L be a normed vecor space o dimension . Ten, V VN L V VN L V VN L V VN L
Skech o he proo. — P 1), 4) 5) . B , i N vi Vi N N ∏ ∏ ∥vi ∥Vi ∥vi ∥Vi ∥v v N ∥ 1).
i
i
F 1) 2) .
□
Proposition 2.14. — Le V and W be nie-dimensional normed vecor spaces. Te operaor norm on HC V W coincides wih he -norm on V C W hrough he canonical isomorphism V C W HC V W Proo. — T T [36, §1.1, Tè 1] E V , F C G W .
□
Remark 2.15. — L L . I P L L ≃ C L L C. Proposition 2.16. — Le W be an hermiian vecor space and le r be a posiive ineger. Endow he exerior powers r W wih he hermiian norm dened in § 3 on page 9. Ten he canonical map W r r W decreases he norms. SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
22
CHAPER 1. GEOMERIC INVARIAN HEORY AND ARAKELOV GEOMERY
Proo. — F w W r w H q (1) 10 ⟨ w w⟩ Ei
R ∑
R ∑
R ∑
.
R
w
w r
⟨w w r w w r ⟩ Ei ∥w ∥Ei ∥w r ∥Ei ∥w ∥Ei ∥w r ∥Ei
∥w ∥Ei ∥w r ∥Ei
□
Applicaion o he lower bound o he heigh on he quoien 2.5. — L T 2.1 bi . D Y q P F SL E , f D, M D Y OFD. F D MD . 2.6. Application o the First Main Teorem o Invariant Teory. — A Y . S K ,
F Eb Eb oK oK Enbn
F i n Ei K Ei oK K E b K E b K K Enbn . T SLE SD F MD . T i n ei K Ei Dbi . C E Db E Db ( D 2.7 2.9) : ϕ E Db SD E b
E Db E Db K K EnDbn .
Lemma 2.17. — For every i n le i Ei be non-zero. A se o generaors o he SLE-invarian elemens o he vecor space SD E b PE b OD
is given by he image hrough ϕ ◦ o he elemens Db e
f
Dbn e n
where n ranges in SDb SDb SDbn . MÉMOIRES DE LA SMF 152
2. LOWER BOUND OF HE HEIGH ON HE UOIEN
23
Trough he idenicaion Y MD oK K ≃ PE b ODSLE : { ∑ } ϕ ◦ f h P Eb OFSL E MD v D SD b v V YCv K
Proo. — 1) T q C 2.10 ( SD E b SLV q ϕ). 2) L Q YQ K ′ K. S MD , (1) SD b SLE ϕ ◦ f — MD — Q. B : ∑ h MD Q ϕ ◦ f M v Q ∑
v VK
v VK
D
y YCv
.
ϕ ◦ f M
Dv
y
□
2.7. Size o the invariants. — C oK
EDb EDb EnDbn
E Db (. E Db ) oK EDb K C ∥∥ e n Dbn e n E e Db e En
(. ∥∥ ). C oK EoK EDb K C ∥∥ E EDb H E Db E Db
D oK E EDb . F K C SD Eb C ( § 3 9). Lemma 2.18. — Wih he noaion inroduced above, he map E Db ◦ ϕ denes an homomorphism o oK -modules n ⊗ ϕ ◦ E EDb Ei Dbi ei SD Eb i
which decreases he norms.
Proo. — T E Db ◦ ϕ oK . T ϕ ◦ : 1) E Db SD E b ; n n ⊗ ⊗ 2) E Db Ei Dbi ei E Db ; Ei Dbi ei i
i
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CHAPER 1. GEOMERIC INVARIAN HEORY AND ARAKELOV GEOMERY
24
3) Ei ei Ei ;
4) E EDb E Db
n ⊗ i
Ei Dbi ei
.
F K C : , 1) § 3 9 ; 2) P 2.135) R 2.15; 3) Ei ei ≃ Ei ei P 2.133) P 2.16; 4) 3) P 2.14. □ Lemma 2.19. — Le n SDb . For every K C: ∥ ∥ e Db e NDb N
Proo. — F i n vi viei Ei . C R R r i j i n j Dbi r i j ei i j. F R R v R
ei n Db i e i ⊗ ⊗ ⊗ i
j
vir i j ei
n Db i e i ⊗ ⊗ i
j
vir i j ei vir i j ei ei
T v R R R EDb . F T EDb T R R TR v R . W : ∑ TR ∥T ∥ TR ∥T ∥ F R R
R R
R R
R r ii j i j
B T T R R TR v R . T ∥ T ∥ ∥T ∥ ∥ T ∥ ∥T ∥ R R TR { } #R e Db enDbn ∥T ∥ ∥T ∥ T , T , T , R R TR ( T
R R v R ).
□
2.8. End o the proo o Teorem 2.1. — F i n i Ei . F n SDb n ⊗ Db e Db e Ei Dbi ei f n n EE Db i
MÉMOIRES DE LA SMF 152
2. LOWER BOUND OF HE HEIGH ON HE UOIEN
25
S ϕ ◦ oK , v, n ∏ Db e ∥ i ∥ iEi vi ∥ϕ ◦ f ∥ MD v y y YCv
i
A L 2.18 2.19 K C: n ∏ Db e ∥ϕ ◦ f ∥ MD y ∥ϕ ◦ f ∥ ∥ ∥ ∥ i ∥ iEi i y YC
e Db e NDb N
n ∏ i
i
Db e
∥ i ∥ iEi i
A L 2.17, K Qh P Eb OFSL E n ∑ bi ∑ ∥ i ∥ Ei v e b e bNN e i v V i K
bi ∑ bi Ei Ei ei i n
Ei ei . Ei
□
SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
CHAPER 2 DIOPHANINE APPROXIMAION ON P1 VIA GEOMERIC INVARIAN HEORY
I R’ T ( ) G I T C 1. L . I S 1 R’ T ( M T, T 1.12). M , R’ T q ( M E L B, T 1.7) M T . I S 2 G I T . A , F F M T. T ( ) S 3 4. F, S 5, , F F. O H D D’ L EVN (T 2.2). W W. T “GI ” T DG’.
1. Statement o the results 1.1. Roth’s Teorem with moving targets and the Main Eective Lower Bound 1.1.1. Heigh and disance on he projecive line. — I . Deinition 1.1. — 1) F K x x x PQ absolue logarihmic heigh ∑ ∥x x ∥ v hx K Q v V K
CHAPER 2. DIOPHANINE APPROXIMAION ON P VIA GI
28
VK K , v, x v x v ∥x x ∥ v x v x v
v v
2) L K v VK K. T v-adic spherical disance P , Cv x x x y y y , v x y
x y x y v ∥x x ∥ v ∥y y ∥ v
3) L x y K P . T v x y v K. F S VK ( ) ∑ v x y R S x y v S
I S v v x y.
Proposition 1.2 ( [8, T 2.8.21]). — Le x y be disinc K-raional poins o P . Ten, V x y hx hy K Q K Deinition 1.3. — L a P K ′ K, S VK VK v S v K ′ Cv K . D a v Cv P v : ∑ S a x v a v x v S
1.1.2. Roh’s Teorem wih moving arges. — I R’ T : heorem 1.4. — Le K be a number eld, S VK a nie subse, K ′ a nie exension o K and a real number. For every place v S x an embedding v K ′ Cv which respecs K. Tere is no sequence o couples x i ai wih i N made o a K-raional poin x i o P and a K ′-raional poin ai o P disinc rom x i saisying he ollowing properies: 1) he sequence hx i is unbounded; 2) hai ohx i as i goes o inniy; 3) or all i N he ollowing inequaliy is saised: S ai x i hx i K Q V’ R’ T , K:
MÉMOIRES DE LA SMF 152
1. SAEMEN OF HE RESULS
29
heorem 1.5 (c. [69, T 1]). — Le K be a number eld, S VK a nie subse, q a posiive ineger and a real number. q Tere is no sequence o q -uples x i ai ai (i N) made o pairwise disinc 1 K-raional poins o P saisying he ollowing properies:
1) or all q, hai ohx i as i goes o inniy; 2) or all i N he ollowing inequaliy is saised: q ∑ S ai x i hx i K Q
T 1.5 T 1.4 K G K ′ K ai ai . W . L q q P 1.2, x i ai . H q . 1.1.3. Main Eecive Lower Bound. — A , R’ T R’ T — . T K x x n v a an . A B [7], . T “M E L B” G I T. I r r n : P n . R’ T: Deinition 1.6. — L q n t . 1) C ∆n t n n n t . 2) L tqn n q q ∆n tqn
L Rn . T tqn n .
1. i.e. i N a i
, ai
, x i , a i
q.
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30
3) L Rqn q q n n Rqn
4) F n r r r n r r r n .
W M E L B (c. [7, T 2] n ):
heorem 1.7 (M E L B). — Le K ′ be a nie exension o K o degree q and le S VK be a nie se o nie places. Le n be a posiive ineger, le n be a real number and le r r r n be an n-uple o posiive real numbers such ha r i r i Rqn or all i n . Ten, or all i n and or all couples x i ai made o a K-raional poin x i o P and a K ′-raional poin ai o P such ha Kai K ′, he ollowing inequaliy holds: { { }} ∑ r i v ai x i tqn ′ K Cv i n K Q v S n n q ∑ q∑ n r i hx i r i hai q r i i where, or every place v VK , he embeddings K ′ Cv are mean o be K-linear.
Remark 1.8. — W M E L B . F , R 1.13 M E L B . A R’ T . 1.1.4. Deducing Roh’s Teorem rom he Main Eecive Lower Bound. — L M E L B (T 1.7) R’ T (T 1.4). L R ( [47]). T R’ T . Lemma 1.9. — Le q n be posiive inegers. Ten, tqn n n q In paricular,
n
n tqn
Proo. — A [8, L 6.3.5] : ∆n n n T
MÉMOIRES DE LA SMF 152
tqn n.
□
1. SAEMEN OF HE RESULS
31
Proo o Teorem 1.4. — A q x i ai i N T 1.4. U q K ′, Kai K ′ i N q K ′ K . F . B ( “M’ ”, [8, 6.4.2]) I N , v S v , i I , S ai x i vS ai x i v ai x i v #S ( v ’ ) ∑ v v S
U q x i ai i I i N. n , n r T 1.7. A x i ai i n , v S, { { { }} } v r r a x a x i v i i v i i i ′ K Cv
i n
i n
n n n ∑ ∑ r i hx i r i qhai C q i i { } ∑ r i v ai x i tqn i n K Q v S { } tqn r i S ai x i i n K Q C q . B , i n,
,
S ai x i K Qhx i
n n { } n ∑ ∑ r i hx i q r i hx i r i qhai C i n i i
tqn
S x i ’ hx i hx i ( Rqn ). T n r r i hx i r j hx j i j n. D q r hx :
n q ∑ hai n r C tqn q n i hx i r hx
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CHAPER 2. DIOPHANINE APPROXIMAION ON P VIA GI
32
n S hai ohx i i hai hx i . T r i r i hx . T, n tqn q n
L n , L 1.9, n n tqn .
□
1.2. Statement o the Main Teorem 1.2.1. More combinaorial daa. — I . Deinition 1.10. — L q n , r r r n n t . 1) C Rn : n { } ∏ ⃞ r n Rn i r i i n r i {
∇r t n ⃞ r { ∆r t n ⃞ r
r r
i } n t rn } n t ⃞ r ∇r t rn
A Z Zn : ⃞ rZ , ∇rZ t ∆rZ t. F r ⃞ n , ∇n t ∆n t. 2) I n L Rn , n n R, n t n n
3) D qr
n ∏ i
∇n t
i j n
{r } j
ri
∆n t
q
4) D uqr t q n { { } } ∆n uqr t qr q ∆n t
Lemma 1.11. — Te uncion n n R is sricly concave 2 . Moreover he ollowing properies are saised: 2. T , t t n , n t t n t n t
MÉMOIRES DE LA SMF 152
1. SAEMEN OF HE RESULS
1) For all t , n t 2) n t or all t n;
33
tn t n n
3) n n t n t or all t n;
4) Te uncion n is increasing on n and decreasing on n n. Proo. — S 1) 3) . W n 1) . F n n t n t n t n n
2) F n n n n . 4) F 3) n .
□
1.2.2. Main Teorem. — K , : heorem 1.12 (M T). — Le K ′ be a nie exension o K o degree q and le S VK be a nie subse. Le n be an ineger, t a t x non-negaive real numbers and le r r r n be an n-uple o posiive real numbers. I he ollowing inequaliy is saised, (SS)
n uqr t a n t x qr
hen, or all i n and or all couples x i ai made o a K-poin x o P and K ′-poin ai o P such ha Kai K ′, he ollowing inequaliy holds: { { }} ∑ r a x q ∆n t a t a i v i i K ′ Cv i n K Q v S
Cqr t a t x where
Cqr t a t x
n ∑ i
∇n t x
r i hx i q Cqr t a t x
n ∑ i
r i hai Cqr t a t x r
n q ∆uqr t a n t x
q ∆n t a q ∆uqr t a n t x Cqr t a t x ∆n uqr t a ∇n t x q ∆n t a q Cqr t a t x
T 1.12 (SS) ( , n uqr t a n t x qr ) q ∆n t a . T T 1.12 . T , q ∆n t a , T 1.7 T 1.12 , .
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CHAPER 2. DIOPHANINE APPROXIMAION ON P VIA GI
34
Remark 1.13. — T Cqr t a t x T 1.11 C 4 T 2.1 C 1 M T. D , T 1.12 Cqr t a t x ∆n uqr t a ∇n t x q ∆n t a q
( R 2.8).
1.3. From the Main Teorem to the Main Eective Lower Bound. — I M E L B (T 1.7) M T 1.12. n S r i r i Rqn i n , qr .
1.3.1. Choice o he parameers. — T M E L B T 1.12 t a tqn W uqr uqr tqn . Lemma 1.14. — Wih he noaion inroduced above: uqr qr n, hence uqr ; 1) ∆n n 2) n qr n ; ∆n uq r
3) n uqr qr ;
n uqr qr n n . 4) n
Proo. — 1) B n tqn . T, ∆n uqr qr n
S ∆n t t n n t n uqr n qr
2) T uqr (1) n uqr n n n n qr n n n ∆n uq r n C nn n . 3) F L 1.111), n uqr qr n qr n n
qr . 4) S 3).
MÉMOIRES DE LA SMF 152
□
1. SAEMEN OF HE RESULS
35
F t x , , (SS) . M , n n n, q w qr n n uqr n w qr qr n
Lemma 1.15. — Wih he noaion inroduced above: 1) ∇n w qr ; 2) n ; ∇n w q r
n 3) ∆n uqr n w qr .
n n . 2) F (1): t n n,
Proo. — 1) L 1.143) w qr n
∇n t
n ∇n t
3) B w qr D 1.102): ∆n uqr n w qr ∆n uqr qr n uqr n qr ∆n uq r
qr
n n
qr
q L 1.14 (2). T n □ qr n. 1.3.2. Applicaion o he Main Teorem. — L 1.143) T 1.12 ta tqn t x w qr n w qr . L t x w qr L 1.15: { { }} ∑ r a x tqn i v i i K ′ Cv i n K Q v S n n ∑ ∑ n q r i hx i q r i hai r Cqr
n Cqr q. T . □ i
i
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CHAPER 2. DIOPHANINE APPROXIMAION ON P VIA GI
36
2. From the Fundamental Formula to the Main Teorem 2.1. Interlude on the index. — L K . 2.1.1. Index. — L n P P n n K. F i n i P P i . L z z zn K P b b bn n . F i n ti zi P K. Deinition 2.1. — L f OPz P z. T f ∑ f f ℓ t ℓ tnℓn ℓ ℓ ℓn Nn
f ℓ K. T index o f a z wih respec o he weigh b { } b f z b ℓ bn ℓn f ℓ ,
I f , b z .
I b b bn n , n b bn b b b . T s L P z: f s L z b s z b ss z 2.1.2. Higher dimensional Dyson’s Lemma. — T H D D’ L: N [50]. T EV [28] ( ) . L r r r n n . C P:
OP r OP r n OP r n heorem 2.2 (H D’ L). — Le z z q be K-poins o P and t t q be non-negaive real numbers. Suppose ha ▷ i z , i z or every i n and every , ; ▷ here exiss a non-zero global secion f P OP r such ha, or every q, r f z t
Ten he ollowing inequaliy is saised: q ∑
∆n t qr where qr
MÉMOIRES DE LA SMF 152
n ∏ i
i j n
{r } j
ri
q
2. FROM HE FUNDAMENAL FORMULA O HE MAIN HEOREM
37
2.1.3. Index a a single poin. — L z z zn K P, t r r r n n . Deinition 2.3. — L Z qr z t P f r f z t. C P OP r : Kr z t K P OP r Z r z t OP r } { f P OP r r f z t
Proposition 2.4. — Keeping he noaion inroduced above, or every i n le Ti Ti be a basis o K such ha Ti vanishes a zi . n r i ℓi ℓi 1) Te monomials Tz ℓ Ti or ℓ ∇rZ t orm a basis o he K-vecor i Ti space Kr z t. 2) K Kr z t #∇rZ t and
K K r z t ∇n t n r r n
K Z r z t OP r ∆n t n r r n
3) K Z r z t OP r #∆rZ t and Proo. — L .
□
2.1.4. Index a muliple poins. — L r r r n n t N .
F q z z zn K P. S zi , i n.
, zi
Deinition 2.5. — C q z z z q . C P, q Z r z t Z qr z t
C P OP r :
Kqr z t KP OP r Z qr z t OP r { } f P OP r r f z t q
R uqr t q n { { } } ∆n uqr t qr q ∆n t
Proposition 2.6. — Keeping he noaion inroduced above: q 1) Z qr z t OP r Z r z t OP r .
SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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38
2) K Kqr z t (2.1)
n ∏ i
r i q #∆rZ t. In paricular,
K Kq r z t q ∆n t n r r n
3) Suppose uqr t n. Le z P K be a poin such ha or i n and q, one has i z , i z . Ten, or t uqr t, Kqr z t Kr z t
4)
K Kq r z t ∆n uqr t n r r n
R (3) (4) q H D D’ L. Proo. — 1) T Z r z t’ ( [28, L 2.8]). 2) U Kqr z t KP OP r Z qr z t OP r , K Kqr z t K P OP r K Z qr z t OP r K P OP r
q ∑
K Z r z t OP r
q P 2.43).
n ∏ i
r i q#∆rZ t
3) B f Kqr z t Kr z t . T H D D’ L q ∑
∆n t ∆n t qr
∆n t ∆n uqr t. T t uqr t t uqr t.
4) I ∆n uqr t , uqr t n, . A uqr t n. A 3), t uqr t, Kqr z t Kr z t
T G’
K Kqr z t K P OP r K Kr z t K P OP r #∇rZ t
P 2.42) q. T q r t uqr t. □ 2.2. Defnition o the “moduli problem”. — L K VK .
MÉMOIRES DE LA SMF 152
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39
2.2.1. Linear acions on grassmannians. — L E oK . F N GN E N E, i.e. oK GN E oK { O F } X f X S oK f E N
S oK G oK E. T, N , oK G GN E N , P N E q ON E. T P GN E P N E Gq.
2.2.2. Back o he Main Eecive Lower Bound. — L K ′ K q . L n . L P PoK n n oK . L r r r n n OP r P,
OP r OP r n OP r n
F i n x i K PoK ai K ′ PoK Kai K ′. C x x x n , a a an P a a HK K ′ Q L t x t a K Kr x t x f PK OP r r f x t x Kqr a ta f PK OP r r f a t a
PK P. 3 S f K a , a a. D kr t x kqr t a K Kr x t x Kqr a ta . I , PK OP r K : Kr x t x Gkr tx P OP r Kqr a ta Gkq r ta P OP r 3. H f , K , a, K ′ , f K ′ . A, Q K q r a t a f P OP r oK Q r f a t a K ′ Q G , K Kq r a t a . I , Kq r a t a K Q K r a t a
K ′ Q
K r a t a f P OP r oK Q r f a t a .
SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
CHAPER 2. DIOPHANINE APPROXIMAION ON P VIA GI
40
n T oK SLnoK P PoK . W
Fr t x
k∧ r t x
P OP r
Fqr ta
kq∧ r t a
P OP r
P , q SLnoK : Gkr tx P OP r P Fr t x Gkq r ta P OP r P Fqr t a 2.2.3. Te geomeric invarian heory daa. — W F F : Pr Kr x t x Kqr a ta Xr Gkr tx P OP r oK Gk t P OP r q r
a
G SLnoK Lr Xr P
: jr Xr P Fr t x oK P Fqr t a P Fr t x oK Fqr t a
(T P G S .) F K C
Fr t x C Fqr ta C
k∧ r t x
n ⊗
kq∧ r t a
i
Sr i C
n ⊗ i
Sr i C
q ∥∥ Fr tx ∥∥ Fq r ta ( § 3 9). E Fr t x C Fqr t a . T SUn . D Lr Xr . 2.3. Proo o the Main Teorem 2.3.1. — I T 1.12 (T 2.7) 5.2 ( P 3.1, 3.2 4.1) 3 4. T 1.12, , n r r r n . E , r f M E L B r .
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41
2.3.2. Semi-sabiliy condiions. — T 1.12 F F Pr , Pr . I S 5.2 : heorem 2.7. — Le n be a posiive ineger and r r r n be a n-uple o posiive inegers. Le t x t a be real numbers wih t a tqn . I he inequaliy n uqr t a n t x qr
is saised hen here exiss a posiive ineger q n r t a t x such ha, or every ineger , he K-poin P r X r K is semi-sable under he acion o SLn wih respec o he polarizaion given by he Plücker embeddings. 2.3.3. Applying he Fundamenal Formula. — T (SS) T 1.12. T T 2.7 q n r t a t x , , K P r K r x t x Kq r a ta
. T F F (, , C 1.7 C 1) P r q: h L r P r
∑ v P r h X r L r G K Q v S
. D n r r n , (2.1)
∑ v P r n K Q v S r r n
h L r P r n r r n
h X r L r G n r r n
I q . T Pr x i ’ ai ’. I S’ L. H q A . T . I . T q .
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CHAPER 2. DIOPHANINE APPROXIMAION ON P VIA GI
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2.3.4. Upper bound o he heigh. — T P Xr P Fqr t a P Fr t x . T: Kqr a ta h Lr Pr h Fr tx Kr x t x h Fq r t a
S A ( P 3.13.2) : n ∑ h Fr tx Kr x t x
h
Fq r t a
∑
ℓi hx i
i ℓ ∇rZ t x
n n ∑ ∏ Kqr a ta r i kqr t a q r i hai r q i
i
A r : (2.2)
h L r P r n r r n
∇n t x
n ∑
r i hx i
i
n ∑ q ∆n t a q r i hai r q i
2.3.5. Upper bound o he insabiliy measure. — L v K. I v : { } r k t t a x v Pr qr a a i v i i ′ i n
K Cv
n ∑ i
∑
ℓ ∇rZ t x
kr t x kqr t a ℓi r i v ai x i
kqr t a
n ∑ i
r i kqr t a r kr t x r
. T S 4 ( P 4.1). I v , f , P 2.6 2), 2.42) 2.64): { v P r q ∆n t a t a r i v ai x i n i n r r n K ′ Cv n } n t x ∆n uqr t a ∑ r i v ai x i i I v , r ∆n uqr t a ∇n t x . B D 1.102), n uqr t a ∆n uqr t a MÉMOIRES DE LA SMF 152
2. FROM HE FUNDAMENAL FORMULA O HE MAIN HEOREM
43
(SS) n t x ∆n uqr t a . B v ai x i ∑ w ai x i w v
w K ′ v. S P 1.2, ∑ ∑ ∑∑ w ai x i w ai x i K ′ Q hx hai v S w VK ′
v S w v
(2.3)
∑ v P r , n K Q v S r r n
{ } ∑ r a x q ∆n t a t a i v i i K ′ Cv i n K Q v S
n n t x ∆n uqr t a ∑ r i qhx i hai i r ∆n uqr t a ∇n t x
2.3.6. Lower bound o he heigh on he quoien. — F i n Ei oK bi r i kqr t a kr t x . A T 2.1 C 1 G SL E oK oK SL En GL Fqr ta Fr t x
n ⊗ i
Eibi
n r i kq r ta kr tx ⊗ oK Fqr t a Fr t x i
T Ei , Ei i n. T Gq jr Xr P Fqr t a Fr t x T 2.1 h Xr Lr G h P Fqr t a Fr t x O G kqr t a kr t x r kqr t a kr t x
kqr t a kr t x q ( § 3 9). T S’ , kqr t a n r r n
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44
kr t x . T , r , : (2.4)
h X r L r G ∆n uqr t a ∇n t x r n r r n
Remark 2.8. — T T 2.1 C 1 T 1.11 C 4, h Xr Lr G kqr t a kr t x
,
h X r L r G n r r n
M T f (2.1) q (2.2), (2.3) (2.4). 3. Upper bound o the height 3.1. Rational point Proposition 3.1. — Wih he noaion inroduced in Secion , n ∑ ∑ ℓi hx i h Fr tx Kr x t x i ℓ ∇nZ rt x
Proo. — L T T K . F i n x i x i K x i P K. S x i . F n ℓ ⃞ r n ⊗ T ℓ Tr i ℓi Txℓii i
Tx i x i T x i T . A K Kr x t x T ℓ ℓ ∇rZ t x . L v K. T H q (1) 10 ∧ ∑ T ℓ ∥T ℓ∥P OP r v Fr t x v
ℓ ∇rZ tx
ℓ ∇rZ t x
F n ℓ ⃞ r n ∑ T ℓ ∥Tr i ℓi Txℓii ∥v P OP r v
i
n ∑ i
r i ℓi ∥T ∥v
n ∑ i
C .
MÉMOIRES DE LA SMF 152
ℓi ∥Tx i ∥v
n ∑ i
ℓi ∥x i ∥v
□
3. UPPER BOUND OF HE HEIGH
45
3.2. arget points Proposition 3.2. — Wih he noaion inroduced in Secion ,
n n ∑ ∏ h Fq r ta Kqr a ta r i kqr t a q r i hai r q i
i
T . 3.2.1. — Eq oK P OP r n ⊗ Sr i oK P OP r i
D P OP r oK . T oK P OP r P OP r T ℓ
n ⊗ i
Tr i ℓi Tℓi
. A, v, n n ∑ ∑ r i ℓi ∥T ∥ v ℓi ∥T ∥ v T ℓ P OP r v i
I , (3.1)
i
∑ ∑ P OP r T ℓ P OP r v v VK
ℓ ⃞
Z r
3.2.2. — E K Kqr a ta oK P OP r . T oK Kqr a ta P OP r Kqr a ta
q P OP r , C P OP r Kqr a ta
q P OP r C. D C oK . W (3.1): Kqr a ta K Qh Fq r ta Kqr a ta C P OP r C
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3.2.3. — D E K Z qr a ta OP r Ω G K ′ K. E Ω E K Ω oΩ . A P 2.6: q (3.2) E E K Ω
, K ′ Q, E Z r a t a OPΩ r 4 . F i n ai ai Ω ai P K. S a K ai , , ai . F K ′ Q, Ω E n ⊗ Tr i ℓi T ℓi Ta ℓ i
ai
Ta T ai T ℓ ℓ ℓn ∆rZ t a . L E i
oΩ E T ℓ’. Eq T ℓ’ . D E oΩ . F, (3.2), K E K Ω oΩ oΩ E ’. D EΩ . 3.2.4. — T PK OP r E Z qr a ta OP r C oK K E. A q (P ‘S q’ 11), ∑ C C ∥ ∥ v EΩ (3.3) h Fq r ta Kqr a ta K Q Ω Q v V Ω
, v VΩ , ∥ ∥ v v , ∥ ∥ v
,f C Ω
∥f ∥ Ev ∥ f ∥ Cv
T ∥∥ v . T oΩ E EΩ .
3.2.5. — I v . F q PΩ OP r E E K Ω E . D ∥ ∥ v . W : ▷ v : ∥∥ v ∥ ∥ v ; q { } ▷ v : ∥∥ v q ∥ ∥ v . q
4. H a Ω PΩ PK K Ω Z r a t a PΩ t a a .
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F q i n ϕ i ϕi
47
Ω
T T T Ta T a i T i
C ϕ r
Sr ϕ
Sr n ϕ n
Ω
PK OP r K Ω Sr Ω Ω Ω Sr n Ω
i n ϕ i i . W ◦ϕ r PΩ OP r E Z r t a : T ℓ
n ⊗ i
Tr i ℓi Tℓi
Ta ℓ ℓ ∆rZ t a
B Ta ℓ’ oΩ E . T ∥ ◦ ϕ r ∥ v
∥ ∥ v ∥ϕ r ∥ v B oK V V V, n ∑ r i ϕ i Eo v ∥ϕ r ∥ v ϕ r EP OP r v K
i
T . B ϕ i ϕ i T ϕ i T ai . T, ▷ v : ϕ i Eo v ∥ϕ i T ∥v ∥ϕ i T ∥v Ω ai ai v
▷ v :
ϕ i Eo v ϕ i T v ϕ i T v Ω ai ai v
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K: n {∑ ∑ ∑ } ∥∥ v r i ai ai v r Ω Q q ′ v VΩ K Q
v VΩ
∑
∑
i
n ∑
v VΩ K ′ Q i
Ω Q
∑
r i ai ai v r Ω Q q
n ∑
K ′ Q i
r i hai r q
D Ω Q, hai hai K ′ Q (3.3), n ∑ r i hai r q h Fq r ta Kqr a ta C q i
U C P OP r Kqr a ta P OP r
n ∏
r i
Kqr a ta kqr t a
i
.
□
4. Upper bound o the instability measure 4.1. Notations and frst reductions. — L v K. Proposition 4.1. — Wih he noaion inroduced in Secion 2.2, i he place v is nonarchimedean, { } r k t t a x v Pr qr a a i v i i ′ i n
K Cv
n ∑ i
∑
ℓ ∇rZ t x
kr t x kqr t a ℓi r i v ai x i
whereas, in he v archimedean case, he previous inequaliy holds wih kqr t a
n ∑ i
r i kqr t a r kr t x r
added on he righ-hand side.
T i n K ′ Cv :
1) x i x i Kv x i P Kv ∥x i x i ∥ v . 2) ai ai Cv ai P Cv ∥ai ai ∥ v ; 3) q i Cv ai x i ai x i .
MÉMOIRES DE LA SMF 152
4. UPPER BOUND OF HE INSABILIY MEASURE
49
4.1.1. Elemens o SL measuring disances
Deinition 4.2. — F i n q дi Cv : ( ) ( ) ⊤ x i x i ai ai дi GL Cv xi xi a x i a x i ai ai i
C n д
д дn
i
GL Cv n .
Proposition 4.3. — Wih he noaion inroduced above, or all i n and q,
1) дi
ai x i ai x i ;
2) дi
v v ai x i ;
3) For a non-zero vecor y y Cv such ha ∥y y ∥ v ,
v ai y v x i y v д yT yT i v v ai y v x i y v
where T T denoes he canonical basis o Cv and y P Cv he line generaed by y y .
Proo. — 1) C: x i x i ai ai . 2) C 1).
3) T дi Cv Cv , ( T , T ) ( ) ai ai xi xi
T .
□
4.1.2. Proo o Proposiion 4.1. — I P 4.1 P 4.6 4.7: . Deinition 4.4. — F h h hn GL Cv n ∥h w x ∥ Fr tx v , ▷ v h Kr x t x ∥w x ∥ Fr tx v ∥h w a ∥ Fq r ta v , ▷ v h Kqr a ta ∥w a ∥ Fq r ta v w x Fr t x K (. w a Fqr t a K) P Kr x t x (. Kqr a ta ). Deinition 4.5. — F i n K ′ Cv дi дi i , . S д д дn .
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50
E v Pr { } v Pr n v h Kr x t x v h Kqr a ta h SL Cv { } д K x t д K a t v r x v qr a ′ K Cv
T Fr t x Fr t a GLnoK
kr t x r kr t a r . B P 4.32), i
v ai x i n kr t x ∑ д Kr x t x v д Kr x t x r i v ai x i v i
n kqr t a ∑ v д Kqr a ta v д Kqr a ta r i v ai x i i
O P 4.1 :
Proposition 4.6. — Wih he noaion inroduced above, i v is non-archimedean, n ∑ ∑ v д Kr x t x ℓi v ai x i i ℓ ∇rZ t x
whereas, i v is archimedean, he preceding inequaliy holds wih kr t x r be added on he righ-hand side. Proposition 4.7. — Wih he noaion inroduced above, i v is non-archimedean, { } v д Kqr a ta kqr t a t a r i v ai x i i n
whereas, i v is archimedean, he preceding inequaliy holds wih n kqr t a ∑ r i kqr t a r i added on he righ-hand side.
4.2. aylor expansion at the single point: proo o Proposition 4.6. — K S 4.1. 4.2.1. — L i n
Ti x i T x i T Kv
S x i x i Kv Ti . I v Ti ov Ti , Ti ov ov . I v Ti Kv Ti , Ti Kv . S Ti x i i n, P 2.41) K Kr x t x n ⊗ T ℓ Tiri ℓi Tiℓi i
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4. UPPER BOUND OF HE INSABILIY MEASURE
51
ℓ ℓ ℓn ∇rZ t x . T Fr t x oK Kv , ∧ w T ℓ ℓ ∇rZ t x
P Kr x t x . I v T ℓ’ ov Kr x t x Kv P OP r ov
T
∥w ∥ Fr tx v
I v T ℓ’ ∥w ∥ Fr tx v
∑
ℓ ∇rZ t x
∥T ℓ∥P OP r v
r B ℓi r i , i
r ℓi i ℓ ∇rZ t x i kr t x r
∥w ∥ Fr tx v
∑
n ∑
∑
n ∑
ℓ ∇rZ t x i
∑
ℓ ∇rZ t x
n ∑
r ℓi i
r i
i
4.2.2. — F ℓ ∇rZ t n ∑ д T ℓP OP r v r i ℓi ∥дi Ti ∥v ℓi ∥дi Ti ∥ v i
T H q, ∑ д T ℓP OP r v д w Fr tx v ℓ ∇rZ t x
∑
n ∑
ℓ ∇rZ t x i
r i ℓi д Ti v ℓi д Ti v
F i n P 4.33) v ▷ дi Ti v v ;
▷ ∥дi
Ti ∥ v v ai x i .
S v :
n ∑ ∑ v д Kr x t x д w Fr tx v ℓi v ai x i i ℓ ∇rZ t x
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I v : v д Kr x t x д w Fr tx v kr t x r n ∑ ∑ ℓi v ai x i kr t x r i ℓ ∇rZ t x
.
□
4.3. aylor expansion at the algebraic points: proo o Proposition 4.7. — K S 4.1. 4.3.1. — I v ov Kv f fkq r ta ov : Kqr a ta K Kv P OP r oK ov
I v f fkq r ta Kqr a ta Kv . W Fqr a ta oK Kv , w
kq∧ r t a
f
P Kqr a ta . I ∥∥ v Cv P OP r oK Cv . W H’ q (1) 10 kq∑ r t a v д Kqr a ta д w Fq r ta v д f v
:
Lemma 4.8. — Le f be a non-zero elemen o Kqr a ta . Wih he noaion inroduced above, i v is non-archimedean, { } ∥д f ∥ v ta r i v ai x i i n ∥ f ∥v
whereas, i v is archimedean, he preceding inequaliy holds wih ∑ r i r i n
added on he righ-hand side.
4.3.2. — F i n :
Ti ai T ai T Cv
S ai ai Cv Ti . I v ov Cv Ti ov MÉMOIRES DE LA SMF 152
4. UPPER BOUND OF HE INSABILIY MEASURE
53
Ti , Ti ov ov . 5 I v Ti Cv Ti , Ti Cv . F n ℓ ℓ ℓn ⃞ r n ⊗ T ℓ Tiri ℓi Tiℓi i
T T ℓ’ Cv P OP r K Cv . I v T ℓ’ ov P OP r oK ov . I v T ℓ’ ℓ ⃞ r , n ∏ ri ∥T ℓ∥ v rℓ ℓi i
4.3.3. — I v , 4.2.2 n n ∑ ∑ ℓi v ai x i ℓi v ai x i д T ℓ v i
i
, v , q kqr t a r . 4.3.4. — W f ℓ f ℓT ℓ f ℓ Cv . I v , } { ∥ f ∥ v f ℓ v ℓ ⃞ rZ I v ,
∥ f ∥ v
∑
ℓ ⃞
Z r
f ℓ v rℓ
S v ai x i ℓ ∇rZ t a : n n { ∑ ∑ } ℓi r i v ai x i ℓi v ai x i r i n i i i { } ta r i v ai x i i n
B f r f a t a , , f ℓ ℓ ∆rZ t a . I : { } f ℓ v д T ℓ v д f v ℓ ∇rZ t a
ta
i n
{ } r i v ai x i ∥ f ∥ v
.
5. S Ti ov ov Ti ov , (.[11, 1.6.1 P 2]). O Ti Kv . I Ti , .
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54
4.3.5. — S v . P 4.33) q : ∑ д f f ℓ v д T ℓ v v ℓ ∇rZ t a
(4.1)
i n
{
v ai x i r i
} ta
∑
ℓ ∇rZ t a
f ℓ v
n ∏
r i ℓi
i
C ℓ ℓ P OP r J’ q: √ n n n ∏ ∑ ∏ ∏ ∑ r i ℓi f ℓ v r i f ℓ v r i ℓi i
ℓ ∇rZ t a
P OP r f : ∑
ℓ ∇rZ t a
f ℓ v
n ∏ i
i
n
i r i
r i ℓi
U ba ba ba b , .
i
. T
ℓ ∇rZ t a
ℓ ∇rZ t a
ℓ ∇rZ t a
n { ∏ } r r i ℓi ℓ i
∑
ℓ ∇rZ t a
n { ∏ } r r i ℓi ∥ f ∥v ℓ
f ℓ v rℓ
i
n { ∏ } r r i ℓi r A (4.1) ℓ ℓ ∇rZ t a i □
5. End o the proo: semi-stability in the general case 5.1. Basic acts about the semi-stability o subspaces 5.1.1. Insabiliy coefcien. — L K G K K X q Gq L. L x K X . L Gm G G ( ) x Gm X x x
B X , x q x A X . D x K x . S Gm , Gm K x L L x
L x Z. I insabiliy coefcien o x L. 6 6. W [48, D 2.2].
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5. END OF HE PROOF: SEMISABILIY IN HE GENERAL CASE
55
heorem 5.1 (HM ). — Suppose ha K is perec and L is ample. Wih he noaion inroduced above, he poin x is semi-sable i and only i L x
or every one-parameer subgroup Gm G.
T M [48, T 2.1] K . T K [45, T 4.2] R [59]. 7 5.1.2. Insabiliy coefcien o linear subspaces. — L V K r . C r Gr V P Gr V P r V .
S K G V . T Gr V , P r V q r O P V . S P Gq , O Gr V Gq . Deinition 5.2. — L Gm G . 1) L W V r . S W O W O.
2) F p Z Vp v V v p v
L p (. p ) (. ) p Vp . 3) F p Z V p q p Vq . S , V p Z Vp . I , p p V p V p p
Proposition 5.3. — Le W V be a linear subspace o dimension r . Se W p W V p or every ineger p. 1) Te subspaces W p orm a decreasing lraion o W and ∑ W p K W p K W p p Z
p K W
p∑
K W p
p p
7. I [45, T 4.2] T 5.1 K’ M’ [48, A C 2, B].
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2) Le w w r be a basis o W . For every i r le w i be he insabiliy coefcien o he poin w i PV . Ten he vecor w i wries as w i w i w i wih w i V . I he elemens w w r V are linearly independen, hen r ∑ w i W i
3) Wih he noaions o (2), here exiss a basis w w r o W such ha heir componens o minimal weigh w w r V are linearly independen.
Proo. — T [48, C 4, §4]. S [67, § 2, L 2]. □ Proposition 5.4. — Le W , W be subvecor spaces o V . Ten: 1) (I ) I W is conained in W , hen (5.1) W W p K W K W
2) (G ) (5.2) W W W W W W
Proo. — 1) C. 2) F p G K W p K W p K W p W p K W p W p
C W p W p W W p.
□
Remark 5.5. — T P : V W W W W ( V ). I, 2) . 5.2. Asymptotic semi-stability: proo o Teorem 2.7
5.2.1. — G S 2.2. T [63, §2 L 2]. I P r X r . F K ( P PK n ). heorem 5.6. — Le n be a posiive ineger and r r r n be a n-uple o posiive inegers. Le t x t a be real numbers wih t a tqn . I he inequaliy n uqr t a n t x qr
is saised hen here exiss a posiive ineger q n r t a t x such ha, or every ineger , he K-poin P r X r K is semi-sable under he acion o SLn wih respec o he polarizaion given by he Plücker embeddings.
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5. END OF HE PROOF: SEMISABILIY IN HE GENERAL CASE
57
5.2.2. Compuaion o he insabiliy coefciens. — F n r r r n , t i n : ∑ Z t ℓi r i ri ℓ ∇rZ t
Z t . A L 1.11 ri
Deinition 5.7. — L Gm SLn K . 1) F i n, Ti Ti K m i , Gm K, Ti m i Ti
Ti m i Ti
T n m m m n weigh o Ti Ti ( i n) adaped basis or . T m i . I m i Ti Ti . 2) W , Q y P Ti y i y . D y q K P i y i n insabiliy poin o ( ).
Proposition 5.8 (I f ). — Le Gm SLn be a oneparameer subgroup. Wih he noaion inroduced above, or i n, n ∑ Z Kr x t x i x m i ri t x i
Proposition 5.9 (I f ). — Le be a posiive real number. Under he assumpions o Teorem here exis a posiive real number and a posiive ineger he wo o hem possibly depending on n, q, r , t a and t x saisying he ollowing properies: or every one-parameer subgroup Gm SLn , every ineger and every real number , n ∑ m i Zri uqr t a n r i r r n qr Kq r a ta i
Proo o Teorem 2.7. — A HM (T 5.1) Gm SLn , P r K r x t x K r a t a
L . L , P 5.9. U , i n, :
(SS ′)
Zri uqr t a Zri t x n r i r r n qr
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F . P 5.8 5.9 P r : n ∑ i
] m i Zri uqr t a n r i r r n qr i x Zri t x
S m i , (SS ′). T . □ Proo o Proposiion 5.8. — C Ti Ti (i n). S i x i n ( ). U i K Ti i Ti x i . A P 2.41), K Kr x t x n ⊗ Tiri ℓi Ti i Ti ℓi T ℓ i
ℓ ℓ ℓn ∇r t x . T n ⊗ m i r i ℓi Tiri ℓi Ti m i i Ti ℓi i T ℓ i
S m i , T ℓ ni m i r i ℓi , T ℓ
n ⊗ i
T T ℓ ℓ
Tiri ℓi Tiℓi
∇rZ t x ,
Kr x t x
n ∑
∑
i ℓ ∇rZ t x
m i ℓi r i
P 5.32)
n ∑ i
Z m i ri t x
5.2.3. Proo o Proposiion 5.9 Lemma 5.10. — Le Gm SLn be a one-parameer subgroup. Suppose 8 uqr t a , n Le a be a K-raional poin o P . Ten, or , Kqr a ta Kr a uqr t a n ∑ kqr t a kr uqr t a P OP r m i r i i
Proo. — T uqr t a , n , D 1.104), ∆n uqr t a qr q ∆n t a
8. I uq r t a n C (SS) T 2.7 n n n .
MÉMOIRES DE LA SMF 152
□
5. END OF HE PROOF: SEMISABILIY IN HE GENERAL CASE
59
S a K i a , i a i n K ′ Q. A P 2.63), Kqr a ta Kr a uqr t a
( Q K). G’ f (P 5.4 2) Kqr a ta Kr a uqr t a : Kqr a ta Kr a uqr t a Kqr a ta Kr a uqr t a
W 5.1.2, b P OP r b ni m i r i ( r Tnr n ). T (P 5.41) T Kqr a ta Kr y uqr t a P OP r .
□
Lemma 5.11. — Le be a posiive real number. Under he assumpion o Teorem 2.7 here exis a posiive real number and a posiive ineger he wo o hem possibly depending on n, d, r , ta and t x such ha, or every ineger and every real number : K P OP r 1) n r r n Z #∇ r uqr t a 2) q ∆n t a qr ; n r r n 3)
kq r t a n r r n
q ∆n t a
Proo. — 1) C. 2) F uqr t a 3) P 2.6 2). □ Proo o Proposiion 5.9. — L L 5.11. . L a PK q i a i n. A P 5.8 a : n ∑ K r a uqr t a m i Zri uqr t a i
A L 5.11 P 2.42): kq r t a k r uqr t a P OP r n r r n qr
T L 5.10 : n ∑ m i Zri uqr t a n r i r r n qr Kq r a ta i
.
□
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60
6. Semi-stability on P P and the Wronskian determinant
I n W . S qr q r r r r
W r r .
6.1. — T : heorem 6.1. — Le r r r be a couple o posiive inegers such ha r r . Le t x t a be non-negaive real numbers such ha t x q ∆ t a q ∆ t a q r
I he inequaliy (6.1)
q ∆ t a q r
is saised hen here exiss a posiive ineger q r ta t x such ha, or every ineger , he K-poin P r X r K is semi-sable under he acion o SL wih respec o he polarizaion given by he Plücker embeddings. I , ▷ t a tq q;
▷ t x q w w q r
T M E L B n . Proposition 6.2 (I f ). — Le be a posiive real number. Under he assumpions o Teorem 6.1 here exiss a posiive ineger (possibly depending on q, r , ta and t x ) saisying he ollowing properies: or every one-parameer subgroup Gm SL K and every ineger : Kq r a ta r r ⟨m r ⟩ q ∆ t a q ∆ t a q r
where ⟨ ⟩ denoes he sandard scalar produc on R .
A T 2.7 T 6.1 P 6.2. T P 6.2 : Proposition 6.3. — Le f Kr a ta be a non-zero secion. I q ∆ t a q r , hen, or every one parameer subgroup Gm SL K , f ⟨m r ⟩ q ∆ t a q r
he insabiliy coefcien o f being aken as a poin o PP OP r and wih respec o he inverible shea O.
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61
Proo o Proposiion 6.2. — L . A L 5.11 :
kq r t a q ∆ t a r r L f fkq r ta Kq r a ta ( ) f fkq r ta ( P 5.3). T P 5.32) (6.2)
Kq r a ta
kq∑ r t a
f ℓ
kq r t a ⟨m r ⟩ q ∆ t a q r ℓ
q P 6.3. C (6.2).
□
T D’ L: heorem 6.4 (c. C 6.15). — Le f P Or be a non-zero global secion and le b b b be a couple o non-negaive real numbers. Le y be a Q-poin o P . Le q be an ineger and or every q le z be a Q-poin o P. For every i suppose: 1) i z , i z or every , ; 2) i z , i y or every q. For every q suppose t and q ∑ (6.3) ∆ r f z q r
Ten, b f y bi r i and ) ( q ∑ b f y ∆ ∆ r f z q r bi r i
A D’ L [28, . 489]. T (6.3) q EV b f y “ ” ( loc. ci.).
I P 6.3 f ( ): Proposition 6.5. — Le Gm SL K be a one parameer subgroup and x an adaped basis or . Ten, or every non-zero elemen f P OP r , f ⟨m r ⟩ m f y where y is he insabiliy poin o wih respec o he chosen adaped bases.
Proo o Proposiion 6.3. — I f T 6.4 z a K ′ Q, b m y y ( ): t a ( (6.1) ), ∆ t a t a . □
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T T 6.4. I P 6.5, D’ L — i.e. T 2.2 n 9 — . U, D’ L : r z q m y . T T 6.4 . A : Proposition 6.6. — Le b b b and c c c be couples o non-negaive real numbers. Suppose ha is made o posiive real numbers. For all i le fi P OP r i be a non-zero secion. Ten, or all Q-poin z o P , b f f z bi c i c f f z
I , T 6.4 W . 6.2. Homogeneous Wronskian. — I SL K . W [1, 2.8]. L r r .
Deinition 6.7. — L f f Sr K T T K . T homogeneous Wronskian f f : ) ) ( ( r f ℓ Wf f j r T Tj j ℓ I S r K , , r T T ( r ).
T [1, 2.9] W. I W’ [8, P 6.3.10] f f Wf f . T W Sr K ∧ Sr K S r K W
9. T D [27], ( [7], [68] [70]).
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Proposition 6.8 (. [1, 2.3, 2.5 2.8]). — Te ollowing properies are saised: 1) I T′T′ is a basis and W ′ Sr K S r K is he Wronskian map aken wih respec o he later basis, hen as linear maps, W ′ T′T′ r W
where T′T′ is he linear map sending Ti on Ti′ or i . 2) Te linear map W is equivarian under he acion o SL K on he vecor spaces Sr K and S r K .
6.3. ensorial rank. — L V V K .
Deinition 6.9. — T ensorial rank v V K V v v v v v vi ℓ Vi i ℓ . D v. T GLV GLV ( ). T v V V v V K V ≃ HK V V
( V V ). v F i vi vi v Vi v ℓ v ℓ v ℓ . T, i , vi vi v . 6.4. Splitting polynomials through the Wronskian. — L r r r i Vi Sr i K . W , P OP r ≃ V V
F f P OP r f . W [7, . 266], s f f F i Ti , Ti K . Deinition 6.10. — L f P OP r f . F ℓ ℓ ℓ ℓi i , : f , f ℓ ℓ ℓ ℓ ℓ T T T T OP r r . T homogeneous Wronskian Wf [∏ r i ] Wf ℓ f ℓ ℓ ri i OP r r .
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L f P OP r f . W f
∑
ℓ
f ℓ f ℓ
fi ℓ P Or i i ℓ . F i W Wi f Wfi fi Ti Ti . A : Proposition 6.11. — Wih he noaions inroduced above, Wf W f W f 6.5. Index o the Wronskian. — I W . Proposition 6.12. — Le f P OP r be a non-zero secion and le b b b be a couple o non-negaive real numbers. Ten, or every Q-poin z o P, ) ( f ∆ t f r b Wf z bi r i f r
where t b f z bi r i .
Proo. — S W ( ) (P 6.8), i Ti , Ti K Ti i z , Ti i z . L t b f z f z. L f f , r r . S T T z, ℓ ℓ , { } b ℓ ℓ f z t ⟨b ℓ ℓ ⟩ } { ℓ bi r i r bi r i t r ⟨ ⟩ ⟨ ⟩ ℓ b ℓ ℓ bi r i r ℓ ℓ bi r i r r
( r r r ℓ r ). L S . S , {∑ } b ℓ ℓ f z b Wf z S
S
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ℓ
{∑
{ }} ℓ t bi r i r bi r i r ℓ
6. SEMISABILIY ON P P AND HE WRONSKIAN DEERMINAN
W t ′ t bi r i u r t ′ ,
F,
65
∑ ℓ ℓ t′ t bi r i bi r i r r ℓ ℓ
∑
r u
r u ∑ ℓ u u r u t ′ t′ r u t ′ r ℓ
C :
Lemma 6.13. — Wih he noaions inroduced above, le t t ′ . Ten, u ∆ t u t′ r r Proo o Lemma 6.13. — :
▷ S u t ′. T, ut ′ u t ′ , r , r r
▷ S u r . T, u u u t u t′ T t t t . B , u t, u r , u t u t t u u t t
q q t t . T . □ T .
□
Proposition 6.14. — Le f P OP r be non-zero and b b b a couple o nonnegaive real numbers. Le q be an ineger and or every q le z be a Q-poin o P. For every i suppose i z , i z or every , . Ten, q ∑
where t
∆ t q r
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Proo. — S f . T b Wf z . ▷ Upper bound. – B P 6.6. S Wf W f W f P 6.6 c r b Wf z bi r i r Wf z I r Wf z . S f . U , Wi f Or i P z : ∑ Wi f i z r i i ) ( q ∑ ∑ r i Wi f i z r i i
r Wf z
q ∑ ∑ r Wf z ri i
F q, P 6.12 ( z z b r ) :
∆ t r Wf z r { } t r f z . S , b Wf z , q r q r , q ∑ ∑ bi r i q r ∆ t ri r i
▷ Lower bound. – P 6.12 z z b :
t
∆ t b Wf y bi r i r { } b f z bi r i .
C b f z :
q ∑ q r ∆ t r r
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( r ). D r : q ∑
67
r
q r r q r q r
∆ t
f , ( r ) — [7, L 2 II.4] . T f . T f . □ Corollary 6.15. — Under he assumpions o Proposiion 6.14, suppose r f z or q and (6.1)
Ten, b f z
q ∑
∆ r f z q r
bi r i and
∆
q ∑ b f z ∆ r f z q r bi r i
Proo. — I b f y bi r i , P 6.14
(6.1).
q ∑
∆ r f z q r
□
6.6. Wronskian as a covariant. — L . F . T W “” SL K , i.e. SL K q W P P OP r d P S r K K S r K
U PP OP r f P OP r . T W W SL K q W O ≃ O U . I , R’ G I T. , O f O Wf SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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f OP r Gm SL K . I y , P 6.5 m Wf y m f y m m A , T 6.4 y y b m . U, Pr ( t x ). I P 6.12 P 6.14.
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CHAPER 3 KEMPF-NESS HEORY IN NON-ARCHIMEDEAN GEOMERY
I K N [46] ( ) . E , q . T , . I S 1 K N . T C 4. I S 2 : ( B ), . C , . C , . W ( B T’ ). S 3 : ( T 3.3). T GI q ( , T 1.6). T , , B .
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I GI q. W , B . T C 4. 1. Statement o the main results 1.1. A result o Kemp and Ness. — L G () V ( ) G ∥∥ V R . S ∥∥ U G. L v V . K N [46] pv G R pv д ∥д v ∥
A , : heorem 1.1. — Wih he noaions inroduced above: 1) Te uncion pv obains is minimum value i and only i he orbi o v is closed. 2) Any criical poin o pv is a poin where pv obains is minimum. 3) I pv obains is minimum value, hen he se where pv obains his value consiss o a single U - Gv cose here Gv is he sabiliser o v in G. 1.2. Interpretation via the moment map. — A GS M, G I T Kä . I PV L U G V . F v V , a L U 1
v L U R
⟨a v v⟩ i ∥v ∥ H ⟨ ⟩ ∥∥, i q L U V V . v a
S v V 2 pv д pv e д G PV . Proposition 1.2. — A non-zero vecor v V is minimal i and only i he linear map v is idenically zero. (F [48, T 8.3].) 1. O . 2. T G v v .
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W 2) T 1.1 q: PV M PV PV G O. L Y q PV G. T U Y C
[48, T 8.3]. W U , q U MW q. T GS [40], [41], [42], K [48, C 8] W [72]. 1.3. Present setting. — I : ▷ C k ; ▷ V kf X k G; ▷ ∥∥ u X ( D 2.29) U G ( D 2.20 2.21). AL [3] X f ( S ) u X C R U . S 1) 3) T 1.1 : Example 1.3. — C C C t x y tx y
T ℓ ∥x y∥ x y U. H t t C , t
t C . T “” . M t t “” U.
I KN , G I T. 1.4. Algebraic setting. — L k . L G k f k X S A . D Y AG X Y AG A.
T G I T f ( [48, T 1.1 C 1.2] , [43] [63, T 3] ).
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heorem 1.4. — Te k-scheme Y is o nie ype and he morphism saises he ollowing properies: 1) is surjecive and G-invarian; 2) le K be a eld exension o k and K X K X k K YK Y k K be he morphism obained exending scalars o K; hen or all poins x x ′ X K, K x K x ′
i and only i G K x G K x ′ , ∅
he orbis being aken in X K . 3) or every G-sable closed subse F X is image F Y is closed; 4) he srucural morphism ♯ OY OX induces an isomorphism ♯ OY OX G
I Y q X G k, i.e. G ′ X Y ′ q Y . F , Y quoien X G quoien morphism projecion (on he quoien). 1.5. Analytic setting. — S k . K G (. X , . Y ) k kf G (. X , . Y ). H, q ; B. W S 2.1: . T k G k X k X Y ( ) G . L G k X X k G X . Deinition 1.5. — T orbi x X X G x x
A F X G -sable (. G -sauraed) x F , G x (. G x ) F . I . I , x y X , y G x x G y
( x y ; [5, P 5.1.1]).
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1.6. Analytic topology o the GI quotient. — O 1)–3) T 1.4 k : heorem 1.6 (c. P 3.1 3.9). — Wih he noaion inroduced above, he morphism X Y saises he ollowing properies: 1) is surjecive and G -invarian; 2) or every x x ′ X : x x ′
i and only i G x G x ′ , ∅
3) i F is a G -sable closed subse o X , hen is projecion F is a closed subse o Y . I 1) 2) “” (T 1.4 1)–2)). I 2), G x x X C X : Z . T q K N. A N [51]. T 1.6 q, : Corollary 1.7. — Wih he noaion inroduced above, he ollowing properies are saised: 1) or every poin x X here exiss a unique closed orbi conained in G x; 2) or every G -sauraed subses F F ′ X : F F ′ , ∅
i and only i
F F ′ , ∅
3) a subse V Y is open i and only V X is open; 4) le U be an open subse o X ; hen U is G -sauraed i and only i U U i U saises one o his wo equivalen properies, hen is projecion U is an open subse o Y . I Y q 3 X q : x RG x ′ G x G x ′
I , ♯ OY OX G , , ,
♯ OY OX G
. T q S ( [62, C 5, P 43)]) L’ S T ( , ). N, — S , 3. A S S .
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( ) R . Qestion. — I , ,
?
♯ OY OX G
1.7. A variant o the result o Kemp-Ness Deinition 1.8. — A u X invarian G U G : t G k X t U, u t u t
G k X X G X .
heorem 1.9 (c. T 3.3). — Le u X be a plurisubharmonic uncion which is invarian under he acion o a maximal compac subgroup o G. For every x X ,
x ′ x
ux ′
x ′ G x
ux ′
Deinition 1.10. — L u X . A x X : ▷ u-minimal on -bre ux ux ′ x ′ X x x ′; ▷ u-minimal on G-orbi ux ux ′ x ′ G x. T u (. u G) X u (. X G u). Corollary 1.11 (c. C 3.4). — Le u X be a plurisubharmonic uncion which is invarian under he acion o a maximal compac subgroup o G. Ten, 1) a poin x is u-minimal on -bre i and only i i is u-minimal on G-orbi; 2) X u X G u; 3) i u is moreover coninuous, he se o u-minimal poins on -bres X u is closed. I K N q T 1.9: Corollary 1.12. — Wih he noaion inroduced above, le u be opologically proper. Le x X be a u-minimal poin on is G-orbi (hus on is -bre). Ten here exiss a poin x G x such ha is orbi is closed and ux ux. Proo. — I x ′ G x . I f □ x x ′ ( u ). T q T 1.9 U K N. T S 3.5.
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1.8. Metric on GI quotients. — L X k G. L L G . S L ∥∥L : , U X s U L , ∥s ∥LU U R x U : ▷ ∥s ∥LU x sx ; ▷ ∥s ∥LU x ∥s ∥LU x k; ▷ V U , ∥s ∥LV ∥s ∥LU V . N L. L X X Y q X G. L X Y q . F D M D Y , ϕ D M D L D X G ( [48, T 1.10]). D M D : y Y t M D y, ∥t ∥M D y ∥ t ∥L D x x y
O , i.e. ( P 4.5). heorem 1.13 ([75, T 4.10], c. T 4.6 C 3). — Make he ollowing assumpion: ▷ he meric ∥∥L is invarian under he acion o a maximal compac subgroup o G; ▷ or every analyic open subse U X and every secion s U L ha does no vanish on U , he uncion ∥s ∥L U R is plurisubharmonic. Ten, he meric ∥∥M D is coninuous. Remark 1.14. — I , ∥∥ L FS KN. Z FS ( [73, T 2.2] [15, A A]). T Kä ∥∥ MD [74, T 2.2]. Remark 1.15. — I , . M , X k ◦ k ◦ G. S X q G L. L X , G L X, G L. T L ∥∥ L ( § 4.1.3). T, ∥∥ L G G ( D 2.21) ,
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L , ∥s ∥ L U X s U L ( C 4.4).
heorem 1.16 (c. T 4.6). — Under he assumpions o Teorem 1.13 suppose ha k is non-rivially valued and algebraically closed. Le x X k be a semi-sable k-poin o X and t x MD be a non-zero secion. Ten,
x ′ x
∥ t ∥L D x ′ ∥ t ∥L D д x д G k
where he supremum on he le-hand side is ranging on k-poins x ′ in he bre o x. S k Qp F t F . W R 1.15 X X Y q X . F D MD Y X , ϕ D MD LD X L ∥∥ MD MD .
heorem 1.17 (c. T 4.11). — Le k be eiher a nie exension o Qp or o he orm F t or a eld F and le K be he compleion o an algebraic closure o k. Le y be a K-poin o Y and le t y M D be a secion. Ten, ∥t ∥ MD y ∥ t ∥ LD x x y
where he supremum on he righ-hand side is ranging on K-poins x o X in he bre o y. 2. Preliminaries to the local part L k k . 2.1. Analytic spaces 2.1.1. Overview. — O k. T : 4 1) T : C . 2) T : R R q X X X X . 5 4. T R C . 5. N q X R X OX : ▷ X q X q ;
▷ X X , U X OX U OX U OX f U OX ♯ f f ♯ OX OX R .
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3) T : k ( ) k B . R B [5], [6]; [57, §1.2], B’ [23]. W k. I , k X B [5, §1.5, §3.4 §3.5], P [54], N [52, §2], . 2.1.2. Underlying opological space. — L X k . Deinition 2.1. — T X X x x X ( ) x R k k ( x x). T X , U X, 1) U x X x U X ; 2) f U OX , f U R f x f x T analyic opology X .
N x X x.
heorem 2.2. — I X is non-empy, he opological space X is non-empy, locally separaed and locally compac. Moreover, 1) i is Hausdor i and only i X is separaed over k; 2) i is compac i and only i X is proper over k. Proo. — T [5, §1.5]. S 1) 2) () () [5, T 3.4.8 3.5.3]. □ Deinition 2.3. — L x X . T complee residue eld x x x . T , Hx. T : , f X Y k, f f X Y . F X X X , X X .
Remark 2.4. — T X x : x , q x k . D :
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1) I , G’M f C () C. T X X X X C X C . 2) I , X X X X C GCR
3) I , X B. I X . 6 Proposition 2.5. — I k be an algebraically closed eld and he absolue value on k nonrivial, hen he se o k-poins X k is dense in X . A Z : Proposition 2.6. — Le X be a k-scheme o nie ype. For every consrucible se Z X , X Z X Z where on he le-hand side he closure is aken wih respec he analyic opology and on he righ-hand side o he Zariski one. Proo. — S [39, E. XII, C 2.3] [5, P 3.4.4] . T □ X ≃ X C GCR.
2.1.3. Srucural shea. — L X . B X Akn n.
Deinition 2.7. — L U X . A analyic uncion U f U x U x x U : 1) f x x; 2) U x U д kt tn U , y U , f y д y ( t tn Ank ). T k U OX U . T U ▷ OX U k X . F x X x . Deinition 2.8. — T n-dimensional analyic afne space k An OAn An k k
6. F , X Ak , Gauss norm ∥ ∥ k t R , a i t i a i , k t . I f , ∥ ∥ Ak .
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Remark 2.9. — I , C An C n C q . I , R An Cn GCR q R f Cn f z f z. I n f B. S [5, §1.5] [54].
E Ak , k X . 1) I X f, j X Ank n. L I OAn X I OAn I . C k X , OX j OAn I
O OX j. 2) F k X X iN X i f . T OXi X i OX X . O OX .
Deinition 2.10. — T k X X OX analyicaion o X . X X . A f Y X k k f Y X . I f f . 2.1.4. Exension o scalars
Deinition 2.11. — A analyic exension K k K q k K.
L K k. L X k X K X k K K K. L X K K K X K . Deinition 2.12. — T X K X k X K k X K X
exension o scalars map. I , X .
Proposition 2.13. — Te map X K k is surjecive and opologically proper. Since he opological spaces X and X K are locally compac, he map X K k is closed.
Proo. — I , k R K C, XCR q G . I [6, §1.4]. □
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T K : k f Y X f K YK X K K f .
Deinition 2.14. — A K-poin X x x x X x x K.
L x X X x K . T x x X . L x K K X K S K S x X K X K .
Deinition 2.15. — T x K K ( K K) X K poin associaed o x and he embedding x K.
2.1.5. Fibres. — L f Y X k . L x X , K x x K X K x. Proposition 2.16. — Keep he noaions jus inroduced. Ten: 1) Te map o scalars exension Y K k YK Y induces a homeomorphism f x Y K k YK X K x K
2) For every analyic exension K ′ o k and every poin x ′ X K′ such ha X K ′ k x ′ x, he map induced by K ′ K k , is surjecive.
Y K ′ k f K′ x ′ f x
Proo. — 1) I G . I , [6, §1.4]. 2) L Ω x ′ x ′ x Ω′ X Ω x ′. T x Ω′ x Ω x x Ω x ′ ( x ′ x). T Y ΩK ′
Y K ′ k
Y ΩK
Y K k
f Ω x Ω′ f K′ x ′ f x
f Ω x Ω f K x K f x
K x x x K X K x. T : , (1); YΩ X Ω x Ω YK X K x K
T Y K ′ k f K′ x ′ f x .
□
Proposition 2.17. — Wih he noaions inroduced above, le y y Y be poins such ha f y f y . Ten, here exiss an analyic exension Ω o k and Ω-poins x Ω , x Ω X Ω such ha:
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1) Ωk yi Ω yi or i ; 2) f Ω yΩ f Ω yΩ .
Proo. — T . Firs sep. – S X S k k . T : f Ω yi Ω i yΩ , yΩ X Ω y y . Second sep. – L x X f y f y K x . L x K X K x. A P 2.16 Y K k YK X K x K f x
. T yK yK YK 1) K k yiK yi i ; 2) f K yK f K yK .
C K Y ′ YK X K x K , X ′ x K □ f K YK X K . 2.2. Maximal compact subgroups. — L k . 2.2.1. Subgroups. — L G k (i.e. k ). L m G k G G G G . Deinition 2.18. — A H G subgroup : 1) m H H G k G H ; 2) H H ; 3) e Gk H .
A H compac G .
L K k H G . T H K G K k H G K
G K . L G k G k X X . Deinition 2.19. — L H G x X . T H -orbi o x, H x, H x G k X
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2.2.2. Archimedean deniion. — L k R C G () k . Deinition 2.20. — I k C maximal compac subgroup G U GC GC. I k R maximal compac subgroup G U G CR U GC. O f H H C Z. I : ▷ H C Z ; ▷ H C . I U G, U G ≃ U R C U UR. A T G R ( , U) T U T C. 2.2.3. Non-archimedean deniion. — L k G k. C G G. A Gk [18], [19], G [5, C 5] [57], [58]. L H f k ◦ H H k ◦ k . C U H h H f д f k ◦ H k ◦ H k ◦ H.
Deinition 2.21. — A H G maximal compac subgroup H U G k ◦ G k ϕ G k ◦ k G.
T U G ( D 2.18), G . T . Proposition 2.22. — Wih he noaions inroduced above: 1) he se o k-raionals poins U Gk U G Gk coincides wih he se o k ◦ poins Gk ◦ ; 2) or every analyic exension K o k, one has Kk U G U Gk ◦ K ◦
as subses o G K ; 3) i k is algebraically closed and non-rivially valued, he se U Gk is dense in U G. Proo. — 1) L ϕд A k k д Gk. T д U G ϕд f f k ◦ G, ϕд ϕд k ◦ G k ◦ . MÉMOIRES DE LA SMF 152
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2) L f f N k ◦ k ◦ G. F д G , f x f k ◦ G fi x i N
T f f N K ◦ K ◦ G. 3) T U G f B. T [5, P 2.1.15]. □ T [26] [34] , , Z. M : heorem 2.23. — Le G be a reducive k-group. Ten, here exis a nie separable exension k ′ o k, a Z-reducive group scheme G and an isomorphism o k ′-group schemes G k k ′ ≃ G Z k ′
T C 3.1.5 T 3.6.53.6.6 [26]. 2.3. Plurisubharmonic unctions. — I . I , , . I , P R [60], [61], R B [4], K [44], F J [31] T [66] ( ). T [66, C 5]. M , P . T : , M P . H, . O CL D [21] B, F J [17]. 2.3.1. Harmonic uncions. — L k Ω Ak .
Deinition 2.24. — A h Ω R harmonic x Ω U x Ω, N i N fi U OU i R N ∑ i fi h U i
N N , . I C . I T ( [66, D 2.31] [loc. ci., Tè 2.3.21]).
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Proposition 2.25. — Le Ω be an open subse o he analyic afne line Ak . Te ollowing properies are saised: 1) Harmonic uncions give rise o a shea o R-vecor spaces on Ak . 2) I f is an inverible analyic uncion on Ω hen f is an harmonic uncion. 3) Le f Ω ′ Ω be an analyic map beween open subses o Ak ; or every harmonic map h on Ω he composie map h ◦ f is harmonic on Ω ′. 4) Le K be an analyic exension o k and ΩK Kk Ω. For every harmonic uncion h Ω R composie uncion h ◦ K k ΩK R is harmonic. 5) (M P) I he open se Ω is conneced, hen an harmonic uncion h on Ω atains a global maximum i and only i i is consan. 6) I Ω is conneced, every non-consan harmonic uncion h Ω R is an open map.
Proo. — S 1) –4) q . 5) T M P ( ) [25, C I, 4.14]; P 2.3.13 [66]. 6) I f Ω . T Ω I R ( ) . : I , ; b R I , h b M P. T □ 2.3.2. Subharmonic uncions. — L k Ω Ak .
Deinition 2.26. — A u Ω subharmonic Ω ′ Ω h Ω ′ u Ω′ h , , . I q . T Ω q ΩC . I T ( [66, D 3.1.5], [loc. ci., C 3.1.12] [loc. ci., C 3.4.5] . Proposition 2.27. — Le Ω be an open subse o he analyic afne line A . Te ollowing properies are saised: 1) Harmonic uncions are subharmonic. 2) I u v are subharmonic uncions on Ω and are non-negaive real numbers, hen u v and u v are subharmonic uncions. 3) I f is an analyic uncion on Ω hen f is subharmonic. 4) Le K be an analyic exension o k and ΩK Kk Ω. For every subharmonic uncion u Ω R composie uncion u ◦ K k ΩK R is subharmonic. MÉMOIRES DE LA SMF 152
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5) Le f Ω ′ Ω be an analyic map beween open subses o Ak ; or every subharmonic map u on Ω he composie map u ◦ f is subharmonic on Ω ′. 6) (M P) I he open se Ω is conneced, hen a subharmonic uncion h on Ω atains a global maximum i and only i i is consan. 7) I ui i I is a locally bounded amily o subharmonic uncions on Ω, he is regularised upper envelope 7 is subharmonic. 8) Le u un be subharmonic uncions on Ω and ϕ Rn R be a convex uncion ha is non-decreasing in each variable. Exend ϕ by coninuiy ino a uncion ϕ n Ten he uncion ϕ ◦ u un Ω is subharmonic. Proo. — 1) F . 2) S [66, P 3.1.8]. 3) F . 4) I q [25, C I, T 4.12]. I [66, C 3.4.5]. 5) I [25, C I, T 5.11]. I [66, P 3.1.14]. 6) F . 7) S [66, P 3.1.8]. 8) S [25, C I, T 4.16]. □ Proposition 2.28. — Le v R be a uncion. Te composie map v ◦ t Gm
is subharmonic i and only i one o he ollowing condiions are saised: ▷ v is idenically equal o ; ▷ v is real-valued and convex. Proo. — I v . I v , v ◦ t (8) : v hi i I
hi ai bi v. F i I hi t ai t b (). T (7) , v t i I hi t () , . 7. N u i i I .
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S v . S t v ◦ t , v . L a b ϕ f va ϕa
vb ϕb
O v ϕ a b. S a b v ϕ , a b. B v ϕ , a b. T ϕ t t Gm Ω t Gm a t b
. 8 A v ◦ t , v ϕ ◦ t M P Ω. S , . M, v, v ϕ va ϕa vb ϕb
v ϕ .
□
2.3.3. Plurisubharmonic uncions. — L X k . Deinition 2.29. — A u X plurisubharmonic K k, Ω AK Ω X K , u ◦ Ω Ω. I ; . Proposition 2.30. — Le X be a k-analyic space. 1) I X is an open subse o he afne line Ak he u is plurisubharmonic on X i and only i i is subharmonic. 2) I u v are plurisubharmonic uncions on X and are non-negaive real numbers, hen u v and u v are plurisubharmonic uncions. 3) I f is an analyic uncion on X hen f is plurisubharmonic. 8. I . I Ω Ω e b a C C x Ak a t x b F , C x Ak t x
. T Ω , . T C , , . O C k , [5, P 3.1.8], [5, T 3.2.1].
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4) Le K be an analyic exension o k and u X be a plurisubharmonic uncion. Ten he composie uncion u ◦ K k X K is plurisubharmonic. 5) Le f X ′ X be an analyic map beween k-analyic spaces; or every plurisubharmonic map u on X he composie map u ◦ f is plurisubharmonic on X ′. 6) I ui i I is a locally bounded amily o plurisubharmonic uncions on X , he is regularised upper envelope is plurisubharmonic. 7) Le u un be plurisubharmonic uncions on X and ϕ Rn R be a convex uncion which is non-decreasing in each variable. Exend ϕ by coninuiy ino a uncion ϕ n
Ten he uncion ϕ ◦ u un X is plurisubharmonic.
Proo. — F P 2.27.
□
2.3.4. Consrucion o invarian uncions. — L n .
Deinition 2.31. — A exended norm An An R k k K k, K Ank , K k
u
R u K Ank K K n An k
K K n . I k R C u Cn . I k u , K k, K n . Remark 2.32. — 1) I k C Cn . 2) I k R Rn . 3) I k ux r t x r n tn x r r n t tn An . I k n , d
x x x k n
T k n q. T d k n : k n . Proposition 2.33. — Le k be non-archimedean. 1) A non-archimedean norm on k n exends o a coninuous non-archimedean norm u on An in a way such ha, i are norms on k n , hen k u x u x d x An k
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2) A coninuous non-archimedean exended norm is a plurisubharmonic uncion. Proo. — 1) T . S , , v vn k n r r n x v x nvn r x r n x n L ϕ ϕ n k n k . F x An k
u x r ϕ x r n ϕ n x
T u . T u u [35, P 2.1] k. T ( [11, 2.6.2, P 3]). 2) L u . I , k, k . T k n u : v vn k n r r n x v x nvn r x r n x n ( [11, 2.4.1 D 1 2.4.4 P 2]). L ϕ ϕ n . S k n An q : x Akn , k ux r ϕ x r n ϕ n x I u .
□
Proposition 2.34. — Suppose k algebraically closed. Le G be a reducive k-group and X an afne k-scheme o nie ype aced upon by G. Le U be a maximal compac subgroup o G. Ten, here exiss a coninuous, U-invarian, plurisubharmonic and opologically proper uncion u X .
Proo. — U X f Gq , X An G X . I k C f U. I k , G k ◦ U. L k n x д Gk ◦ д x
S Gk ◦ ( , G ) x . T Gk ◦ k n P 2.33 q u An . T u , k Gk ◦ . S Gk ◦ U, u U . □ 2.4. Minima on fbres and orbits. — I .
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2.4.1. Minima on bres Deinition 2.35. — L f X Y u X . T u-minima on f -bres f u Y y Y ux
f uy
f x y
L k f X Y k . L f X Y k f . L K k f K X Y K K. Proposition 2.36. — Le u X be a uncion. Wih he noaions jus inroduced: f K u ◦ X K k f u ◦ Y K k Proo. — T , y K YK , Y K k f K y K f y
y Y K k y K ( P 2.162)).
□
2.4.2. Minima on orbis. — L X k k G. Deinition 2.37. — L H G u X . T u-minima on H -orbis u H X , x X , u H x ′ ux ′
I H
G
uG u
x H x
G
.
Remark 2.38. — L G k X X k G X . L G k X X k . D H H G k X X
. W , , u H H u ◦
Proposition 2.39. — Le u X be a uncion. Le K be an analyic exension o k and consider he subgroup H K G K k H o G K . Ten, u ◦ X K k H K u H ◦ X K k Proo. — T P 2.36 R 2.38.
□
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Proposition 2.40. — Le u X be an upper semi-coninuous uncion. Ten, 1) he uncion uG X is upper semi-coninuous; 2) i u is coninuous, he subse X G u x X uG x ux is closed.
Proo. — 1) I uG x uд x д GC. I , P 2.39, . S k k . I k Gk G . A u x X , uG x ′ ux uд x x G x
д G k
C X uд x uд x д Gk. □ 2) F uG u. 3. Kemp-Ness theory L k . F : ▷ I f X Y k, f k f X Y f ; ▷ L X k k G. I x X G x G x. 3.1. Set-theoretic properties o the analytifcation o the quotient. — T () () T 1.6. L 1.41.5. Proposition 3.1. — Wih he noaion inroduced above: 1) he morphism X Y is surjecive and G-invarian; 2) or every x x ′ X , x x
i and only i G x G x , ∅
3) or every poin x X here exiss a unique closed orbi conained in G x. I , x x ′ X q G x q G x ′ . B P 3.1 :
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Corollary 3.2. — Wih he noaions inroduced above, le X ′ be a G-sable closed subscheme o X and Y ′ is caegorical quoien by G. Ten he induced morphism o k-analyic spaces Y ′ Y is injecive. Proo o Proposiion 3.1. — 1) C P 1.4. 2) C x x X :
x x G x G x , ∅
S G x G x y. B G , x y x S x x . F x , x k . A P 2.17 K k K x K , x K X K : ▷ K k x iK x i i ;
▷ K x K K x K ( K X K YK K ). S , AGKK AG k K G K K G . T f K YK q f K X K K G K . T, K, x x X k. L x x X k X X k X . : ▷ X Y k k X Y ; ▷ G x i k x i X G x i (i ) . F i : G x i G x i T x x x x . A T 1.4 : G x G x , ∅
F i G x i X Z Z (P 2.6): G x i G x i G x i
S G x , G x G x , G x . 3) F 2): x X y y G x. S G y y . I y y , (2) f G y G y , . □ SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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3.2. Comparison o minima 3.2.1. Saemens. — L 1.4–1.5 T 1.9: heorem 3.3. — Wih he noaion inroduced above, le u X be a plurisubharmonic uncion which is invarian under he acion o a maximal compac subgroup o G. For every poin x X , ux ′ ′ ux ′ ′ x G x
x x
Corollary 3.4. — Le u X be a plurisubharmonic uncion which is invarian under he acion o a maximal compac subgroup o G. Ten, 1) a poin x is u-minimal on -bres i and only i i is u-minimal on G-orbis; 2) X u X G u; 3) i u is moreover coninuous, he se o u-minimal poins on -bres X u is closed. Proo o he Corollary. — C u u G. S 3) P 2.402). □ I T 3.3 : heorem 3.5. — Wih he noaion previously inroduced, or every poin x X here exiss a poin x ha belongs o he unique closed orbi conained in G x and such ha ux ux. L T 3.3. Proo o Teorem 3.3. — F x X ,
y x
uy uy y G x
I q. L x x ′ X x ′ x. A T 3.5 x ′, x ′ q G x ′ ux ′ ux ′. B , x ′ x ′ x
G x ′ q G x. T, ux ′ ux ′
uy uy uy
y G x ′
y G x
y G x
q u. S x ′ , uy uy y x
T 3.3.
y G x
T T 3.5.
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3.2.2. Parabolic subgroups conaining desabilizing one-parameer subgroups. — D . L k k G f k X . L S X G X . T K ( [45, T 3.4]): heorem 3.6. — Le x X k be k-poin o X such ha G x S , ∅
Ten, here exiss a parabolic subgroup P PS x o G saisying he ollowing propery: or every maximal orus T P here exiss a one-parameer subgroup T Gm T such ha he limi t x exiss 9 and belongs o S.
t
3.2.3. Desabilizing one-parameer subgroups: archimedean case. — L G () U GC . T, R U U R C G U R U. M, T G R T C U T C. L X S A f G. L S X G Z . Lemma 3.7. — Le x X C be a poin such ha G x mees S. Ten, here exiss a oneparameer subgroup Gm G saisying he ollowing properies: ▷ he limi poin t x exiss and belongs o S; t
▷ he image o U is conained in U.
T [46] X AnC G S . I Gq f X An f S.
Proo. — W KN. A T 3.6 P G : T P T Gm T T t x
t
S. L P G U . L T P P R. A T G. T T 3.6 Gm T q . □ 9. N k x Gm X , t t x k x A X , t t x x
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3.2.4. Desabilizing one-parameer subgroups: non-archimedean case. — L k k ◦ . S k . L G k ◦ G . A G U Gmk ◦ G Gm U G. L X S A f k G S X G . Lemma 3.8. — Le x X k be a poin such ha G x mees S. Ten, here exiss a oneparameer subgroup Gmk ◦ G such ha he limi poin on he generic bre t x
t
exiss and belongs o S.
Proo. — B T 3.6 P G : T G T Gm T T t x
t
S. D P G G: k ◦ [34, E XXVI, Tè 3.3–C 3.5]. B q P G P. L T P T . 10 L Gm T T 3.6. S k , T q Gmk ◦ T q . □
3.2.5. End o proo o Teorem 3.5. — L x X U G u. U k : ▷ archimedean case: k C; ▷ non-archimedean case: k x k. L S q G x. A L 3.73.8 Gm G : ▷ x t x S; t
▷ U U.
10. B [34, E XII, Tè 1.7] ( ) q , S N s S s G S S s s s s. I Q H H [10, C 11.3]: , Q q H . O , [34, E XIX, C 2.6]. T P () S k ◦ : k P .
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L ux ux. T t t x k x Ak X x x . T t ux u x Ak t ut x
( u ◦ x ) U. B M P, u x t u x ux t
A P 2.28 v x R
v x t u x t
q . I , v x u x t ux t
v x . I , ux v x v x ux
T 3.5.
□
3.3. Analytic topology o the quotient 3.3.1. Saemen. — I 3) T 1.6: Proposition 3.9. — Le F X be a closed G-sable subse o X . Ten is projecion F is closed in Y . C C 3.2: Corollary 3.10. — Le X ′ be a G-sable closed subscheme o X and le Y ′ be is caegorical quoien by G. Te induced morphism o k-analyic spaces Y ′ Y is a homeomorphism ono a closed subse o Y . T P 3.9. 3.3.2. Minimal poins on afne cones. — T P 3.9 f . D . L Ad B Bd A d
d
() k k A , B (i.e. k ). L X S A Y S B . L ϕ A B D k, k d , Bd Ad D
T ϕ k f X Y . L f X Y k f . SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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Deinition 3.11. — L u X R . 1) A x X u-minimal on f -bre x ′ f x ′ f x ux ′ ux. T u f X f u.
2) L h A k X X A. T u X R -homogeneous z A k X , u hz z u z , A k X .
Proposition 3.12. — Wih he noaion inroduced above, le u X R be a coninuous, -homogeneous and opologically proper uncion. I X f u is closed in X , hen he resricion o f o X f u, is opologically proper.
f X f u Y
Proo o Proposiion 3.12. — T , k R . 11 C b bn B b , Y f An S kt tn kt tn k kt tn t . f f X f u X . A , f X . T q x i i N X ux i i . O f x i i N An ux i , i N. S k , i N i k i ux i . D q X xi xi i B u xi f . M x i ’ x ux . B q 12 , q x i x . B : ▷ x u f ; ▷ u x . 11. A K k K K R . T : . O k R . A ux i k . 12. I ; q [55].
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T . I f X An f fn f D ( ϕ D). T, i N n, f x i
f x i f x i i ux i
T f x i , f x i i . B ux i i ,
f x i
i n
i n
f x i ux i
f x . S x u f , S A X . x . □ T u u x u 3.3.3. Reducing o he case o he afne spaces. — G P 3.9 1.41.5. O X f Ank . L X S A X S A kf ( ) k G i X X Gq . F Y q X G X Y q . T k X
i
Y
X
j
Y
, j q. C 3.2 f j Y Y . I , F X , F j ◦ iF
S P 3.9 k X . I F G X , iF G X iF Y . T F j ◦ iF Y . O f Gq i X Ank . L X Ank S kt tn G. S G X , G k A kt tn . I , G AG AG A k . 3.3.4. Using Kemp-Ness heory. — T P 3.9 k: k C k G k ◦ G.
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L U u X R , , , U ( P 2.34). P 3.12 u u X u . S u , , , C 3.4, u u G, . N P 3.12 X u Y
. A u , . T X u Y , X u . O : G F X , (3.1) F X u F
X u Y , . L (3.1). C. O x F u x ′ F x x ′. S F G , G x x. L x ′ G x u G: u . S u G , x ′ u ; x x ′ x, (3.1). □ 3.4. Continuity o minima on the quotient. — L X f k k G. L Y q X G X Y q . L u X G. C uncion o u-minima on -bres u Y y Y uy ux x y
Proposition 3.13. — Te map u Y is upper semi-coninuous. I he uncion u is coninuous and opologically proper, hen: 1) he resricion o o X u X G u is opologically proper and surjecive ono Y ; 2) he uncion u is coninuous on Y . Proo. — u, V y Y uy . T 3.3 , x X , u x
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I :
V y Y uy x X u x x X uG x
T uG X , x x ′ G x ux ′ ( P 2.40). T q U V G X . B C 1.7 (4), V U Y . S u . 1) T X u Y u. I X u Y . L K Y . T u , K: uy y K
T K x X u x , f x X u x X u
. B u u X u, x X u u x x X u ux
T X u u . 2) T X u X , X u Y ( ). T q u
X u
u
u .
X u
□
3.5. Comparison with the result o Kemp-Ness 3.5.1. Special plurisubharmonic uncions. — I . W q T 1.9 KN , . L X . Deinition 3.14. — A u X special plurisubharmonic D X , D z C z , u ◦ .
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Proposition 3.15. — Special plurisubhamornic uncions enjoy he ollowing properies: i is a posiive real number and u is a special plurisubharmonic uncion u, u is special plurisubharmonic; i u v are special plurisubharmonic, hen u v is special plurisubharmonic; i X is a conneced analyic curve and f is a non-consan holomorphic uncion hen f is a special (pluri)subharmonic uncion; i f X ′ X is a holomorphic map wih discree bres and u is a special plurisubharmonic uncion on X , hen f u is special plurisubharmonic on X ′; srongly plurisubharmonic uncions are special plurisubharmonic. Example 3.16 1) T 4) : p ℓp , p ∥x ∥ℓp x p x n p
Cn . I p , ( n ) Kä OPn . { } 2) T ℓ ∥x ∥ℓ x x n . L G f X . heorem 3.17. — Le u X C be a special plurisubharmonic uncion invarian under he acion o a maximal compac subgroup U o G. Le x X C be a poin which is u-minimal on is G-orbi. Ten, 1) he orbi G x is closed; 2) le G x be he sabilizer o x; he ollowing inclusion is an equaliy: kд k U д G x C д GC д x X G u
I , G U. Corollary 3.18. — Le u X C be a coninuous opologically proper, special plurisubharmonic uncion, invarian under he acion o a maximal compac subgroup U o G. Ten, he coninuous map induced by , X u U Y
is a homeomorphism.
Proo. — 1) B x S q G x. A L 3.7 Gm G : ▷ x t x S; t
▷ U U.
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L ux ux. T t t x x Ak X x . C ux t u x C u x t ut x
T u x () U. A M P, u x ux t
P 2.28 v x R
v x t u x t
q . A u x , I , v x I . S v x u x t ux t
v x . T ux v x v x ux x. 2) S G T . A 1) T x x . R X T x T T Tx ( Tx x) x , x T X
t t x
. T u x t ut x T C U. I T C U Rn ( n T ) : n T C U C U Rn z zn z zn
S u x U, ( ) v Rn R u x . M, u x , v Rn . T x u v Rn . : , Rn , v t t . L G. L д GC д x u G. B C’ k U t T C д kt T G T C U T . S u U, uд x ukt x ut x SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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t x u G. B k ′ U T C t ′ Tx C t k ′t ′. T д kt kk ′t ′ kд k U д G x C .
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4. Metric on GI quotients 4.1. Extended metrics 4.1.1. Deniion. — L X k L . C L X , VL SX S OX L F k S S VL x s S x X S s S x L. C k X VL X VL.
Deinition 4.1. — A ∥∥L VL R exended meric on L K k, ∥ ∥L
∥∥L K VL K VL R L: K x X K s x L ∥s ∥L x ∥x s∥L K K x L. A VL . Remark 4.2. — L ∥∥L L. F U X ∥s ∥L U R , x ∥s ∥L x. O , , U X s U L , ∥s ∥LU U R x U : ▷ ∥s ∥LU x sx ; ▷ ∥s ∥LU x ∥s ∥L x k; ▷ ∥s ∥LU V ∥s ∥LV V U . T ∥s ∥LU L. I , . I , .
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4.1.2. Consrucions. — U (, …) . F . L ∥∥L L, K k, x X K K X s x L. F ϕ x L s x L, ∥ϕ ∥L K x
t x L
ϕt ∥t ∥L K x
∥s n ∥L n K x ∥s ∥L K xn
T ∥ϕ ∥L K x x ϕ VL : L . A L n n ∥∥L . L K k. I ∥∥L L ∥∥L ◦ V L K k VLK R
L K L X K X k K.
4.1.3. Exended meric associaed o inegral models. — S k X . L X k ◦ X , , k ◦ k X ≃ X k ◦ k. L L X L X ≃ L. T . D ∥∥ L . L K k x X K K X . T K ◦ K x K ◦ x S K ◦ X. T K ◦ x L K ◦ : s x L. E s x L ≃ x L K ◦ K s s q K. S: ∥s ∥ L K x K
T ∥s ∥ L K x x s VL ∥∥ L VL R
L. T : ∥∥ L L . I K ∥∥L ◦ K k LK ◦ L XK ◦ X k ◦ K ◦ . S k ( k ◦ ) L . T S X L d VL SX S OX L X d
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, S X OX . F x X VL X u Lx
f X L
f x
F f fn k ◦ X L, u Lx fi x i n
Proposition 4.3. — Suppose L generaed by is global secions and L very ample. Wih he noaions inroduced above, ∥∥ L u L ◦ I ∥∥ L VL , VL . Proo. — L K K ◦ . L x K X x q K ◦ X x. S L , f X L x f K ◦ x L. C s x L s x f . F K s s : ∥s ∥ L K x K . T, f x s K f x s K ∥s ∥ L K x K K F f X L,
f x s K f x s K ∥s ∥ L K x K K
f x s k ◦ . T .
□
Corollary 4.4. — I L is ample, hen he coninuous map ∥∥ L VL is plurisubharmonic. 4.2. Extended metric on the quotient 4.2.1. Deniion o he exended meric. — L X k k G G L. T k G X L d G A X L d AG d
d
. D X . T AG A G k X Y P AG MÉMOIRES DE LA SMF 152
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Y q X G. S AG , k Y : D M D Y ϕ D M D L D X q G. T ϕ D k VM D D VL D X
L ∥∥L L t VM D ∥t ∥M D
∥s ∥L D
D s t
s VL D . X Proposition 4.5. — Wih he noaion inroduced here above, he uncion ∥∥M D is an exended meric on M D . Proo. — L K k . L y Y K K Y t y M D y. S Y K YK ∥∥L , ∥t ∥M D K y
x X K
∥ t ∥L D K x
x y
I ∥∥M D , ∥∥M D . U M D , M D . I f s Y M D y Y , ∥t ∥M D y
T t Y M D M D ϕ D G t X L D G L D X X . F y Y , ∥t ∥M D y ∥ t ∥L D x x X
X □ ∥∥L . heorem 4.6. — Suppose ha: 1) he exended meric ∥∥L is invarian under a maximal compac subgroup o G; 2) he dual exended meric ∥∥L VL R is a plurisubharmonic uncion.
Ten he exended meric ∥∥M D is coninuous.
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4.2.2. Passing o he afne cones. — I . T L MD . S D LD MD . C k : X L d D AGD X M Dd X L d D G AD d
d
d
T k X Y AD AGD . T AGD AD k, S AD Y S AG X D
G ( [63, T 3]). Y q X T : VLD X
VLD
D MD
VM D
L D
X
Y
L D M D . T L D M D , VLD VM D . T ∥∥L D ∥∥M D u L D u M D Y . B ∥∥M D , y Y , X (4.1)
u M D y
u L D x u L D y
x y
( ). Proo o Teorem 4.6. — T u L D ∥∥L D : , , G. A P 3.13 u M D , □ ∥∥M D . 4.3. Compatibility with entire models 4.3.1. Noaion and saemens. — S k ( k ◦ ). L G k ◦ k ◦ X q G L. H S’ k ◦ . Deinition 4.7. — A A japanese K ′ K FA A K ′ A (i.e. A). A A universally japanese A .
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F , k k Qp k Ft F [37, C 7.7.4]. T S [63, T 2] : k ◦ G AG X Ld G A X Ld d
d
. D X , Y q X Y . F D MD Y LD ϕ D MD LD X
Gq .
D k k ◦ ( X k ◦ k X ). L ∥∥ L L L. Deinition 4.8. — W , Ω k . A x XΩ : 1) minimal s x L д GΩ, ∥s ∥ L Ω x ∥д s ∥ L Ω д x
T s. x 2) residually semi-sable 13 x XΩ Ω G k ◦ Ω. X k ◦ Ω Ω
L x X Ω x. T x (. ) Ω x Ω XΩ (. .) heorem 4.9. — Suppose ha k ◦ is universally japanese. Wih he noaions inroduced above, or every semi-sable poin x X he ollowing are equivalen: 1) x is minimal; 2) x is residually semi-sable. Corollary 4.10. — Under he hypoheses o Teorem 4.9, le Ω be an analyic exension o k which is algebraically closed and non-rivially valued. Le x X Ω be a semi-sable poin. Ten, here exiss a semi-sable minimal poin x X Ω lying in he closure o he orbi o x and whose orbi is closed (in X ). I k Qp , B [20, P 1]. W B B . 13. S X , x Ω◦ x S Ω◦ X. T reducion o x , x, x Ω◦ .
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T . A q, M D : 1) ∥∥ MD MD ; 2) ∥∥M D ( 4.2.1).
heorem 4.11. — Suppose ha k ◦ is universally japanese. Wih he noaion inroduced above, he merics ∥∥M D and ∥∥ MD coincide. In paricular, or every analyic exension Ω o k which is algebraically closed and non rivially valued, ∥t ∥ MD Ω y ∥ t ∥ LD Ω x x y
where he supremum is ranging on he semi-sable Ω-poins o X . 4.3.2. Some more noaions. — S D LD MD . B 4.2.2. A (4.1), y Y , u M D y u L D y
u L D x
x y
y Y C u LD , u MD x X f x u M y дy u LD x
S
D
f X LD
LD
д Y MD
MD , P 4.3, ∥∥ LD u LD ◦ L D
∥∥ MD u MD ◦ M D
T Y MD ≃ X L G Y MD X LD . , T, x X (4.1) u MD x u LD x
be a poin ha does no Lemma 4.12. — Wih he noaions inroduced above, le x X , belong o he analyicaion o verex OX o he afne cone X S OX S X OX X X L d D X .
d
Le x be he associaed poin o Te ollowing are equivalen: 1) he poin x is residually semi-sable; x. 2) u LD x u MD
Proo. — U x u LD x . L Ω x x S Ω◦ X x . 1) 2). S MD G f X LD x f Ω◦ x LD . I , f x Ω◦ . T u MD x u LD x MÉMOIRES DE LA SMF 152
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2) 1). T q u MD x G . I f X LD f x Ω◦ , Ω x x . □ Proo o Teorem 4.11. — S ∥∥ L, ∥∥ MD ∥∥M D L MD , LD MD . T q ∥∥M D ∥∥ MD q q u MD u M D . F y Y , q (4.1) u MD y u M D y
u LD x
x y
I q. L y Y . S , x y. u LD X A T 4.9 x x X . L 4.122) u LD x u MD x u MD y I ,
u MD y
u LD x ′
x ′ y
□
Proo o Teorem 4.9. — 1) 2) q (4.1) L 4.122). f k ◦ X 2) . D X LD , , k ◦ AD X Ld D d
U x u LD x . T q x x X k k ◦ X OX S X OX X. A x . T A x K Xk ◦ K. HM x , Ω K Gm Ω G k ◦ Ω
x : , OX S X OX , f X x OX k ◦ Ω t t
A [34, E XI, Tè 4.1] k ◦ k ◦ G k ◦ . S Ω◦ , “H’ L” [34, E XI, C 1.11] T G k ◦ Ω◦ SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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CHAPER 3. KEMPFNESS HEORY IN NONARCHIMEDEAN GEOMERY
T ( ). T, Ω , T Gm Ω◦ . T Ω Gm Ω G Ω U ◦ G Ω Ω G k ◦ Ω◦ . C ϕ x Gm Ω R ϕ x t u LD t x
T ϕ x U. D x R R : t Gm Ω, x t ϕ x t
T x , x u LD G , : x u LD x S u LD x x . f x , x. T Gm Ω◦ Ω◦ E X LD Ω◦ . T E , E Em
, m Z, Em f E t f t m f . F m Z m Z
um x f x f Em
T , t Gm Ω, { } ϕ x t u LD t x t m um x m Z
t , { } x m um x m Z um x ,
B u LD x m, um x . F, m , um x x. S : ▷ m m um x m ; ▷ m m um x um x ; ▷ m m um x um xm um xm . T x ] { u x } [ m m m T T 4.9 T 1.17. □
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Proo o Corollary 4.10. — U , k Ω k . M, x X . C Z Z G x X, k ◦ Z S k ◦ . T Z G. I, x
G G k ◦ X X
x S Ω◦ X x G k ◦ X X G X. T X Z Z k ◦ .
Claim. — Te srucural morphism X Z S k ◦ is surjecive.
k x ( Proo o he claim. — x X ). S u LD x OX X , u LD y G x ( a priori k). T y X y D 4.8, , T 4.9, . I , y S y◦ X X Z,
X Z
X
S y◦
S k ◦
y
I X Z S k ◦ .
□
S , ( k ). T ( T 4.9) . □
T [38, 17.6.2 18.5.11 (’)] , , : Lemma 4.13. — Le k be a complee non-archimedean eld, which is non-rivially valued and algebraically closed. Le S be a a scheme o nie ype over k ◦ . I he srucural morphism S S k ◦ is surjecive, hen here exiss a secion s S k ◦ S o .
Proo. — I f S f, S S A A k ◦ . C k A A k ◦ k. S A , A A. F f A : ∥ f ∥ A f A k T S S k ◦ ∥∥ A A. T ∥∥ A A. L A A ∥∥ A. SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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L S S S . T specrum ( [5, §1.2]) B k A s S f s ∥ f ∥ A f A MA
, [5, T 1.2.1], MA S A f . M, B k A Sk B ( [57, 1.2.4]): k , k MA □ MA. I , .
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I GI q, F F ( , q C 1), C 1. T . S 1 : , C 3 F F (T 1.5) . I S 2 q. F : , S 4 . S, F F, GI q. W . I S 3 GI q . F q B, G Z. H, B. I S 4 C C 1 ( T 2.1). T ( ). I , ( SL ).
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1. Statement o the main results 1.1. Notation. — L K oK . L X oK oK 1 G. S X q G L. A S [63, T 2] oK G, AG X Ld G A X Ld d d . D X , i.e. x X d G s Ld x. T AG A G oK ,
X Y P AG
Y q X G [63, T 4]. S AG , oK Y : D MD Y , ϕ D MD LD X
q G. “” , K C L X C ∥∥ L . S : ▷ S: Kä ∥∥ L ( ); q U X C s U L ∥s ∥ L ; ▷ I: ∥∥ L G C. S ∥∥ L K C . D L . L K C . D MD : y Y C t y MD ∥t ∥ MD y ∥ t ∥ LD x x y
heorem 1.1 (c. T 1.13 C 3). — Under he assumpions made on he meric ∥∥ L (semi-posiiviy and invariance under he acion o a maximal compac subgroup) he meric ∥∥ MD is coninuous.
T ∥∥ MD K C .
1. L S . A S G reducive ( S -reducive group) : 1) G f, S , 2) s S Ω S ( Ω ) G s G S s Ω.
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Deinition 1.2. — W , M D . C h M YQ R , Q YQ, h M Q h M D Q D D. T h M heigh on he quoien wih respec o X, L and G. 1.2. Instability measure. — L v VK K. I v ∥∥ Lv L.
Deinition 1.3. — L x Cv X. T (v-adic) insabiliy measure v x
д GCv
x L
∥д s ∥ Lv д x ∥s ∥ Lv x
. T s. s T x minimal v ( ∥∥ Lv G) , v x . Proposition 1.4. — Le x be a Cv -poin o X. Ten, 1) he insabiliy measure v x akes he value i and only i he poin x is no semisable; 2) i v is a non-archimedean place over a prime number p, he insabiliy measure v x akes he value i and only i he poin x is residually semi-sable, ha is, he reducion 2 x o x is a semi-sable F p -poin o X oK F p under he acion o G oK F p . T q T 1.9 T 4.9 C 3. 1.3. Statement and proo o the Fundamental Formula heorem 1.5 (F F). — Le P X K be a semi-sable K-poin. Ten he insabiliy measures v P are almos all zero and ∑ h L P v P h M P K Q v V K
Proo. — L P . T,
(1.1)
X K
K X t P MD
∑ K Qh MD P ∥t ∥ MD v P ∑
v VK
v VK
P ′ P
∥ t ∥ LD v P ′
2. S X , x ov X, ov Cv . p F p x X reducion x .
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q ∥∥ MD T 1.17 C 3 . A T 1.16 C 3: ∑ ∥ t ∥ LD v д P K Qh MD P д GCv
v VK
∑
v VK
д GCv
D
∑
v VK
∑
∥ t ∥ LD v д P ∥ t ∥ LD v P
∥ t ∥ LD v P
v P K Qh L P v VK
v P ( t G , д t t д GCv ). T F F. □ 1.4. Te case o a projective space. — L F oK . S oK G F , K C, ∥∥ F U G C. T oK G X P F q L O. F K C O X C FS ∥∥ O ∥∥ F . B , ∥∥ O G C . L L 1.1. L v K. L x F oK Cv x Cv X. B , v x
д GCv
∥д x ∥ Fv , ∥x ∥ Fv
∥∥ Fv F oK Cv ( ∥∥ Fv ). I F F : Corollary 1.6. — Le v be a non-zero vecor in F oK K and le P v be he associaed K-poin o X. I he poin P is semi-sable, hen: ∑ ∥д x ∥ Fv h M P h OF v д GCv ∥x ∥ Fv v VK ∑ ∥д x ∥ Fv v VK
д GCv
I B [20, P 5].
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M . L K C K. A P F C L U
G C P F C . F x F C, x , a L U , ⟨a x x⟩ , i ∥x ∥ ⟨ ⟩ ∥∥ F , i () q L U F C F C . xa
Proposition 1.7. — Wih he noaions inroduced above, he poin x P F C is minimal i and only i he linear map x is idenically zero.
F P 1.2 C 3 . 1.5. Lowest height on the quotient. — S ∥∥ MD MD Y . S h X L G h M Q Q Y Q
B h X L G C F F .
Corollary 1.8. — Le P X K be a semi-sable K-poin. Ten he insabiliy measures v P are almos all zero and ∑ h L P v P h X L G K Q v V K
F q. 1.6. Te lower bound o Bost, Gasbarri and Zhang. — L N e e N . C oK
G GLeoK oK oK GLe N oK S SLeoK oK oK SLe N oK
K C U Ue Ue N G C
SU SUe SUe N S C
L F oK G GL F , oK , : K C ∥∥ F U . B 1.4.
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L E E EN N oK Ei ei i N . F E F “” E ( S 3.1 ). T F E , E GE GL E oK oK GL EN GL FE
. C:
SE SL E oK oK SL EN XE P FE L E OFE FS F E YE q XE SE LE T homogeneous o weigh a a a N ZN , oK T t t N Gm T , t E t N EN t a t Na N F
heorem 1.9 (c. T 3.8). — Wih he noaions inroduced above, i he represenaion is homogeneous o weigh a a a N ZN and he subse o semi-sable poins X is no empy, hen: 1) here exiss an isomorphism E YE Y; 2) or every D divisible enough here exiss a canonical isomorphism o hermiian line bundles, ha is an isomeric isomorphism o line bundles, E M D E E M D
N ⊗ i
f E Ei ai D ei
where f E YE S oK is he srucural morphism; N ∑ 3) h XE LESE h X LS ai Ei . i
Corollary 1.10. — Wih he noaion o Teorem 3.8, or every K-poin P o XE which is semi-sable under he acion o SE: h L P E
N ∑ i
ai Ei h X LS
F N G [32, T 1] B [12, P 2.1] Z [75, P 4.2]. 1.7. An explicit lower bound. — I , q. L N E E EN MÉMOIRES DE LA SMF 152
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N oK . C oK
G GL E oK oK GL EN S SL E oK oK SL EN
K C , U U∥∥ E U∥∥ EN G C
SU SU∥∥ E SU∥∥ EN S C
L F oK G GL F , , K C ∥∥ F U . C S F. B 1.4. F n n n ∑ i ℓn n n i
T ℓn n , S’ , ℓn n n .
heorem 1.11 (c. T 4.1). — Wih he noaions inroduced above, le N ] ⊗ ϕ E Ei ai oK Eibi F i
be a G-equivarian and generically surjecive homomorphism o hermiian vecor bundles. Ten, N ∑ ∑ h P F OFS bi Ei bi ℓ Ei i Ei
i
wih equaliy i b b N .
A, : Conjecture 1.12. — Under he same hypoheses o Teorem 1.11,
N ∑ bi Ei h P F OFS i
T T 1.11 oK . T [22] [16].
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2. Examples o height on the quotient 2.1. Endomorphisms o a vector space 2.1.1. Semi-sable endomorphisms. — L k E k () n. C k GLE f k X EE SA A Sk EE . F ϕ E E Pϕ T , Pϕ T T E ϕ T n ϕT n n n ϕ
T f ϕ n ϕ f ϕ, i.e. A, SLE. Proposition 2.1 ( [49, P 2]). — Te afne space Ank ogeher wih he map EE Ank ϕ ϕ n ϕ
is he caegorical quoien o X by SLE. In paricular, he invarians n generae he k-algebra o invarians ASLE . Proposition 2.2 ( [49, P 4]). — Le ϕ E E be a linear map. Ten:
1) he orbi o ϕ is closed i and only i ϕ is semi-simple i.e. i can be diagonalized; 2) he closure G ϕ o he orbi o ϕ conains he orbi o he semi-simple par ϕ o ϕ.
Corollary 2.3. — For every non-zero endomorphism ϕ E E he associaed k-poin ϕ PEE is semi-sable i and only i ϕ is no nilpoen. 2.1.2. Arihmeic siuaion. — L K oK . L E oK . C S SL E E E. E oK E E ( § 3 9). T ∥∥ E E SU∥∥ E G C. B 1.4 ( F E E G S). heorem 2.4. — Wih he noaions inroduced above, le ϕ be an endomorphism o he Kvecor space E oK K. Suppose ha he corresponding K-poin ϕ o PE E is semi-sable ha is, he endomorphism ϕ is no nilpoen. Ten, Ω Qh M ϕ is equal o ∑ ∑ v n v n v VΩ .
ΩC
where n are he eigenvalues o ϕ couned wih mulipliciies and Ω is a number eld conaining hem.
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T q Y PE E SL E Sn ( , q Pon K P n oK ). C “” A h Pon , i.e. , K Ω K Ω n , ∑ ∑ v n v n Ω Q v V Ω Q ΩC Ω
.
T, h M h Y q Pon Y Sn . K T T 2.4.
2.1.3. Reducion o local saemens. — L v K E Cv n. L e en E q E ∥∥E x v x n v ∥x e x n en ∥E x v x n v
v v
Eq Cv EE ∥∥ EE : ∥ϕ ∥ EE
∥ϕx∥E ∥x ∥E x , ∥ϕe ∥E ∥ϕen ∥E
v v
I ∥∥E o E o e o en E ( o Cv ). T ∥∥ EE o EE EE. I ∥ϕ ∥E ϕ ◦ ϕ ϕ ϕ ∥∥E . Proposition 2.5. — Wih he noaion inroduced above, or every endomorphism ϕ o E,
д SLECv
∥дϕд ∥ EE
v n v v n v
v v
where n are he eigenvalues o ϕ (couned wih mulipliciies). Proo o Teorem 2.4. — I f F F C 1.6 P 2.5. □
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I P 2.5, f ( e en ),
.
n
n
2.1.4. Compuing minimal endomorphisms. — I ∥ϕ ∥ EE ∥дϕд ∥ EE д SLECv
Proposition 2.6. — Le n Cv . Wih he noaions inroduced above, he endomorphism ϕ n is minimal. Proo o Proposiion 2.6: he non-archimedean case. — L v . P 1.42) f ϕ ϕ F v PEE o F p . L n Cv . U ϕ n v n v
T ϕ F p PEE ϕ n F p E o F p F p e F p en
, i n, i F p i . T ϕ , . T, P 1.43), ϕ , . □ Proo o Proposiion 2.6: he archimedean case. — L suE L L SU∥∥E . A X C suE : ϕ A suE ϕ A
⟨A ϕ ϕ⟩ EE , i ∥ϕ ∥ EE
A ϕ Aϕ ϕA L , ⟨ ⟩ EE ∥∥ EE i q . A P 1.7, ϕ ϕ A A suE. T q : ⟨Aϕ ϕ⟩ EE ⟨ϕA ϕ⟩ EE
A EE
Lemma 2.7. — Wih he noaion inroduced above, or every endomorphism ϕ o E he ollowing condiions are equivalen:
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1) ⟨Aϕ ϕ⟩ EE ⟨ϕA ϕ⟩ EE or all A EE; 2) ϕ ϕ ϕϕ , where ϕ denoes he adjon endomorphism o ϕ wih respec o he norm ∥∥ EE ; 3) Te endomorphism ϕ is diagonalisable on a orhonormal basis. Proo o Lemma 2.7. — T q 2) 3) . T q 1) 2) . F i j n Ai j E Ai j ek ik e j k n ( ik K’ ). W ϕ ni j ϕ i j Ai j . W , i j n, n n ∑ ∑ ϕ jk Aik ϕAi j ϕ ki Ak j Ai j ϕ k
k
T Ai j EE , i j n, ⟨Ai j ϕ ϕ⟩
n ∑
k
O , , ϕ ϕϕ
n ∑
i j
⟨ϕAi j ϕ⟩
ϕ jk ϕ ik n
i j ϕ ji Ai j .
ϕ ki ϕ k j
k
T
ϕ ϕ
⟨Ai j ϕ ϕ⟩Ai j
n ∑
n ∑
i j
⟨ϕAi j ϕ⟩Ai j
I q.
□
T L : n □ ϕ n . 2.1.5. A varian. — L . ∥∥ EE EE : ∥ϕ ∥ x ,
I SU∥∥E .
I
∥ϕx∥E ∥x ∥E
Proposition 2.8. — Wih he noaion inroduced above, ∥дϕд ∥ n д SLEC
where n C are he eigenvalues o ϕ couned wih mulipliciies
Proo. — F i n vi i : д SLE C i n, дϕд д vi i ∥д vi ∥E E
I , д SLE C,
∥дϕд ∥ n
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д SLEC
∥дϕд ∥ n
S n SLE ϕ, ∥дϕд ∥ n n
д SLEC
.
□
T , P 2.5. 2.2. Te lowest height on the orbit is not the height on the quotient 2.2.1. Te quesion. — L S 1.1 P K X. S G: (2.1)
h L д P
д G Q
P ′ P
h L P ′
Q X P. S v P v VK , F F h L P h M P, : (2.2) h L P ′ h M P ′ P P
C q: (2.3)
h L д P h M P
д G Q
Qestion. — W q (2.1), (2.2) (2.3) ? T G GLnZ F : 1) I ( n ) q (2.2) (2.3) F 3 . 2) I ( n ) F ( , F Zr Rr ) f. T Gm , , Gm GL F . 3) I , GLnZ F EZn . E F ( § 3 9) Rn . I q (2.2) (2.3) q Q P F ( P F ). N, q (2.1) q . 3. A E K E E.
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2.2.2. Linear acion on a non-rivial hermiian vecor bundle. — C Gm AZ t x x t x tx
S, t Gm S x x A S. C G Gm X PZ G L O. F k, X k P k
L Y q X ( S Z) X Y q . F r r r ∥∥r R ∥x x ∥r r x r x
E L FS ∥∥r , U. D Lr X h Mr q Y Lr . Proposition 2.9. — Wih he noaions inroduced above: 1) Te poin P Q is semi-sable and
h Mr r r
2) For every t Gm Q,
h Lr t t wih equaliy or t .
r r
I , q (2.2) (2.3) r r . T ( P 2.10 2.13). 2.2.3. A negaive example when he hermiian vecor bundle is rivial. — C Z E Z x x x x x x
L Gm Z
t x x x t x tx t x
C Gm X PZ O. E O FS ∥∥. Proposition 2.10. — Consider he poin P . Wih he noaions inroduced above: 1) Te poin P is semi-sable and h M P SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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126
2) For every t Gm Q,
h O t P
wih equaliy or t .
T q (2.2) (2.3) , E . Proo. — 1) T P . F p , P P P Fp
PFp . I P q Q. O , p , no : , PF . F t Gm C , t t t t T t , t t Gm C
F F F ∑ h M P v VQ
t Gm Cv
t v
2) L K t Gm K. F v , t v t v t v
v , t v t v t v
T, K, P F, K Qh O t P } ∑ { ∑ ∑ t t t t K C t K C K C
Lemma 2.11. — Le N be a posiive ineger. For every x x N R,
N {∑ i
xi
N } ∑ x i e x i e x i e x i N i i
N ∑
Proo o he Lemma. — L RN R x x N
N {∑ i
MÉMOIRES DE LA SMF 152
xi
N } ∑ x i e x i e x i e x i i i
N ∑
2. EXAMPLES OF HEIGH ON HE UOIEN
127
T . T R RN , , R R x N x x e x e x e x T . T x R, .
x N
□
L P 2.10. O N K C, N K Q, L x i t i i N . T, h O д P . □ 2.2.4. Endomorphisms. — L n E Z E Zn . L e en E. A S 2.1 G SLnZ F E E 2.1.2. F K n K, n E K K n ( ), n
n Proposition 2.12. — Le n K and suppose ha hey are no all zero. Wih he noaion inroduced above, h OF n h M n
In paricular, or all non-zero semi-simple endomorphism ϕ o K n , h OF д ϕ h M ϕ д SLn Q
T q T 2.4. T q (2.2) (2.3) , , . H, q (2.1) q . F , n ϕ E ( ), ( ) ϕ
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128
Proposition 2.13. — Wih he noaions inroduced above: 1) Te endomorphism ϕ is semi-sable and h M ϕ
2) For every д SL Q,
wih equaliy or д .
h O д ϕ
Proo. — 1) T q T 2.4. 2) L K д SL K ( ) a c д b d a b c d K ad bc . W , ) ( ab a дϕд b ab
F v K: ∥дϕд ∥ v ab v ab v av b v av b v
O , K C,
∥дϕд ∥ ab ab a b a b
S a , . S b v v VK , ∑ ∑ K Qh O д ϕ a av v VK
K C
T P F: { ∑ } { } ∑ ∑ av av a v VK
v VK
K C
P , K Qh O д ϕ { } ∑ ∑ (2.4) a a K C
K C
Lemma 2.14. — Le N be a posiive ineger. For every x x N R, N } { ∑ x i i
N ∑ i
e x i N
T L L 2.11.
MÉMOIRES DE LA SMF 152
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129
L a , . O N K C
N K Q, L x i ai i N . A (2.4), h O д ϕ T a b , . □ 3. Te approach o Bost, Gasbarri and Zhang 3.1. wisting by principal bundles. — I B, G Z B , G. 3.1.1. Te algebraic consrucion. — L G S. L X S () G P G 4 ( Z ). B , wis o X by P q X S P () G д x p дx pд S S ′ д GS ′ x p X S ′ PS ′. C, X P : 1) P S i I S i P , i I , pi S i P . 2) G X S S i X S S i j X S S i j
x дi j x
S i j S i S j дi j q S i j G p j pi . I X P X S. T : 1) L G . T G P S X P . M , H G ( , S S ′, H S ′ GS ′) H P G P . 2) L G F S VF P P VF F . C S U S U F P MS U VF P T F P S, S G P VF P VF P . T F P F P. 4. A principal G-bundle P S () P S G P : 1) P S G P S P S ; 2) P Z : S i I S i i I pi S i P .
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130
CHAPER 4. HEIGHS ON GI UOIENS: FURHER RESULS
3) L L G X VLP P VL X . C L P X P X P U X P U L P MS U VLP T L P G P X P VL P X P VLP .
Example 3.1. — L N e e N . C S : G GLeS S S GLe N S S SLeS S S SLe N S
L E E E N N S Ei ei i. E G P E FS E S S FS E N
i S FS Ei rame bundle Ei : S f S ′ S, FS Ei S ′ I OS ′ OSei′ f Ei
T () G P E . L F S G GLF . C G X PF L OF . D F E F P E . T: G E GLE S S GLE N S E SLE S S SLE N
X E PF E
L E OF E G E , S E , X E L E G P E , S P E , X P E L P E . 3.1.2. Te hermiian consrucion. — L . L G CG GC . D G CG G. Deinition 3.2. — A principal hermiian G-bundle P C P G C P PC G, .
C P CG C P C P
p u p pu
L X X C X C X X C. S G X CG C X . MÉMOIRES DE LA SMF 152
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131
Deinition 3.3. — L P P C P G. T wis o X by P X P X P C X P , X P X P C X P C X C P CG X P C X C PC GC
N G.
L p C P x X C x p x p X P C X C PCGC. T X C X P C
x x p
C X C X P . L P P . T G : 1) L G . T G G CG P G P CG P G P CG P G P C. L H H C H H G C H H C CG . T H P H P C H P H P P ( G P ) H P C. 2) L F F ∥∥F ( ) . S G F ∥∥F CG . C VF VF DF DF v F ∥v ∥F L VF P VF P DF P VF P . T q ∥∥F P F P DF P v F P ∥v ∥F P
T F P F P ∥∥F P F P . 3) L X G G L. S L q ∥∥L CG . C VL VL DL VL L X DL x s x X C s x L ∥s ∥L x R X C , DL VLC. M CG . L VL P VL P DL P VL P . T q ∥∥L P L P DL P x s x X P C s x L P ∥s ∥L P x
T L P L P ∥∥L P L L ∥∥L P .
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132
Example 3.4. — L N e e N . C G GLeC C C GLe N C
S SLeC C C SLe N C
CG U Ue Ue N
C S SU SUe SUe N
L E E E N N , E i Ei ∥∥Ei
Ei ∥∥Ei . S C Ei ei i. E G P E P E C E P E FC E C C FC E N
C E OE OE N
, i, FC Ei Ei OE i orhonormal rame bundle E i , i.e. Cei E i ( Cei ). L F F ∥∥F G GLF . S ∥∥F U. C G X PF L OF FS ∥∥L . D F E F E ∥∥F E F P E . T: 1) G U P E G E U E G E GLE S S GLE N
U E U∥∥E U∥∥E N
2) S SU P E S E SU E S E SLE S S SLE N
SU E SU∥∥E SU∥∥E N
3) L L ∥∥L P E L E L E ∥∥L E X E PF E L E OF E
∥∥L E FS ∥∥F E
3.2. Statement and proo o the result 3.2.1. Seup. — L K . L N e e N . C oK
G GLeoK oK oK GLe N oK
S SLeoK oK oK SLe N oK
K C U Ue Ue N G C
SU SUe SUe N S C
L F oK G GL F , oK , : K C ∥∥ F U .
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3. HE APPROACH OF BOS, GASBARRI AND ZHANG
133
T G F G X P F G L O. F K C O X C FS , U . C X S q Y XS X S, Y P AS A X Ld SodK F d
d
L X Y q . F D MD Y LD . E S 1.1. S S G, d , oK S Ld , X Ld S X Ld
G X Ld . T X G. M G q Y , D , MD G. F K C ∥∥ MD U . 3.2.2. wising by hermiian vecor bundles. — L E E EN N oK Ei ei i N . A S 3.1.1 N oK E E EN E. G E 3.1, FE GE, SE, XE, LE F, G, S, X, L E. T:
GE GL E oK oK GL EN XE P FE
SE SL E oK oK SL EN LE OFE
T oK GE FE LE GE X G . C XE E E S , LE. D XESE q XE E D , MD E LED . Proposition 3.5 (C GI q ). — Wih he noaions inroduced above: is he wis o X by E; 1) Te se o semi-sable poins XE 2) Te quoien XESE is he wis o he quoien Y XS by E; 3) Te inverible shea MD E is he wis o MD by E. Skech o he proo. — A . Remark 3.6. — L V oK G. D VE E. T VESE VE SE VS E VS E.
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CHAPER 4. HEIGHS ON GI UOIENS: FURHER RESULS
134
T : , VS E VESE q S oK . P f □ V X Ld d .
L K C . B , , G C GL FC
, , ∥∥ F U . T S 3.1.2. G E 3.4, FE C ∥∥ FE U E U ∥∥ E U ∥∥ EN T LE FS ∥∥ LE ∥∥ FE . T ∥∥ LE SU E SU ∥∥ E SU ∥∥ EN S C
L ∥∥ MD E MD S 1.1. T ∥∥ MD MD U : ∥∥ ′MD E
MD E M D E.
Proposition 3.7. — Wih he noaions inroduced above, he merics ∥∥ MD E , ∥∥ ′MD E coincide. Proo. — T i Cei Ei C i N . T . □ 3.2.3. Saemen. — K . B homogeneous o weigh a a a N ZN oK T t t N Gm T , t E t N EN t a t Na N F heorem 3.8. — Wih he noaions inroduced above, le E E EN be a N -uple o hermiian vecor bundles over oK such ha Ei ei or every i. I he represenaion is homogeneous o weigh a a a N ZN and he subse o semi-sable poins X is no empy, hen: 1) here exiss an isomorphism E YE Y; 2) or every D divisible enough here exiss an isomorphism o hermiian line bundles, ha is an isomeric isomorphism o line bundles, E M D E E M D
N ⊗ i
f E Ei ai D i ei
where f E YE S oK is he srucural morphism; N ∑ 3) h XE LESE h X LS ai Ei . i
MÉMOIRES DE LA SMF 152
3. HE APPROACH OF BOS, GASBARRI AND ZHANG
135
Corollary 3.9. — Wih he noaion o Teorem 3.8, or every K-poin P o XE which is semisable under he acion o SE, h L P E
N ∑ i
ai Ei h X LS
3.2.4. Proo o Teorem 3.8. — L . Proposition 3.10. — Le V be non-zero oK -module which is a and o nie ype. Le r G GL V be a homogeneous represenaion o weigh b b b N . I he submodule VS o S-invarian elemens o V is non-zero, hen: 1) ei divides bi or every i N ; 2) he induced represenaion r G GL VS is given by r д дN
N ∏
дi bi ei
i
or every oK -scheme T and or every д дN GT . Proo. — C r G GL VS. S S FS , r oK N GL VS r GS Gm
T r b b b N , , oK T t t N Gm T , O ,
bN r t E t N E t b t N
r t E t N E rt e t Ne N bN rt e t Ne N t b t N
S 1) 2) .
□
Corollary 3.11. — Under he hypoheses o Teorem 3.8: Te acion o G on Y is rivial; For every ineger D divisible enough he oK -group scheme G acs on he bres o MD hrough he characer N ∏ д дN дi ai D ei i
More precisely, or every oK -scheme T , every д дN GT , every poin y YT and every secion s T y MD , N ∏ д дN y s y дi ai D ei s i
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136
Proo. — P D MD . T, O ≃ M Gq. T j D Y P Y MD j D D Y MD S X LD S SoDK F
S a a a N , SoDK F Da Da Da N
I P 3.102) V SoDK F G Y MD N ∏ д дN дi ai D ei i
A 1) 2) .
□
T 3.8 C: Proo o Teorem 3.8. — A P 3.52), q YE q Y E: G , E YE Y
S, P 3.53) P 3.7, M D E M D E. S G MD N ∏ дi ai D ei д дN i
E M D E E M D
.
N ⊗
f Ei ai D i ei
i
□
4. Lower bound o the height on the quotient 4.1. Statement. — I T 1.11. L 1.5. L N E E EN N oK . C oK
G GL E oK oK GL EN
S SL E oK oK SL EN
K C , U U∥∥ E U∥∥ EN G C
SU SU∥∥ E SU∥∥ EN S C
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4. LOWER BOUND OF HE HEIGH ON HE UOIEN
137
L F oK G GL F , , K C ∥∥ F U . C S F 1.4.
heorem 4.1. — Le a a a N and b b b N be N -uples o inegers. Wih he noaions inroduced above, le ϕ
N ⊗ i
E Ei ai Eibi F
be a G-equivarian and generically surjecive homomorphism o hermiian vecor bundles. Ten, N ∑ ∑ Ei h P F OFS bi bi ℓ Ei i
wih equaliy i b b N .
i Ei
T ϕ Gq v v K ( , ϕ oK ). F , f T 4.1 ϕ , : heorem 4.2. — Le a a a N and b b b N be N -uples o inegers and se
F
N ⊗ i
Wih he noaions inroduced above,
E Ei ai Eibi
N ∑ bi Ei h P F OFS i
wih equaliy i b b N .
∑
i Ei
bi ℓ
Ei
T S 4 T 4.2. 4.2. ensor products o endomorphisms algebras 4.2.1. Noaion. — I T 4.2 bi i N , , N ⊗ F E Eiai i
N a a a N .
heorem 4.3. — Wih he noaion inroduced above, h P F OS B q:
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138
Proposition 4.4. — Wih he noaion inroduced above, h P F OS
Proo. — T E Eiai ≃ E Eiai ai . A T 3.8 f K Q Ei , , i N , Ei Z Ei Zei . C E′ Ea ENa N . I E′ Za e a N e N . W Sq F ≃ E E′. C ϕ E′ ( E′) : ϕ
T ϕ P F SL E′ ( S) . I h M q Y P F S P F Y q , F F (C 1.6) h M ϕ h OF ϕ .
□
I q. I f : heorem 4.5. — Le ϕ F oK K be a non-zero vecor such ha he associaed K-poin ϕ o P F is semi-sable. Ten, ∑ ∥д ϕ ∥ Fv д SCv ∥ϕ ∥ Fv v V K
I, T 4.3 T 4.5 F F ( C 1.6) K ′ K P F K ′. T T 4.5. 4.2.2. Te case o a non-nilpoen endomorphism. — C ϕ N a i K oK K i Ei
N ⊗ i
MÉMOIRES DE LA SMF 152
E Eiai ≃ E
N ⊗ i
Eiai
4. LOWER BOUND OF HE HEIGH ON HE UOIEN
139
W ϕ . T ϕ S. A, : oK
H SL Ea ENa N
F( ). A C 2.3, ϕ H. T ϕ S , v K, (4.1)
д SCv
∥д ϕ ∥ Fv
h HCv
∥hϕh ∥ Fv
T F N N N ⊗ ⊗ ⊗ Eiai Eiai oK Eiai E i
i
i
T T 2.4 ϕ ∑ ∥hϕh ∥ Fv Ω K
q
v VK
∑
v VΩ .
h HCv
v n v
∑
ΩC
n
n ϕ ( ) Ω . S q , (4.1) , ∑ ∑ ∥д ϕ ∥ Fv ∥hϕh ∥ Fv v VK
д SCv
T 4.5 .
v VK
h HCv
4.2.3. Te case o a non-vanishing invarian linear orm. — S S f P F O F ϕ. F M T I T. T . F i N S ai ai . F i Eiai . S E Eiai , i SL Eai . T , N , N S a S a N
N N F S.
W F M T I T (c. [71, C III], [22, T 3.1, C] [2, A 1]):
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140
heorem 4.6 (F M T I T). — Te subspace o elemens o F oK K which are invarian under he acion o S oK K is generaed, as a K-linear space, by he elemens , while ranges in S a S a N .
F N N S a S a N
F
N ⊗ i
E Eiai ≃ F
N ⊗ i
E Eiai
L T 4.5. S S ϕ, T 4.6, N N S a S a N ϕ , . B , ϕ ϕ ◦
S ϕ ◦ . T, : ∑ (4.2) д ϕ ◦ Fv
д SCv
v VK
R : 1) F v K F oK Cv , ∥ ◦ ∥ Fv ∥ ∥ Fv
2) T S ( S ). A q , д SCv , ∥д ϕ ∥ Fv д ϕ ◦ Fv
S , q (4.2) ∑ ∥д ϕ ∥ Fv v VK ′
д SCv
T 4.5 .
4.2.4. Te general case. — L . B D S f P F OD SD F
ϕ. C D V P F P FD
ϕ ϕ D
T ϕ D P FD . S FD SD F Sq , P , f S f ′ FD oK K. 5 5. L k V W k G. A Gq ϕ V W ϕ V G W G . I ϕ k V G W G [49, 181–182].
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4. LOWER BOUND OF HE HEIGH ON HE UOIEN
141
U f ′ S f ′ P FD O
ϕ D . T ϕ D ∑ ∥д ϕ D ∥ FD v д SCv
v VK
F v K д SCv , D ∥д ϕ D ∥ FD v ∥д ϕ ∥ Fv
T 4.5.
□
4.3. Te general case 4.3.1. Noaion. — I T 4.2. A T 4.3 : heorem 4.7. — Le ϕ FoK K be a non-zero vecor such ha he associaed K-poin P x o P F is semi-sable. Ten, ∑ ∥д x ∥ Fv д SCv ∥x ∥ Fv v V K
T T 4.7.
4.3.2. Te case o a non-vanishing invarian linear orm. — S S P. I , S N ⊗ F E Ei ai Eibi i
. T P 3.101) ei Ei bi i N . T P F T 4.7 T 4.3. Deinition 4.8. — F i N . 1) F ei , i oK i Ei E Ei Ei
, ϕ E Ei v v Ei , ϕ v v ϕv v ϕv v
2) F ei , , i oK ,
i Eiei E Eiei Ei i E Eiei Ei Eiei SOCIÉÉ MAHÉMAIUE DE FRANCE 2017
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142
: ϕ E Eiei v vei Ei , ϕ v vei ∑ ϕv v ei v vei Sei
E oK E Ei Ei E Eiei Ei ( , ) ∥∥ Ei . Proposition 4.9. — Fix an ineger i N . Wih he noaions inroduced above: or ei , i is a GL Ei -equivarian isomorphism o hermiian vecor bundles; or ei , , i is GL Ei -equivarian and, or an embedding K C, x ,
∥i x∥ ei ∥x ∥
where he supremum is ranging on he elemens o Eiei C, he norm in he numeraor is he one o E Eiei Ei and he norm in he denominaor is he one o Eiei . Proo. — T i GL Ei q . L K C W Ei C
w C W ei
∥∥W ∥∥ Ei
f i
L x xw W . 1) I w ϕ W : f ϕ x x ϕx x ϕx x W W ∥ϕx ∥W ∥ϕx ∥W ∥ϕ ∥ EW ϕ x x EW W
f ( f ) . 2) S w , . F w R r rw r w ( w) x R x r x rw
W R w w x R W w . F t W w ∑ tR xR t R w w
L x xw W x xw Sw x x x w . W , t W w , f ∑ ∑ f t t R x R x x xw R w w Sw
MÉMOIRES DE LA SMF 152
4. LOWER BOUND OF HE HEIGH ON HE UOIEN
: f t EW w W
∑
R w w
t R w ∥t ∥W w
b e i
F i N i i b i E ≃ Eei bi ei , : bi
ei
i
ei ,
i
bi e i
143
□
,
Eibi E Ei bi Ei bi Eibi E Ei bi Ei bi ei
F i N oK E Ei ai bi ′ Fi E Ei ai bi
ei
oK bi e i
i i
bi e i
E Ei ai Eibi Fi′ Ei
S T N Gq oK F F′ D,
F′
F′
FN′ .
D
N ⊗ i
bi e i
Ei
E F′ D Ei . P , Gq P F P F′ D
T, P ( 5), P K P F′ D S. I x F oK K P, F F (C 1.6) P 4.9 : ∑ ∥д x ∥ Fv h M P д SCv ∥x ∥ Fv v V ∑
K
v VK
д SCv
∑ bi ∥д x∥ F′ Dv ℓei ∥x∥ F Dv i e i
S S D, P F′ D P F′
Sq. M,
OF′ ≃ OF′ D f D
f P F′ S oK .
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144
L Y′ q P F S ′ P F′ Y′ q . D h M′ q Y′ ( S OF′ ). A F F, ∑
v VK
д SCv
∑ ∥д x∥ F′ Dv h M′ ′ ◦ P bi Ei ∥x∥ F Dv i N
, ,
N ∑ bi ∑ bi Ei h M P h M′ ′ ◦ P ℓei i e i i
T T 4.3 h M′ , T 4.7 . 4.3.3. Te general case. — S S s P F OD
P. O 4.2.4 — , D P F P FD . T . T T 4.7, T 4.2. □
MÉMOIRES DE LA SMF 152
BIBLIOGRAPHY
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This text consists o two parts. In the frst one we present a proo o ThueSiegel-Roth’s Theorem (and its more recent variants, such as those o Lang or number felds and that “with moving targets” o Vojta) as an application o Geometric Invariant Theory (GIT). Roth’s Theorem is deduced rom a general ormula comparing the height o a semi-stable point and the height o its projection on the GIT quotient. In this setting, the role o the zero estimates appearing in the classical proo is played by the geometric semi-stability o the point to which we apply the ormula. In the second part we study heights on GIT quotients. We generalise Burnol’s construction o the height and refne diverse lower bounds o the height o semi-stable points established to Bost, Zhang, Gasbarri and Chen. The proo o Burnol’s ormula is based on a non-archimedean version o KempNess theory (in the ramework o Berkovich analytic spaces) which completes the ormer work o Burnol. Ce texte est constitué de deux parties. Dans la première nous présentons une preuve du théorème de Thue-Siegel-Roth (et des variantes plus récentes, comme celle de Lang pour le corps de nombres et celle with moving targets de Vojta) basée sur la théorie géométrique des invariants (GIT). Le théorème de Roth est déduit d’une ormule reliant la hauteur d’un point semi-stable et la hauteur de sa projection dans le quotient GIT. Dans ce cadre, le rôle du « lemme des zéros » présent dans la preuve classique est joué par la semistabilité géométrique du point auquel on applique la ormule. Dans la deuxième partie nous étudions la hauteur sur les quotients GIT. Nous généralisons la construction de Burnol de cette hauteur et nous améliorons plusieurs minorations de la hauteur de point semi-stables précédemment établies par Bost, Zhang Gasbarri et Chen. La preuve de la ormule de Burnol porte sur une version non-archimédienne de la théorie de Kemp-Ness (dans le langage de la géométrie analytique de Berkovich), qui complète le travail antérieur de Burnol.