Moments, Monodromy, and Perversity. (AM-159): A Diophantine Perspective. (AM-159) 9781400826957

Table of contents :
Contents
Introduction
Chapter 1: Basic results on perversity and higher moments
(1.1) The notion of a d-separating space of functions
(1.2) Review of semiperversity and perversity
(1.3) A twisting construction: the object Twist(L,K,F,h)
(1.4) The basic theorem and its consequences
(1.5) Review of weights
(1.6) Remarks on the various notions of mixedness
(1.7) The Orthogonality Theorem
(1.8) First Applications of the Orthogonality Theorem
(1.9) Questions of autoduality: the Frobenius-Schur indicator theorem
(1.10) Dividing out the "constant part" of an ɩ-pure perverse sheaf
(1.11) The subsheaf Nncst0 in the mixed case
(1.12) Interlude: abstract trace functions; approximate trace functions
(1.13) Two uniqueness theorems
(1.14) The central normalization F0 of a trace function F
(1.15) First applications to the objects Twist(L,K,F,h): The notion of standard input
(1.16) Review of higher moments
(1.17) Higher moments for geometrically irreducible lisse sheaves
(1.18) Higher moments for geometrically irreducible perverse sheaves
(1.19) A fundamental inequality
(1.20) Higher moment estimates for Twist(L,K,F,h)
(1.21) Proof of the Higher Moment Theorem 1.20.2: combinatorial preliminaries
(1.22) Variations on the Higher Moment Theorem
(1.23) Counterexamples
Chapter 2: How to apply the results of Chapter 1
(2.1) How to apply the Higher Moment Theorem
(2.2) Larsen's Alternative
(2.3) Larsen's Eighth Moment Conjecture
(2.4) Remarks on Larsen's Eighth Moment Conjecture
(2.5) How to apply Larsen's Eighth Moment Conjecture; its current status
(2.6) Other tools to rule out finiteness of Ggeom
(2.7) Some conjectures on drops
(2.8) More tools to rule out finiteness of Ggeom: sheaves of perverse origin and their monodromy
Chapter 3: Additive character sums on An
(3.1) The theorem
(3.2) Proof of the LΨ Theorem 3.1.2
(3.3) Interlude: the homothety contraction method
(3.4) Return to the proof of the LΨ theorem
(3.5) Monodromy of exponential sums of Deligne type on An
(3.6) Interlude: an exponential sum calculation
(3.7) Interlude: separation of variables
(3.8) Return to the monodromy of exponential sums of Deligne type on A^n
(3.9) Application to Deligne polynomials
(3.10) Self dual families of Deligne polynomials
(3.11) Proofs of the theorems on self dual families
(3.12) Proof of Theorem 3.10.7
(3.13) Proof of Theorem 3.10.9
Chapter 4: Additive character sums on more general X
(4.1) The general setting
(4.2) The perverse sheaf M(X, r, Zi's, ei‘s, Ψ) on P(e1, ..., er)
(4.3) Interlude An exponential sum identity
(4.4) Return to the proof of Theorem 4.2.12
(4.5) The subcases n = 1 and n = 2
Chapter 5: Multiplicative character sums on A^n
(5.1) The general setting
(5.2) First main theorem: the case when χ^e is nontrivial
(5.3) Continuation of the proof of Theorem 5.2.2 for n = 1
(5.4) Continuation of the proof of Theorem 5.2.2 for general n
(5.5) Analysis of Gr^0(m(n, e, χ)), for e prime to p but χ^e = 1
(5.6) Proof of Theorem 5.5.2 in the case n ≥ 2
Chapter 6: Middle additive convolution
(6 .1 ) Middle convolution and its effect on local monodromy
(6.2) Interlude: some galois theory in one variable
(6.3) Proof of Theorem 6.2.11
(6.4) Interpretation in terms of Swan conductors
(6.5) Middle convolution and purity
(6.6 ) Application to the monodromy of multiplicative character sums in several variables
(6.7) Proof of Theorem 6.6.5, and applications
(6.8) Application to the monodromy of additive character sums in several variables
Appendix A6: Swan-minimal poles
(A6.1) Swan conductors of direct images
(A6.2) An application to Swan conductors of pullbacks
(A6.3) Interpretation in terms of canonical extensions
(A6.4) Belyi polynomials, non-canonical extensions, and hypergeometric sheaves
Chapter 7: Pullbacks to curves from A^1
(7.1) The general pullback setting
(7.2) General results on Ggeom for pullbacks
(7.3) Application to pullback families of elliptic curves and of their symmetric powers
(7.4) Cautionary examples
(7.5) Appendix: Degeneration of Leray spectral sequences
Chapter 8: One variable twists on curves
(8.1) Twist sheaves in the sense of [Ka-TLFM]
(8.2) Monodromy of twist sheaves in the sense of [Ka-TLFM]
Chapter 9: Weierstrass sheaves as inputs
(9.1) Weierstrass sheaves
(9.2) The situation when 2 is invertible
(9.3) Theorems of geometric irreducibility in odd characteristic
(9.4) Geometric Irreducibility in even characteristic
Chapter 10: Weierstrass families
(10.1) Universal Weierstrass families in arbitrary characteristic
(10.2) Usual Weierstrass families in characteristic p ≥ 5
Chapter 11: FJTwist families and variants
(11.1) (FJ, twist) families in characteristic p ≥ 5
(11.2) (j^-1, twist) families in characteristic 3
(11.3) (j^-1, twist) families in characteristic 2
(11.4) End of the proof of 11.3.25: Proof that Ggeom contains a reflection
Chapter 12: Uniformity results
(12.1) Fibrewise perversity: basic properties
(12.2) Uniformity results for monodromy; the basic setting
(12.3) The Uniformity Theorem
(12.4) Applications of the Uniformity Theorem to twist sheaves
(12.5) Applications to multiplicative character sums
(12.6) Non-application (sic!) to additive character sums
(12.7) Application to generalized Weierstrass families of elliptic curves
(12.8) Application to usual Weierstrass families of elliptic curves
(12.9) Application to FJTwist families of elliptic curves
(12.10) Applications to pullback families of elliptic curves
(12.11) Application to quadratic twist families of elliptic curves
Chapter 13: Average analytic rank and large N limits
(13.1) The basic setting
(13.2) Passage to the large N limit: general results
(13.3) Application to generalized Weierstrass families of elliptic curves
(13.4) Application to usual Weierbtrass families of elliptic curves
(13.5) Applications to FJTwist families of elliptic curves
(13.6) Applications to pullback families of elliptic curves
(13.7) Applications to quadratic twist families of elliptic curves
References
Notation Index
Subject Index

Citation preview

Annals of Mathematics Studies

Number 159

Moments, Monodromy, and Perversity A Diophantine Perspective

N icholas M. K atz

PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2005

Copyright © 2005 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom'- Princeton University Press, 3 Market Place, Woodstock, Oxfordshire 0X20 1SY All Rights Reserved

The Annals of Mathematics Studies are edited by Phillip A. Griffiths, John N. Mather, and Elias Stein

Library of Congress Cataloging-in-Publication Data Katz, Nicholas M., 1943 Moments, monodromy, and perversity : a diophantine perspective / by Nicholas M. Katz, p. cm. — (Annals of mathematics studies v. 159) Includes bibliographic references and index. ISBN-13: 978-0-691-12329-5 (acid-free paper)— ISBN-13: 978-0-691-12330-1 (pbk. : acid-free paper) ISBN-10: 0-691-12329-2 (acid-free paper)— ISBN-10: 0-691-12330-6 (pbk. : acid-free paper) 1. Geometric group theory. 2. Diophantine analysis. 1. Title. II. Annals of mathematics studies no. 159. QA183.K38 2005 512’.2— dc22 2004062828 British Library Cataloging-in-Publication Data is available

The publishers would like to acknowledge the author of this volume for providing the camera-ready copy from which this book was printed.

pup.princeton.edu

1 3 5 7 9

10 8 6 4 2

Contents

Introduction

C h a p t e r 1: Basic resul ts on p e r v e r s i t y and higher m o m e n ts (1.1) The notion of a d-separating space of functions (1.2) Review of semiperversity and perversity (1.3) A twisting construction: the object Twistd^K,*?,h) (1.4) The basic theorem and its consequences (1.5) Review of weights (1.6) Remarks on the various notions of mixedness (1.7) The Orthogonality Theorem (1.8) First Applications of the Orthogonality Theorem (1.9) Questions of autoduality: the Frobenius-Schur indicator theorem (1.10) Dividing out the "constant part" of an i-pure perverse sheaf (1.11) The subsheaf NncstQ in the mixed case (1.12) Interlude: abstract trace functions; approximate trace functions (1.13) Two uniqueness theorems (1.14) The central norm alization Fq of a trace function F (1.15) First applications to the objects Twist(L, K, 7, h): The notion of standard input (1.16) Review of higher m om e nts (1.17) Higher m om e nts for geometrically irreducible lisse sheaves (1.18) Higher m om e nts for geometrically irreducible perverse sheaves (1.19) A fu n d a m e n ta l inequality (1.20) Higher m o m e n t estimates forTwist(L,K,7,h) (1.21) Proof of the Higher Mom ent Theorem 1.20.2: combinatorial preliminaries (1.22) Variations on the Higher Mom ent Theorem (1.23) Counterexamples

C h a p t e r 2: How to a p p l y t he results of C h a p t e r 1 (2.1) How to apply the Higher M oment Theorem (2.2) Larsen's Alternative (2.3) Larsen's Eighth Mom ent Conjecture (2.4) Remarks on Larsen's Eighth M oment Conjecture

1

9 9 12 13 13 21 24 25 31 36 42 44 45 47 50 52 60 61 62 62 64 67 76 87

93 93 94 96 96

VI

Contents

(2.5) How to apply Larsen's Eighth M oment Conjecture; its current status (2.6) Other tools to rule out finiteness of GgGom (2.7) Some conjectures on drops (2.8) More tools to rule out finiteness of Ggeom : sheaves of perverse origin and their m o n o drom y

C h a p t e r 3: A d d i t i v e c h a r a c t e r s u m s on (3.1) The theorem

97 93

102 104

An

111 111

(3.2) Proof of the J!,^ Theorem 3.1.2

112

(3.3) Interlude: the hom othety contraction method (3.4) R eturn to the proof of the theorem

113 122

(3.5) (3.6) (3.7) (3.8)

123 129 136

Monodromy of exponential sums of Deligne type on A n Interlude: an exponential sum calculation Interlude: separation of variables R eturn to the m o n o drom y of exponential sums of

Deligne type on A n (3.9) Application to Deligne polynomials (3.10) Self dual families of Delignepolynomials (3.11) Proofs of the theorems onself dual families (3.12) Proof of Theorem 3.10.7 (3.13) Proof of Theorem 3.10.9

138 144 146 149 156 158

C h a p t e r 4: A dd i t i v e c h a r a c t e r s u m s on m o r e g ener al X 161 (4.1) The general setting 161 (4.2) The perverse sheaf M(X, r, Zj's, ej‘s, i|j ) on e ) 166 (4.3) Interlude An exponential sum identity (4.4) R eturn to the proof of Theorem 4.2.12 (4.5) The subcases n = l and n = 2

C h a p t e r 5: M u l t i p l i c a t i v e c h a r a c t e r s u m s on A n (5.1) The general setting

174 178 179

185 185

(5.2) First m a in theorem: the case when %e is nontrivial (5.3) Continuation of the proof of Theorem 5.2.2 for n = l (5.4) Continuation of the proof of Theorem 5.2.2 for general n

188 191 200

(5.5) Analysis of Gr^(JR(n, e, %)), for e prime to p but X e = 1 (5.6) Proof of Theorem 5.5.2 in the case n > 2

207 210

C h a p t e r 6: Mi ddle a d d i t i v e c o n v o l u t i o n ( 6 .1 ) Middle convolution and its effect on local m on o d ro m y (6.2) Interlude: some galois theory in one variable (6.3) Proof of Theorem 6.2.11

221 221 233 240

Contents

V ll

(6.4) Interpretation in terms of Swan conductors (6.5) Middle convolution and p u rity ( 6 .6 ) Application to the m o n o drom y of m ultiplicative character sums in several variables (6.7) Proof of Theorem 6.6.5, and applications ( 6 .8 ) Application to the m o n o drom y of additive character sums in several variables

245 248

A p p e n d i x A6: S w a n - m i n i m a l poles (A6.1) Swan conductors of direct images (A6.2) An application to Swan conductors of pullbacks (A6.3) Interpretation in terms of canonical extensions (A6.4) Belyi polynomials, non-canonical extensions, and hypergeometric sheaves

281 281 285 287

C h a p t e r 7: P u ll b ac k s to c u r v es f r o m (7.1) The general pullback setting (7.2) General results on GgGom for pullbacks

295 295 303

(7.3) Application to pullback families of elliptic curves and of their sy m m e tric powers (7.4) Cautionary examples (7.5) Appendix: Degeneration of Leray spectral sequences

308 312 317

C h a p t e r 8: One v a r i a b l e t wi sts on c u r v e s (8.1) Twist sheaves in the sense of [Ka-TLFM] (8.2) M onodromy of twist sheaves in the sense of [Ka-TLFM]

321 321 324

253 255 270

291

C h a p t e r 9: Wei er s tr as s sheaves as i n p u t s 327 (9.1) Weierstrass sheaves 327 (9.2) The situation when 2 is invertible 330 (9.3) Theorems of geometric irreducibility in odd characteristic 331 (9.4) Geometric Irreducibility in even characteristic 343

C h a p t e r 10: W e i e r s t r a s s f a m i l i e s (10.1) Universal Weierstrass families in arb itra ry characteristic (10.2) Usual Weierstrass families in characteristic p > 5

349 349 356

C h a p t e r 11: FJTwi st f a mi li e s a n d v a r i a n t s ( 1 1 .1 ) (FJ, twist) families in characteristic p > 5

371 371

(11.2) (j~^, twist) families in characteristic 3

380

(11.3) ( j ” ^ f twist) families in characteristic 2

390

Contents

V 111

(11.4) End of the proof of 11.3.25: Proof that Ggeom contains a reflection

401

C h a p t e r 12: U n i f o r m i t y res ul ts (12.1) Fibrewise perversity: basic properties (12.2) Uniform ity results for m onodrom y; the basic setting (12.3) The Uniform ity Theorem (12.4) Applications of the U niform ity Theorem to twist sheaves (12.5) Applications to m ultiplicative character sums (12.6) Non-application (sic!) to'additive character sums (12.7) Application to generalized Weierstrass families of elliptic curves (12.8) Application to usual Weierstrass families of elliptic curves (12.9) Application to FJTwist families of elliptic curves (12.10) Applications to pullback families of elliptic curves (12.11) Application to quadratic twist families of elliptic curves

407 407 409 411

C h a p t e r 13: A ve r ag e a n a l y t i c r a n k a n d large N l i m i t s (13.1) The basic setting (13.2) Passage to the large N limit: general results (13.3) Application to generalized Weierstrass families of elliptic curves (13.4) Application to usual Weierbtrass families of elliptic curves (13.5) Applications to FJTwist families of elliptic curves (13.6) Applications to pullback families of elliptic curves (13.7) Applications to quadratic twist families of elliptic curves

416 421 427 428 430 433 435 439

443 443 448 449 450 451 452 453

References

455

Notation

461

Subje ct

I nd ex I nd ex

467

Introduction

It is now some th ir ty years since Deligne first proved his general equidistribution theorem [De-Weil II, Ka-GKM, Ka-SarRMFEM], thus establishing the fu n d a m e n ta l result governing the statistical properties of suitably "pure" algebro-geometric families of character sums over finite fields (and of their associated Lfunctions). Roughly speaking, Deligne showed that any such fam ily obeys a "generalized Sato-Tate law", and that figuring out which generalized Sato-Tate law applies to a given fam ily am ounts essentially to com puting a certain complex semisimple (not necessarily connected) algebraic group, the "geometric m on o drom y group" attached to tha t family. In our earlier books [Ka-GKM], [Ka-ESDE], and [Ka-TLFM], computations of geometric m on o drom y groups were carried out either directly on an open curve as p aram eter space, or by restriction to a well-chosen open curve in the param e ter space. Once on an open curve, our m a in tool was to compute, when possible, the local m on o drom y at each of the missing points. This local m o n o drom y inform ation told us that our sought-after semisimple group contained specific sorts of elements, or specific sorts of subgroups. We typically also had a m od ic um of global inform ation, e.g., we m ight have known that the sought-after group was an irreducible subgroup of GL(N), or of the orthogonal group 0(N), or of the symplectic group Sp(N). It was often then possible either to decide either exactly which group we had, or to show that our group was on a very short list of possibilities, and then to distinguish among those possibilities by some ad hoc argum ent. In this book, we introduce new techniques, which are resolutely global in nature. They are sufficiently powerful tha t we can sometimes prove th at a geometric m on o drom y group is, say, the symplectic group Sp(N), w ith o ut knowing the value of N; cf. Theorem 3.1.2 for an instance of this. The price we pay is that these new techniques apply only to families which depend on very m a n y parameters, and thus our work here is nearly disjoint from our earlier "local m onodrom y" methods of analyzing one-parameter families. However, it is not entirely disjoint, because the new techniques will often leave us knowing, say, that our group is either SO(N) or 0(N), but not knowing which. In such cases, we sometimes prove that the group is in fact 0(N) by restricting to a suitable curve in the param e ter space and then proving tha t the local m o n o drom y at a particular missing point of this curve is a reflection: since SO(N) contains no reflections, we m ust have 0(N). Our work is based on two vital ingredients, neither of which yet existed at the tim e of Deligne's original work on equidistribution. The first of these ingredients is the theory of perverse sheaves,

2

Introduction

pioneered by Goresky and MacPherson in the topological setting, and then brilliantly transposed to algebraic geometry by Beilinson, Bernstein, Deligne, and Gabber. The second is Larsen's Alternative, discovered by Larsen ten odd years ago, which very nearly characterizes classical groups by their fourth m om ents. This book has two goals, one "applied" and one "theoretical". The applied goal is to calculate the geometric m o n o d ro m y groups attached to some quite specific universal families of (L-functions attached to) character sums over finite fields. The theoretical goal is to develop general techniques, based on combining a diophantine analysis of perverse sheaves and their higher m o m e n ts w ith Larsen's Alternative and other group-theoretic results, w hich can be used to achieve the applied goal, and which are of interest in their own right. Let us begin by describing some of the universal families we have in m ind. Grosso modo, they are of three sorts: families of additive character sums, families of m ultiplicative character sums, and Weierstrass (and other) families of L-functions of elliptic curves over function fields in one variable. In the additive character case, we fix a finite field k and a nontrivial C-valued additive character ijj of k. For any finite extension E/k, we denote by ijjjr the additive character of E defined by iJjjr(x) := ijj(Trace£/k(x)). Fix a pair of integers n > 1 and e > 3. We denote by P(n,e)(E) the space of polynomials over E in n variables of degree < e. We are concerned w ith the families of sums, parameterized by f in P(n,e)(E), given by Sum(E, f, +) := I Xi.... ^ in E xn )). It turns out that these sums are, up to sign, the local traces of a perverse sheaf, say M(n, e, ijj), on P (n , e)/k. On some dense open set, say U(n, e, i|j) of P (n , e)/k, this perverse sheaf is a [shift and a Tate twist of a] single lisse sheaf, say 7Tl(n, e, \ \ >), w hich is pure of weight zero. [When the degree e is prim e to char(k), we can take U(n, e, i}j) to be the open set D(n, e) consisting of "Deligne polynomials" of degree e in n variables, those whose leading forms define smooth, degree e hypersurfaces in P n ~^.] It is the geometric m o n o d ro m y of this lisse sheaf TTL(n, e, i}j) on U(n, e, which we wish to calculate. In the m ultiplicative character case, we fix a finite field k and a nontrivial (C-valued m ultiplicative character X of k x, extended to all of k by %(0) := 0. For a ny finite extension E/k, we denote by %E the m ultiplicative character of E x defined by Xjr(x) := XdMormjr/jJx)),

again extended to all of E by X e ^ := 0* ^ lx a pair of integers n >1 and e > 3. We are concerned w ith the families of sums,

Introduction parameterized by f in P(n,e)(E), given by Sum(E, f, X) := Z Xl....Xn in E X E(f(> 5. We denote by X 2

quadratic character of k*. Fix a pair of integers

62 z 3 and d 3 > 3. For each finite extension E/k, we have the product space ( P ( l , d 2 ) xP ( l , d 3 ))(E) of pairs (g2 (t), g j ( t )) of onevariable polynomials over E of degrees at most 62 and d 3 respectively. We are concerned w ith the families of sums, parameterized by (g2 , 8 3 ) in (P C ljd g )x P ( l , d 3 ))(E), Sum(E, g2 ,

83)

:= Z x, t in k X2,E ( 4 x 3 ' 82(t)x “ 83(t))-

It t u r n s out t h a t t h e s e s u m s a r e , up to sign, t h e local t r a c e s of a p e r v e r s e s h e a f , s a y W(d2» d 3), on P ( l , d 2 ) x P ( l , d 3 ) / k . On t h e d e n s e open set LKdg, d3), of P ( 1 ,d2) x P

(1

that (g2 )^ “ 27(g3)2 has M ax( 3 d 2 ,

, d3 ) /k , def ined b y t h e con di ti on 2d3 )

distinct zeroes in k, this

perverse sheaf is a [shift and a Tate twist of a] single lisse sheaf, say °UT(d 2 > ^ 3 ), which is mixed of weight < 0. The weight zero quotient Gr 0 W (d 2 , d 3) of *UT(d2 , d 3) is related to L-functions of elliptic curves over function fields as follows. For (g2 » g 3 ) in U(d 2 » d 3 )(E), the Weierstrass equation y2 _

4x3

_ g2 (t)x - g3 (t)

defines an elliptic curve over the rational function field E(t), and its (unitarized) L-function is the local L-function of Gr® 9. At the very end of this chapter, we give some results on degeneration of Leray spectral sequences, which are certainly well known to the experts, but for which we know of no convenient reference. In chapter 8 , we indicate how the general theory of Twist(L,K,< 3r,h) developed here allows us to recover some of the results of [Ka-TLFM]. Chapters 9, 10, and 11 are devoted to a detailed study of families of L-functions of elliptic curves over function fields in one variable over finite constant fields. Chapter 9 is devoted to explaining how various classical families of elliptic curves provide appropriate input, na m e ly a suitable perverse sheaf K on an A m , to the general theory. Chapter 10 works out w h at the general theory gives for various sorts of Weierstrass families, and Chapter 11 works it out for other, more neglected, universal families, which we call FJTwist families. In chapter 12, we retu rn to theoretical questions, developing some general if ad hoc methods which allow us to work "over Z" instead of "just" over a finite field. These methods apply nicely to the case of m ultiplicative character sums, and to the various Weierstrass and FJTwist families. W h at they m ake possible is equidistribution statements where we are allowed to work over bigger and bigger finite fields, whose characteristics are allowed to vary, e.g., bigger and bigger prime fields, rath er tha n the more restrictive setting of bigger and bigger finite fields of a fixed characteristic. Unfortunately, the methods do not apply at all to additive character sums. Nonetheless, we believe th a t the corresponding equidistribution statements, about additive character sums over bigger and bigger finite fields whose characteristics are allowed to vary, are in fact true statements. It is just th at we are presently incapable of proving them. In the final chapter 13, we make explicit the application of our results to the arithm e tic of elliptic curves over function fields. We first give results on average analytic rank in our families. We then pass to the large-N limit, e.g., by taking Weierstrass families of type (d 2 > d j) as described earlier, and letting M ax( 3 d 2 , 2 d 3 ) tend to infinity, and give results concerning low-lying zeroes as incarnated in the eigenvalue location measures of [Ka-Sar, RMFEM]. It is a pleasure to acknowledge the overw helm ing influence on this book of the ideas and work of Deligne, Gabber, and Larsen. In the course of working on the book, I visited the Institute for Advanced Study, the University of Tokyo, the University of Minnesota, the University of Paris at Orsay, I.H.E.S., and the University of Paris VI. I than k all these institutions for their hospitality and support.

Chapter 1: Basic results on p e r v e r s i t y and higher m o m e n t s

(1.1) The n ot io n of a d - s ep a ra ti n g space of f u n c t i o n s (1.1.1) Throughout this section, we work over a field k. For each integer m > 1, we denote by A m ^, or just A m if no confusion is likely, the m-dimensional affine space over k. (1.1.2) Let V be a separated k-scheme of finite type. The set H°m k - s c h e m e s ^ ’ of k-morphisms from V to A m is na tu ra lly a k-vector space (addition and scalar m ultiplication on the target). Concretely, it is the k-vector space of m-tuples of regular functions on V. If V is n onem pty, Homj

7

Pr 2 i the upper square of which is Cartesian. Recall that M := R p r 2 iN, for N the object on V x f given by N := p r 1 * L ® h aff*K[dim< 3r - m]. By definition of Fourier Transform, we have FTqJ(M) = R p r 2 j((pr^xM )®i!.ljJ(|:^f))[dim7]. By proper base change, we obtain FT^(M) = R p r 2 iR p r 2 )3 |((prl j 2 ^N) D ! ( ( p r 1 |2 * N ) ® p r 2j3* ( £ +(f v f)))[dim7] = R (p r3: V x f x f "

-> f v ),P,

for P the object on V x f x J v given by P := (prj_j2 ^ N ) ® p r 2 >3 ^ ( £ l[J(f^f))[dim (v, f a v °evalQ(v), a v ), we find th at Rpr^_

3 4 j(Rpr^ 2 4 5 1(cr^P))

is supported in Z, where it

is given by L ( v ) ® £ ^ (_a - h (v ))® F T ^(K )(a ^)(- d im 5 0)The composite Z £ V x ^ q " »Am v i Pr 2,3

f 0v x is precisely the m orphism p. Thus we find that FT^(M) = RpjS, for S the object on V x A m v given by S := L ( v ) ® £ ^ (_a - h (v ))® F T ^(K )(a^)(-dim T 0), as asserted. If the m orph ism p were quasifinite, then we would be done: Rpi preserves semiperversity if p is quasifinite. Although p is not quasifinite, we claim that p is in fact quasifinite over the open set T 0 " x A m ^ - (0 ,

0 ).

To see this, we argue as follows. Suppose that over some extension field L/k, we h a v e a point (a ^ ° e v a l Q ( v ) , a v ) 5* (0, 0) in the image

p(V (L) x A m v (L)). Then a ^ m ust itself be nonzero. We m ust show that there are at most finitely m a n y points w in V(L) such tha t (a v °eval()( v), a v ) = (a v °evalQ( w), a v ), i.e., that there are at most finitely m a n y points w in V(L) such that a v °(evalQ(v) - evalQ(w)) = 0 on 7 q®L. This last condition is equivalent to the condition a v o(evdl(v) - eval(w)) =

0

on 5 0 L ,

simply because 7 = 7 q © A m (k), and eval(v) - eval(w) tautologically kills constants. By the hypothesis that ( 7 , t ) is quasifinitely difference-separating, we know that except for finitely

Chapter 1

18

m a n y w in V(L), the m a p eval(v) - eval(w) : T ® L

A m (L)

is surjective. W hen this m a p is surjective, a v °(eval(v) - eval(w)) is nonzero, because a v is nonzero. So for given L/k and given v in V(L), there are only finitely m a n y w in V(L) for which (a v °eval()(v), a v ) = (a v °evalQ(w), a ^). Thus p is quasifinite over the open set

x

A ms/ - (0, 0}.

Therefore the restriction to T q v * A m ^ - {0, 0} of FT^(M) is indeed semiperverse. Once we know this, then FT^(M) is semiperverse on all of 7 q v

x

A m v if and only if its restriction to

the single missing point (0 , 0 ) is semiperverse on th a t point, i.e., is concentrated in degree < 0. But this restriction is just FV

M )(0,0) = R r c(A m ® k, K[m])® R r c(V ® k, L)(-dimT0).

Thus M is indeed semiperverse on 7= f q x A m . For a ny M on 7, the value at the origin of FT^(M) on 7 ^ is ^ * ( 7 ® ^ , M ldim ?]), so we find Hc* ( 7 ® k k, M[dimT]) = H ^c(A m ®k, K[m])®H*c(V® k, L)(-dimT0)> as asserted. QED (1.4.3) R e m a r k on h y p ot he si s 4) in Theo re m 1.4.2 For a n y K on A m , R r c(Am ®k, K[m]) is the stalk at the origin of FT^(K). As FT^ preserves semiperversity, we see tha t if K is semiperverse on A m , then R r c(Am ®k, K[m]) is concentrated in degree < 0. So hypothesis 4) of the theorem holds if either R r c(V

Hom ^_sc|^em e(V, A m ),

on V which is quasifinitely difference-separating and contains the constants, and a k-morphism h : V -> A m 1) Suppose K is perverse on A m , and suppose Hck A m ®k, K) = 0 for i > m - dim(V). Then for any perverse sheaf L on V, the object Twist(L,K,< 3r,h) on 7 is perverse, and the functor L Twist(L,K,< 3r,h) from the category of perverse sheaves on V to the category of perverse sheaves on 7 is exact. 2) Suppose L is perverse on V, and suppose H ^ V ^ k , L) = 0 for i > 0. Then for any perverse sheaf K on A m , the object Twist(L,K,< T,h) on 7 is perverse, and the functor K »-> Twist(L,K,T,h) from the category of perverse sheaves on A m to the category of perverse sheaves on 7 is exact.

Chapter 1

20

3) Suppose we are given a short exact sequence of perverse sheaves on A m , 0

—» K

» K2 —* K3 —>

0,

and suppose that each Kj satisfies HcKAm ®k, Kj) = 0 for i > m - dim(V). Then for any perverse L on V, the objects Twist(L,Kj,< J ,h ) on 7 are perverse, and we have a short exact sequence of perverse sheaves on 7, 0 —> Twist(L,Kj_,T,h) —> Twist(L,K2 )5 ,h) —> Twist(L,K 3 ,7 ,h) —> 0. 4) Suppose we are given a short exact sequence of perverse sheaves on V, 0 — ■ >—> L 2 ~* L 3 —» 0 , and suppose that each Lj satisfies H ^ V ^ k , Lj) =

0

for i > 0 .

Then for any perverse K on A m , the objects Twist(Lj,K,T,h) on 7 are perverse, and we have a short exact sequence of perverse sheaves on 7 , 0 -> Twist(L 1 ,K,< 5r,h) -> Twist(L 2 ,K,< F,h) -> Twist(L 3 ,K,T,h) -> 0. proof There are two key points. The first is th at the form ation of Twist(L,K,7,h) is a functor Db c (V, Q) x Db c ( A m , Q^) -» Db c ( 7 , (Q^),

(L, K) >-> TwistCL.K.f.h), which is triangulated (i.e., carries distinguished triangles to distinguished triangles) in each variable separately. This is clear from the fact that R (p r2)j is triangulated, and the description in the previous result of p r ^ * L ® h a ff*K[dim< 9ro] on V x f as the pullback by an auto m o rphism of the external tensor product of L on V, (Q^dimTo] on T q , and K on A m , the formation of which is visibly bi-triangulated. The second key point is tha t on any separated scheme X/k, a short exact sequence of perverse sheaves on X, 0 —> N — * N 2 —4 N 3 0, is precisely a distinguished triangle in D^C(X, (Q^) whose terms happen to be perverse, cf. [BBD 1.2.3 and 1.3.6]. W ith these points in m ind, the assertions 1 )through 4) are obvious. For instance, to prove 1 ), we note that for any perverse L on V, indeed for any semiperverse L on V, we have HcKV®k, L) = 0 for i > dim(V). This vanishing follows from the dimension inequalities dim Supp K !(L) < -i for every integer i defining semiperversity, and the spectral sequence

Basic results on perversity and higher m om en ts

21

E2a,b = Hca (V Hca+b(V k, L), cf. the proof of 1.10.5. So under the hypotheses of 1), we get the vanishing H1C((V x A m ) m +1. So for any perverse L on V, Twistd^K,?,h) is perverse, by the Perversity Corollary 1.4.4. A short exact sequence of perverse sheaves on V, 0 —* L'l L 2 —> L 3 > 0 , gives a distinguished triangle in D^C(V, (Q^), so we get a distinguished triangle on 7 , -> T w i s t t L ^ K ^ h ) -> Twist(L2 ,K, Twist(L 3 ,K,< 3r,h) ->, which, having perverse terms, is a short exact sequence of perverse sheaves on 7. The proofs of 2), 3), and 4) are similar, and left to the reader. QED

(1.5) R ev i ew of we ig h ts (1.5.1) In this section, we work over a finite field k, on a separated scheme X /k of finite type. We fix a prime n u m b e r i * char(k), and a field embedding 1 : (Q^ C C. We denote by IzI the complex absolute value of a complex n u m b e r z. For a in (Q^, we will write the complex absolute value of i(oc) as |a|L := |i(oc)|, or simply as loci := |ot|L w hen

(1.5.2)

no

confusion

is l i k e l y .

For an object N in D^C(X, (Q^), its trace function is the

Q^-valued function on pairs (a finite extension E/k, a point x in X(E)) defined by (E, x)

h*

N(E, x) := Ej (-l)iTrace(FrobE)X I K j(N)).

[Here and throughout, Frob^ x is the geometric Frobenius attached to the E-valued point x in X(E).] We view N(E, x), via the fixed 1 , as a (C-valued function, and denote by (E, x) N(E, x) the complex conjugate C-valued function. (1.5.3) Recall [De-Weil II, 1.2.2] that for w a real n u m b e r, a constructible (Q^-sheaf % is said to be pu nctu ally i-pure of weight w if, for each finite extension E/k, and for each point x in X(E), all the eigenvalues a of Frobg x I 9 have |oc| = (#E)W//2. A constructible (Q sheaf 9 is said to be i-mixed of weight < w if it is a successive extension of finite m a n y constructible (Q^-sheaves, each of which is pu nc tu a lly i-pure of some weight < w.

22 (1.5.4)

Chapter 1 Let us say th at a real n u m b e r w "occurs via i" if there

exists an £-adic unit a in Q^ w ith IocI = (char(k))w/"2, or equivalently (take powers or roots, Q^ being algebraically closed), if for some finite extension E/k, there exists an £-ad\c unit p in w ith |p| = (^ E )W//2_ Thus any rational n u m b e r w occurs via i . And if a nonzero constructible Q^-sheaf % is p u nc tu a lly i-pure of weight w, then w "occurs in i". L e m m a 1.5.5 Let w be a real n u m b e r, and Q- a nonzero constructible Q^-sheaf on X /k which is i-mixed of weight < w. The set of real num bers wq < w such that

9

is i-mixed of weight < wq

has a least element, and that element occurs via i. proof Write § as a successive extension of finitely m a n y nonzero constructible (Q^-sheaves 9 i> w ith Qj punctually i- pure of some weight Wj < w. As remarked above, each Wj "occurs in i". The largest of the Wj is the least (1.5.6)

wq.

QED

Recall [De-Weil II, 6.2.2] that an object N in D^C(X, Q^) is

said to be i-mixed of weight < w if, for each integer i, the cohomology sheaf H ](N) is i-mixed of weight < w+i. The object N is said to be i-pure of weight w if N is i mixed of weight < w, and if its Verdier dual D)(/^N is i-mixed of weight < -w. L e m m a 1.5.7 Let w be a real n u m b e r, and N a nonzero object in Dbc(X, Q*) which is i-mixed of weight < w. The set of real num be rs wq < w such that N is i-mixed of weight < wq has a least element, and tha t element occurs via i. proof For each i such tha t K J(N) is nonzero, apply the previous le m m a to HHN), which is mixed of weight < w + i. Denote by

wq

j

the least real n u m b e r such that HHN) is i-mixed of weight - w 0 ,i + l-Then each wq j occurs via I, and the largest of the wq j is the least wq. QED L e m m a 1.5.8 Let w be a real n u m b e r, N a nonzero object in Dkc(X, (Qp) which is i-pure of weight w. Then w occurs via i. proof W ith no loss of generality, we m a y assume that the support of N is X (because for a closed im mersion i : Z —> X, we have **°^Z/k = ^X/k°**)- Then there exists a dense open set U of X which is smooth over k, w ith N|U nonzero, i- pure of weight w, and w ith lisse cohomology sheaves. Then for any i w ith HHNlU) nonzero, HKNlU) is pu nctu ally i- pure of weight w+i, and hence w occurs via i. QED

Basic results on perversity and higher m o m e n t s

23

(1.5.9) The m a in theorem of Deligne’s Weil II is th at for f: X -» Y a k-morphism between separated schemes of finite type, for any N in Dbc(X, Q^) which is i-mixed of weight < w, the object RfiN on Y is i-mixed of weight < w. (1.5.10) Here is a simple application, to the object M constructed in the Semiperversity Theorem. W e i g h t C or oll ar y 1.5.11 ( c o m p a r e [Ka-ACT, 1.6]) Hypotheses and notations as in the Semiperversity Theorem 1.4.2 (and in its proof, for the notion of an embedding i of into C, and real num bers a and b. Suppose in addition that 1 ) k is a finite field, 2) L is i-mixed of weight < a, 3) K is i-mixed of weight < b. Then we have the following results. 1) The object A := pr ^ * L ® h a ff "KtdimT'o] on V x ? is i-mixed of weight < a + b + d im ( < ?Q). 2) The object M := R p r 2 i(A) on aF is i-mixed of weight < a + b + d im ( ‘3rq ). proof 1) By means of the a u to m o rph ism a of V x f = V x ^

qx

Am

given by a(v, f q , a) := (v, f q , h(v) + f()(v )

+

A is the cr*-pullback of the external tensor product of L on V, C^[dimTQ] on 7 q , and K on A m . The object (Q^dim^Q] on 7 q is ipure of weight d im ^ Q , and weights add for external tensor products. 2) This is a special case of Deligne's m a in theorem [De-Weil II, 3.3.1] in Weil II. QED (1.5.12) We now resume our review of weights. A perverse sheaf N on X is called i-mixed (resp. i-pure) if it i-mixed (resp. i-pure) as an object of D^C(X, (Q^). One knows [BBD, 5.3.1 and 5.3.4] that if a perverse sheaf N is i-mixed, then every simple constituent of N as perverse sheaf is i-pure of some weight. More precisely, if a perverse sheaf N is i-mixed, then for any finite set of real num be rs W1 x )l2 = 1 + 0 ((* E )_1 /2 ). 2) If M and N are not geometrically isomorphic, then for variable finite extensions E of k, we have •s x in X(E) M(E, x)N(E, x)| = 0 ( ( * E r 1 /2). 3) If M and N are geometrically isomorphic, then for variable finite extensions E/k, we have 'Z x in X(E) MCE> x ) N(E> x)| = 1 + 0 ((*E)~1 /2 ). 4) If M is isomorphic to D ^ /^ M , then its trace function takes, via i , real values, and for variable finite extensions E/k, we have E x in X(E) M (E ’ x )2 = 1 + 0 ((*E)-1 /2 ). 5) Suppose that there exists real £ with 1 > £ > 0 such that for variable finite extensions E/k we have 2 x in X(E) M (E’ x )2 = 1 + 0 ((*E)-£/2). Then M is isomorphic to D^/^M6)

If M is geometrically isomorphic to D ^ /^ M , then for variable

finite extensions E/k, we have I I X in X(E) M (E ’ x)2 > = 1 + 0 ((^ E )_1 /2 ). 7) If M is not geometrically isomorphic to D ^ /^ M , then for variable finite extensions E/k, we have IZx in X(E) M x)2 =

we

+ 0 ( ( * E r 1 /2 ).

By hypothesis, we have in X(E) M(E’ x)2 = 1 + 0((#E)-£ /2 ). So we find p2deg(E/k) = 1 + 0 ( (* E ) " e/2). Consider the complex power series in one variable T defined by Z n >0 P2nTn = 1/(1 - P 2T). It satisfies 1/(1 - p^T) -1/(1 - T) = a series convergent in |T| 0, we have short exact sequences of perverse sheaves 0 -» M< _w -» M -» Gr°(M) -> 0, 0 -» N< _w -* N -* Gr°(N) -> 0. Here Gr(-)(M) and Gr^(N) are both i- pure of weight 0, and M
w > 0. Write the pullbacks Gr^(M)SGom and Gr^(N)SGom of Gr^(M) and

Gr^(N) to X ® k k as sums of perverse irreducibles w ith multiplicities, say Gr°(M)Seom = Z, mjVj, Gr°(N)geom = Zj njV,, with (Vj)j a finite set of pairwise non-isomorphic perverse irreducibles on X ® k k, and w ith non-negative integers m j and n y [This is possible by [BBD, 5.3.8].] Then we have the following results. 1 ) For any integer n > 1, denoting by kn /k the extension field of degree n, we have Zj m jnj = l i m s u p ^ k

l^x in X(E)

x)N(E, x)|,

Basic results on perversity and higher m om e nts

35

Sj ( m j ) 2 = lim s u p E/k^ z x in X(E) |M(E, x)|2, the limsup taken over all finite extensions E/kn . 2) If Zj (mj)^ = 0, i.e., if Gr^(M) = 0, then there exists real £ > 0 such that for variable finite extensions E/k, we have Z x in X(E) lM(E,

x )|^

= 0 ((^ E )“ 8//^).

3) The following conditions a) and b) are equivalent. 3a) There exists real £ > 0 such that for variable finite extensions E/k, we have Z x in X(E) lM(E, x)l2 = 1 + 0((#E)” £/2). 3b) Gr^(M) is geometrically irreducible, proof From the identities Gr°(M)(E, x) = MCE, x) - M< _W(E, x), Gr°(N)(E, x) = N(E, x) - N< _W(E, x), we get Gr°(M)(E, x)Gr°m)(E, x) - M(E, x)N(E, x) = -M< _W(E, x)N(E, x) - M(E, x)N< _W(E, x) + M< _W(E, x)N< _W(E, x). By L e m m a 1.5.13, we have in X(E) lM < -w(E. x^ < _W(E, x)| = 0 ( ( * E r w ), Z x in X(E) lM < -w^E. X^ ( E , x)| = 0 ((« E )_ w / 2 ), S x in X(E) iM(E’ x)^< -w^E' x^l = 0 ( ( * E r w / 2). So we find Z x jn X(E) M(E,

x )N(E, x )

= Z x in X(E) Gr°(M)(E, x)Gr°(N)(E, x) + 0 ((^E)~w / 2 ). So the corollary is simply the Orthogonality Theorem 1.7.2, applied to Gr^(M) and to Gr^(N). QED Third Corollary 1.8.4 Hypotheses and notations as in the Second Corollary 1.8.3, suppose in addition that Gr^(M) is geometrically irreducible. Then we have the following results. 1) If Gr^(M) is isomorphic to D x / kGr^(M), there exists real £ > 0 such that for variable finite extensions E/k, we have z x in X (E) M(E, x)2 = 1 + 0((*E )-£ / 2 ). 2) Suppose there exists real £ > 0 such that for variable finite extensions E/k, we have Z x in X(E) M ^E’ x ) 2 = 1 + 0 ( ( * E r e/2). Then Gr^(M) is isomorphic to D x / kGr®(M). 3) Suppose tha t M has, via i , a real-valued trace function. Then Gr°(M) is isomorphic to D)(/kGr^(M). 4) If Gr^(M) is geometrically isomorphic to D)(/kGr^(M), then there

Chapter 1

36 exists real

8

> 0 such that for variable finite extensions

E/k, we

have

I^X in X(E) M 2I = 1 + 0((*E )-£ /2 ). 5) If Gr^(M) is not geometrically isomorphic to D x / kGr®(M), then there exists real £ > 0 such th at for variable finite extensions E/k, we have |Xx in X(E)

= 0((-E)-£ / 2 ).

proof Just as in the proof of the Second Corollary 1.8.3, we have Z x in X(E) M(E, x )2 = E x in X (E) Gr 0 (M)(E, x ) 2 +

0

( ( * E ) - ^ / 2 ).

So parts 1), 2), 4), and 5) result from the First Corollary 1.8.2, applied to Gr^(M). For part 3), we notice that if M has a real valued trace function, we have Ex in X(E) M(E-

x)2

=

Ex in

X(E) lM(E> x)l2-

The right hand side is, for some real e > 0, 1 + 0((# E)“£//2,thanks to part 3) of the Second Corollary 1.8.3. Now apply part 2). QED (1.9) Questions of a u t o d u a l i t y : t h e F r o b e n i u s - S c h u r indicator theorem (1.9.1) Let K be an algebraically closed field of characteristic zero. W hen a group G operates irreducibly on a finite-dimensional Kvector space V, we have the following trichotomy: either the representation V of G is not self dual, or it is orthogonally self dual, or it is symplectically self dual. The Frobenius-Schur indicator of the G-representation V, denoted FSI(G, V), is defined as FSKG, V) := 0, if V is not self dual, = 1, if V is orthogonally self dual, = -1, if V is symplectically self dual. [When K is (D and G is compact, Frobenius and Schur discovered in 1906 their integral form ula for the Frobenius-Schur indicator: FSKG, V) = J gTrace(g 2 I V)dg, for dg the total mass one Haar measure on G.] (1.9.2) Now let k be a field, i a prime n u m b e r invertible in k, and U/k a separated k-scheme of finite type, which is smooth and connected, of dimension d = dimU > 0. A lisse, irreducible (Q^-sheaf 9 on U "is" an irreducible Q^-representation of tu^(U), and so we m a y speak of its Frobenius-Schur indicator FSI(tt^(U), %). We will sometimes write FSKU, 9) := FSK tt 1( U ) 1 9).

If U/k is geometrically connected, and if 9 ls geometrically irreducible, i.e., irreducible as a representation of Ti^(U®^k), we m a y also speak of its Frobenius-Schur indicator as a representation of ttj_(U}k k)-

representation, and vanishes otherwise. S im ila rly ,

the

cohom ology

o f tt^ ( U ® k k ) - e q u i v a r i a n t values as

in

C^(d).

It

group

alternating

is o n e - d i m e n s i o n a l

TT^(U®j Q^(d). This pairing, viewed as an element of Hc 2 ckuk k, 7T103TI), is a basis, fixed by Frobk , of whichever of the one-dimensional spaces Hc 2 d (U®| 0 such that for variable finite extensions E/k, we have FS(X, M, E) = c + 0 ( ( * E r e/2). Then there exists an isomorphism Gr^(M) s D x / k (Gr^(M)) on X, and

Chapter 1

42 C is FSlSeo m (X, Gr°(M)). proof We have

FS(X, M, E) = FS(X, Gr°(M), E) + FS(X, M< _w , E). By Le m m a 1.9.5, we have FS(X, M < _w , E) = 0((*E)~W). Now apply the previous theorem 1.9.6 to Gr^(M). QED (1.10) Di vi di ng out t he " c o n s t a n t p a r t " of a n i - p u r e p e r ve r se sheaf (1.10.1) In this section, we work on an X/k which is smooth and geometrically connected, of some dimension d > 0. The object Q^[d](d/ 2 ) is geometrically irreducible and i- pure of weight zero on X/k. We will refer to it as the constant perverse sheaf on X/k. We will refer to its pullback to X ® k k as the constant perverse sheaf on X(g)k k (1.10.2)

A perverse sheaf N on X /k is called geometrically constant

if N§eom is isomorphic to the direct sum of finitely m a n y copies of the constant perverse sheaf on Xk k. Equivalently, N is geometrically constant if and only if it is of the form 9 td] f°r 9 a lisse Q^-sheaf on X which is geometrically constant. A perverse sheaf N on X /k is called geometrically totally nonconstant if N&GOm is semisimple and if none of its simple constitutents is isomorphic to the constant perverse sheaf on Xk k. L e m m a 1.10.3 Let N be perverse and i- pure of weight zero on X/k. Then N has a unique direct sum decomposition ^ = ^cst ® ^ncst w ith Ncsj- geometrically constant (i.e., of the form Q[d] for Q- a lisse (Q^-sheaf on X which is i- pure of weight -d and geometrically constant) and w ith Nncs1; geometrically totally nonconstant. proof Because N is i-pure, N§GOrn is semisimple [BBD 5.3.8]. In its isotypical decomposition, separate out the isotypical component (Ngeo m )cs^ of the constant perverse sheaf. We get Ngeom = (Ngeom)cst ® (Ngeom)ncst) w ith (NSeo m )ncst a sum of nonconstant irreducibles. Each s u m m a n d is stable by Frobenius pullback, so the projections of NSGOm onto the two factors are, by [BBD 5.1.2], endomorphisms of N which are a pair of orthogonal idempotents of N. This gives the existence. Uniqueness is clear, again by [BBD 5.1.2], since the pullback to X(8 >k k of any such decomposition m ust be the decomposition of NSeom we started with. QED L e m m a 1.10.4 Let N be perverse and i- pure of weight zero on X/k.

Basic results on perversity and higher m om e nt s

43

Then Hcd (X 1, and let M have dimension of support d(M) > 0. Look at the spectral sequence E2P,q = HcP(Xk k, K°I(M)) => HcP+(3(X(g)k k, M). The only possibly nonvanishing H ^ M ) have 0 > q > -d(M), and we have d i m S u p p K ‘ d(M)(M) = d(M), dimSuppH~*(M) < i, for 0 < i < d(M) - 1. So we see that Hc]( X ® k k, M) vanishes for i outside the closed interval [-d(M), d(M)]. This proves the first assertion. Suppose now that M is perverse irreducible and nonconstant. If d(M) < d, we are done. If d(M) = d, the spectral sequence shows that Hcd ( X ® k k, M) =

E ^ 26’

~d

is a quotient of E22d> _d = Hc2d(Xk k, H " d (M)). This last group is a birational inv arian t. But on some dense open set U of X ® k k, M|U is 9[d] for a lisse sheaf 9 on U which is irreducible and nonconstant, and so has Hc 2 d (U, 9) = 0- By the birational invariance, Hc 2 d( X ® k k, H~^(M)) = 0. QED L e m m a 1.10.6 Let N be perverse and i- pure of weight zero on X/k. Then Hcd(Xk k, N) is i-pure of weight d. View Hc^ ( X ® k k, N)(d) as a geometrically constant lisse sheaf Q on X which is i-pure of weight -d. Then Ncs^, the constant part of N, is given by Ncst = Q[d]. proof Write N as Ncst © Nncst. By the previous le m m a, Hcd (Xk k, Nncst) = 0. Now write Ncst = K[d] for some lisse, geometrically constant sheaf K on X which is i-pure of weight -d. Then Hcd (Xk k, N) = Hcd ( X ® k k, Ncst ® Nncst) = Hcd(X x)G(E, x). (1.12.7) Now suppose we are given a perverse sheaf N on X/k, which is i-mixed of weight < 0 , and an abstract trace function N'. We say th at N' is an a p p r o x i m a t e t r ac e f u n c t i o n for N if there exists a real n u m b e r e > 0 such th at for variable finite extensions E/k, and for variable points x in X(E), we have N(E, x) - N'(E, x) = 0 ( ( * E ) " e/2 " d / 2 ). L e m m a 1.12.8 Suppose N and M are perverse on X/k, both i-m ixed of weight < 0. Suppose N' and M' are approximate trace functions for N and M respectively. Then there exists e > 0 such th a t for variable finite extensions E/k, we have 1) Z E N'M' - EE NM = 0((*E )-£ /2 ), 2) Z E N'M’ - Z E NM = 0 ((* E )_e/2), 3) FS(X, N', E) - FS(X, N, E) = 0 ( ( * E ) ' £). proof Take a stratification {Za } of X by connected smooth locally closed subschemes to which both N and M are adapted. On a strat Za of dimension d a < d = dim(X), * Z a (E) = 0 ((* E )d a), and we have the estimates N(E, x in Za (E)) = 0 ( ( * E ) " d a / 2 ), M(E, x in Za (E)) = 0 ((* E )“d a / 2 ), simply because N and M are semiperverse, and i-mixed of weight < 0. So we have Zjr iNI = 0 (Z a Z x jn Za (E)

^oc^2)

= 0 (S a (^ E )d a / 2 ) = 0 ((^ E )d / 2 ).

Basic results on perversity and higher m om e n t s

47

Expanding out the sum we are to estimate, we get Z E N'M' - Z E NM = Z E (N' - N)M + E e N(M' - M) + S e (N‘ - N)(M' - M) »

0

((* E)- e / 2 ' d / 2 )XE |M| + 0 ((* E)~e / 2 " d / 2 )ZE INI + 0 ((*E)-£ ’ d )*X(E)

= 0((*E )-£ /2 ). This proves 1). The proof of 2) is entirely similar. For 3), we use the tautological estimate N'(E2,

x)

- N(E2,

x)

= 0 ( ( * E 2r e/2-d/2) = 0((#E)~e_d),

and the trivial bound *X(E) = 0 ((* E )d ). QED (1.13) Two u n iq u e n e s s

th e o r e m s

Theorem 1.13.1 Let X /k be a separated scheme of finite type, of dimension d > 0, F an abstract trace function on X/k, and M and N two perverse sheaves on X. Suppose that both M and N are i-mixed of weight < 0, and that F is an approximate trace function for both M and N. Then Gr°(M) and Gr°(N) are geometrically isomorphic, i.e., we have Gr°(M) s Gr°(N) as perverse sheaves on X ® k k. proof Since F is an approxim ate trace function for both M and for N, we have, thanks to the previous le m m a , the estimates Z E |F|2 - Z E INI2 = 0(( * E)~e/2), XE |F|2 - Z E IM |2 = 0 ((# E)_e/2), S E IF!2 - E E NM = 0 ((* E )_e/2), for some e > 0. Both Gr^(M) and Gr®(N) are geometrically semisimple (because they are i-pure of weight 0). Write their pullbacks Gr^(M)SGom and Gr^(N)Seom to Xk k as sums of perverse irreducibles w ith multiplicities, say Gr°(M)Seom = Ej mjVj, Gr°(N)geom = X; njV;, w ith {V j}j a finite set of pairwise non-isomorphic perverse irreducibles on X ® k k, and w ith non-negative integers m j and nj. Then by Second Corollary 1.8.3, we have limsupjr Ejr |N|2 = Ej (nj)2, limsupjr E F |M|2 = Ej ( m j ) 2 , lim sup ^ |Eg NMl = Ej n j m j . In view of the above estimates, these three limsup's are all equal to limsupjr Eg IFI2. So we find Ej ( n j ) 2 = Ej ( m j ) 2 = Ej n j m j .

Therefore we get

Chapter 1

48 Zj (nj - m j ) 2 =

0,

so nj = m j for each i, as required. QED Here is an a rith m e tic sharpening of this uniqueness result. Theorem 1.13.2 Let X /k be a separated scheme of finite type, of dimension d > 0, F an abstract trace function on X/k, and M and N two perverse sheaves on X. Suppose that both M and N are i-m ixed of weight < 0 , that both are semisimple objects in the category of perverse sheaves on X, and that F is an approxim ate trace function for both M and N. Then Gr^(M) = Gr^(N) as perverse sheaves on X. proof Let (Za ) be a smooth stratification of X to w hich N © M is adapted. Thus each Za is a smooth and connected k-scheme, of dimension denoted d a . Any direct factor of N© M , in particular any simple constituent of NffiM, is also adapted to this stratification. Given a simple constituent W of N© M , there is a unique strat Za , ex = a(W ), which contains the generic point of its support. As N © M is i-mixed of weight < 0, any simple constituent W is i-pure, of some weight w(W) < 0. As we have seen in the proof of the Orthogonality Theorem 1.7.2, for a = oc(W), we have W|Za = cWa [da ], for cWa a semisimple, lisse, Q^-sheaf on Za which is i-pure of weight - w(W) - d a ; moreover, W is the middle extension from Za of W|Za . For

p ^ a(W ), W|Z^ has a trace function which satisfies (WlZpHE, x) = 0 ((*E )(_dcx " w (W)- l ) / 2 )

Let us denote by N(oc) the direct factor of N consisting of the sum (with m ultiplicity) of those of its simple constituents whose supports have generic point in Za . Let us denote by N(cx, 0) the direct factor of N(oc) which is the sum (with m ultiplicity) of those of its simple constituents which are i-pure of weight 0. In other words, N(oc,

0)

is just Gr^(N(oc)), viewed as a direct factor

of

N(oc), or

equivalently, N(oc, 0) is, in this notation, (Gr^(N))(a). Let us denote by N(a, < 0 ) the direct factor of N(oc) which is the sum (with m ultiplicity) of those of its simple constituents which are i-pure of weight < 0 . So we have direct sum decompositions N =

N(a), Gr°(N) = © a N(a, 0), N(oc) = N(oc, 0) © N(a, < 0).

We have N(oc)|Za = Tlcx[doc], TLa lisse on Za , i-mixed of weight < -da ,

For p

N(a,

0)|Za = ^ a>o[da ], 5T.a j o = Gr"d °c(Tlcx),

N(a,

< 0)|Za = Tla


0 we will have I e|K0I2 =1 + 0((*E )-£ /2 ). From the definition of M, M := Rpr2i(pi'i*Lha ff*K[dim‘3r - m]), the Lefschetz trace formula, and proper base change, we have M(E, f) = ((-l)d i m 7 0)Sv in V (E)

v)K(E> h(v) + fCv))-

56

Chapter 1

Key L e m m a 1.15.7 The abstract trace function Mq is given in terms of the trace function of L and the abstract trace function Kq by the formula

M0(E, f) = ((-l)dim,r0)Xv in V(E) L(E’

h(v) +

proof of Key L e m m a 1.15.7 By definition, we have M 0(E, f) = M(E, f) - (1 /* T (E ))Z EM. Our first task is to compute Z EM. We have Z EM = Z f in 7(E)((-l)d i m ? 0)Zv in V (E) L(E, v)K(E, h(v) + f(v)) = ((-l)d i m 7 0)Zv m V (E) L(E, v )Z f in ? ( e )K(E, h(v) + f(v)). Because (7, t ) is 2-separating, it is 1-separating, and so for fixed v in V(E), the E-linear m ap eval(v) : 7(E) -* A m (E), f ^ f(v), is surjective. So the inner sum is given by Z f in T(E)K(E- h(v) + f(v)) = *Ker(eval(v))Za jn ^ r n ( E) K(E, h(v) + a) = * Ker(eval(v))Za jn A m (E) ^(E,

= ( * 7 ( E ) ) ( ( * E r m )ZEK. So we find Z EM = ((-l)d i m 7 0)(ZEL )(* f(E ))((* E )- m )ZEK. Thus we have

(1/*T(E))ZEM = ((-l)di m T 0 )(ZEL ) ( ( * E r m )ZEK. Therefore we find M 0(E, f) = M(E, f) - ((-l)d i m '3r0)(i:EL)((;«E)-m )ZEK = ((-1 )d 'm ^ 0 ) E v jn v(E) E(E, v )KCE, h(v) + f(v)) - ((-l)d i m 7 0)Zv In V (E) L(E, v)((# E)~m )ZEK = ((-1 )d ’m '3r0)Zv in V (E) L(E. v)K0(E, h(v) + f(v)). QED We now calculate in closed form the sum Z E |Mq |2. L e m m a 1.15.8 We have the identity

ZE |M0I2 = (ZE|L|2 ) x( ( E)d‘m ^r0) xZ£ |K0I2 • proof We have |M0(E, f)l2 = (2V in v(E) E(E,

v )Kq (E,

h(v) + f(v)))

*(2 W in V(E) E(E, w )K q CE, h(w) + f(w))). S u m m in g over f in 7(E), we get

ZE lMol2 - ^ v ,w in V(E)

v)L(E, w)

x^f in 7(E)^0^E’

+ f(v))KQ(E, h(w) + f(w)).

Basic results on perversity and higher m om ents

57

We now break up the sum according to w hether v = w, or not. We get X E IM 0I2 =

in V(E) L ^E' v )L(E,

v

)

x£ f jn gr(E)Ko(E, h (v) + f(v))Ko(E, h (v) + f(v)) + Z v * w in V(E) L(E- v)L(E, w) x Zf jn f (E)K o (E' h (v) + f(v))Ko(E, h(w ) + f(w)). We will show th a t the first sum gives the m a in term , and the second sum vanishes. Consider the first sum. Because (7, t ) is 2-separating, it is certainly 1-separating, so for fixed v in V(E), the E-linear m a p eval(v) : 7(E) —> A m (E), f •-> f (v ), is surjective. So the innermost sum in the first sum simplifies to £f in 7(E)^0^E> + f(v))Ko(E, h(v) + f(v)) = * (Ker(eval(v)))Za jn ^ m ( £ ) Kq(E, h(v) + a)KQ(E, h(v) + a). By

an additive change

of

variable a

•->

a

+h(v),

w e r e w r i t e t h i s as

= * ( K e r ( e v a l ( v ) ) ) Z a j n ^\m( E ) K q (E, a ) K o ( E , a) = ^ ( K e r ( e v a l ( v ) ) ) Z E |KQ|2

= ( ( * E)d im 7 - m ) x 2 e |k 0|2 = ((* E )d im 7 0 )* X E|K0l2 . So the first te rm is the product (Z E|L|2 )x ( ( * E ) d im ^ 0 ) x Z E|K0l2 . We will now show that the second sum vanishes. Because (7, t ) is 2-separating, for a given pair v ^ w of distinct points in V(E), the E-linear m a p : 7(E) -» A m (E)* A m (E), f (f(v), f(w)), is surjective. So the innermost sum in the second sum simplifies to Z f in gr(E)K0(E, h(v) + f(v))K0(E, h(w) + f(w)) (eval(v), eval(w ))

= * (Ker((eval(v), eval(w)))) x^a,b in A m (E) Kq (E>

+ a)K()(E, h(w) + b)

= *(Ker((eval(v)# e v a K w ) ) ) ) ! ^ in A m (E) Kq (E> aJKgtE, b) = # (Ker((eval(v), eval(w))))|ZE KqI2 = 0, cf. 1.14.2.

QED for L e m m a 1.15.8

From the above L e m m a 1.15.8, we get ((* E)- d im 7 0)ZE |M0I2 = (E e |L|2 )(Z e |K0I2 ) = (1 + 0 ((^ E )'e/2))(l + 0 ((* E )- £/2)) = 1 + 0 ((^ E )'£/2), which proves part 2) of the Standard Input Theorem. To prove parts 3) and 4), we calculate the Frobenius-Schur

Chapter 1

58 sums for MQ(dim7Q/2).

L e m m a 1.15.9 We have the identity F S ( 7 , M 0 ( d i m T 0 /2), E) = ( ( - l ) d i m T 0 ) x F S( V , L, E ) x F S ( A m , K 0 , E).

proof By definition, we have FS(T, M 0 ( d i m T 0 /2), E) : = ( ( * E r d i m T 0) x F S( T , M 0> E) = ( ( * E ) " d l m i r 0 ) x £ f in h ( v ) + f(v))

(E2 ) L(E2, v )Z f in 7(E) K0 (E2 , h ( v ) + f(v)).

We break the sum into two sums, according to w hether v in V(E2) lies in V(E), or not. We will show that the first sum , over v in V(E), is the m a in term , and the second sum vanishes. The first sum is ( ( - * E ) “ d i m ^ 0 ) x E v in V (E) L^E2> v ^ f in 7 ( E ) K 0^E2 ’

Because (7,

t

+ f^v ^-

) is 1-separating, the inner sum is simply

in 7 (E ) K0 (E2> h ( v ) + f(v)) = * ( K e r ( e v a l ( v ) ) ) x E a m A m (E) ^ 0 ^ E2 ’

+

= ( ( * E ) d i m 7 0 ) x Z a m A m ( E) K 0 (E2 , a) = ( ( * E ) d i m i r 0 ) x F S ( A m , K 0 , E).

So the first sum is the product ( (( -# E )-d i m 7 0 ) x E v in V (E)

l (£

2-

x ( ( * E ) d i m 7 0 ) x F S ( A m , K 0 , E) = ( ( - l ) d i m ‘3r0) x FS( V, L, E ) x F S ( A m , K 0 , E).

It rem ains to show th at the second sum vanishes. For v in V(E2), denote by v its image under the nontrivial au to m o rp h ism of Eg/E. Thus we are s u m m in g over points v in V(E2) w ith v ^ v. So the second sum is = ((_ * E ) _ d ’m ^r0 ) E v ^ y in

v

(E 2 ) L( e 2> v ^ f in F(E) Ko ( E2-

+ f(v »-

We will show tha t already its innermost sum vanishes. For this, we need the following S u b le m m a 1.15.10 For a point v ^ v in V(E2), the E-linear evaluation m a p eval(v), viewed as a m a p from 7(E) to A m (E2), f f(v), is surjective. proof of S u b le m m a 1.15.10 Because (7, t ) is 2-separating, and v ^ v, the m a p (eval(v), eval(v)) : 7 (E 2) cp

A m (E2)x A m (E2), (cp(v), cp(v))

is surjective. Take a point of the form (a, a) in A m (E2) xA m (E2). Then there exists cp in 7 (E 2) such that cp( v) = a, cp( v) = a.

Basic results on perversity and higher m om e nts Pick y in Eg w ith y *

0 and y ^ y. Then {1,

we can write cp in 7 (Eg) uniquely as f + yg,

59

y) isan E-basis of Eg, and w ith f, g in 7(E). Thus

we get the identities f(v) + yg(v) = a, f(v) + yg(v) = a.

Because f, g both lie in 7(E), they are Gal(Eg/E)-equivariant maps from V(Eg) to A m (Eg). So "conjugating" the second identity, we get f(v) + yg(v) = a.

_

Subtracting from the first, we find (y -y)g(v) = f(v) = a. QED for S u b le m m a 1.15.10.

0, so g(v) = 0, and

Thanks to S u b le m m a 1.15.10, we can evaluate the innermost sum Z f in T(E) K0 (E2< h(v) + f(v)) = (* Ker(evaKv): 7(E) -» A m (E2)))xZa in A m (E2) K0(E2, h(v) + a) = ((-E)dimT - 2 m ) x I a m

K0(E2, a)

= ( ( * E) d i m 7 0 ' m )*XE2 Kq = 0, cf. 1.14.2. We now m ake use of L e m m a 1.15.9, according to which we have the identity FS(7, M 0( d im T 0/2), E) = ((-1 )dim E). Suppose first th at at least one of Gr^(K) or Gr®(L) is not geometrically self dual. Applying part 1) of Corollary 1.9.7 to K and L shows that for some real e > 0, we have FS(V, L, E)xFS(Am , K, E) = 0 ((* E )_e/2). Then applying Le m m a 1.12.8, part 3), to K and K q , we get FS(V, L, E)xFS(Am , K0, E) = 0((*E)~£ / 2 ). Therefore we have FS(V, M 0(dim'3r0/2), E) = 0 ((^ E )_e/2). Again by L e m m a 1.12.8, pa rt 3), now applied to M ( d i m 7 Q / 2 ) ncsto a nd to M o ( d i m 7 Q / 2 ) , this n ow gives

FS(V, M ( d im ‘3r0/2)n cst0’ E> = 0((*E)-£ / 2 ). Applying part 1) of Corollary 1.9.7 to M (d im 7 Q /2 )ncs(;o shows now that G r ° (M (d im 7 o /2 ))ncst is not geometrically self dual. If both of Gr^(K) and Gr^(L) are geometrically self dual, then replacing each of K and L by a suitable u n ita r y cxde& twist of itself, we reduce to the case when both Gr^(K) and Gr^(L) are self dual.

Chapter 1

60

Then part 2) of Corollary 1.9.7, applied to both K and L, together with L e m m a 1.12.8, part 3), show th a t for some real e > 0 we have FS(T, M 0( d im < ? q /2), E) = (((- l)d i m 7 0)xFSIgeo m (V > Gr°(L))xFSlSeo m (Am , Gr°(K)) + 0 ((* E )-£//2). Applying L e m m a 1.12.8, part 3) and part 3) of Corollary 1.9.7 to M ( d im (V v )®k; M a,b(G> V) := d i m K (V®a ® ( V v )®b )G. We call M a ^(G, V) the (a, b)'th m o m e n t of (G, V). For each even integer 2n > 2, we denote by M 2 n (G, V) the 2n'th absolute m o m e n t, defined by M 2 n (G, V) := M n>n(G, V). (1.16.2) The terminology "moments" comes about as follows. W hen K is (C and G is compact, there are integral formulas for M a ^(G, V) and for M 2 n (G, V). Denote by X : G —» (C %(g) := Trace(glV), the character of the representation. Then we have M a,b(G>

= J G X (g )a X (g )bdg,

M 2n(G, V) = J Q lx(g)|2 n dg, for dg the total mass one Haar measure on G. [Thus the terminology "moments" and "absolute moments".] (1.16.3) There is one elem entary inequality we will need later. L e m m a 1.16.4 Let K be an algebraically closed field of characteristic zero. Suppose a group G operates completely reducibly on a finite-dimensional K-vector space V. If V * 0, then M 2 n (G, V) > 1 for all n > 1. proof M 2 n (G, V) is the dimension of the G-invariants in V ® n ®(V v )®n = End(V®n ), i.e., M 2 n (G, V) is the dimension of EndQ(V®n ), which always contains the scalars. QED

Basic results on perversity and higher m om e nts

61

(1.17) Higher m o m e n t s for g e o m e tr ic a lly irr e d u c ib le lisse sheaves (1.17.1) We continue to work over a finite field k. As earlier, we fix a prime n u m b e r I * char(k), a field embedding i : C C, and a square root of char(k) in Q^. (1.17.2) Let U/k be a separated k-scheme of finite type, which is smooth and geometrically connected, of dimension d = dimU > 0. Suppose 9 on U is a lisse, geometrically irreducible (Q^-sheaf on U. Then 9 "is" an irreducible representation of TT^(U®k k), and we m a y speak of its higher m o m e nts M a ^ ( ^ ( U ® k k), %) and ^ 2 n ^ TTl ^ ® k ^ ^

ca^ these the geometric higher m om e nts of

9 on U, and write M a,bg0° m (U> 9) := M a , b ^ l ( U® k ^ > 9), M 2nge° m ( U > 9) := M 2 n(Tr1(U ® k k), 9). These m om e nts are birational invariants, in the sense that for any dense open set Uj_ C U, we have Ma/

o m (U, 9) = M a>bSeom (U1, 9IU±),

M 2ngeom(U, 9) = M 2 n *eorn(\J1, 9IU!). These equalities hold simply because Tr^(U-j_®k k) maps onto Tc1(U ® k k). (1.17.3) When 9 *s ^ -pure of some weight w, there is a diophantine analogue of the classical integral formulas for higher m om ents. Theorem 1.17.4 Let U/k be a separated k-scheme of finite type, which is smooth and geometrically connected, of dimension d = dimU > 0. Suppose 9 on U is a lisse, geometrically irreducible (Q^-sheaf on U, which is i-pure of some weight w. Fix a pair (a, b) of non-negative integers. For each finite extension E/k, consider the sum Zjr 9 a 9 b Then we have the following results. 1) We have the estimate |IE 9 a 9 b|= 0 ((* E )d,mU + (a+b)w/2) 2) We have the limit form ula M a,bge0m(U’

= limsupE |IE 9 a 9 b|/(*E)dimU + (a+b)w/2

3) We have the limit form ula dimU + (a+b)w/2 = lim sup^ logdZjr 9 a 9k|)/log(# E). proof 1) Each s u m m a n d is trivially bounded in absolute value by (r a n k 9 ) a+k( # E)^a+b^W//2, and there are *U(E) = 0((#E)d^m ^ ) terms. 2) and 3) By an ocde§ twist, we reduce im m ediately to the case when w = 0. In this case, the contragredient 9 V °f 9 complexconjugate trace function to that of 9- So by the Lefschetz trace

Chapter 1

62 formula, we have

£ e 9a 9b = EEi1 = 0 to 2dimU ((-l)'Trace(FrobEl )®b )). -l )‘Trace(FrobEl Hc>(Uk k, 9® a ®(9 v )®b))9 ® a (9 ® ( 9 "^ )®b is t-pure of weight zero, so by Deligne's result Now Q®a [De-Weil II, 3.3.1], we have

SE 9 a 9 b = Trace(FrobEl Hc2d( U ® k k, 9 ® a ® ( 9 v )®b )) + 0 ( ( * E ) di mU _ 1 / 2 ). Moreover, the group Hc2d(Uk k, 9 ® a ®(9~ )®b ) 1S ^--pure of weight 2dimU, and its dimension is precisely M a j:)Se o m (U, §). So if we view this

v_

as an M a jDSeorn(U, 9)~dimensional C-vector space T via la , —

i,

then the semisimplification of Frob| X. There exists a dense affine open set j: U —> Z, such that U/k is smooth and geometrically connected, of some dimension d > 0, and a lisse C^-sheaf Tl on U, which is geometrically irreducible, such that N is i*ji*Tl[dimU]. We define M a , b gG° m (X ’ N) ;= M a ,b g0° m (U > ft).

M 2 nge° m ( X > N) := M 2 n2Gom(U, Tl). This definition is independent of the auxiliary choice of smooth, geometrically connected dense open set U of Supp(M) on whi ch M is lisse, thanks to the birational invariance of the quantites M a,bgG° m (U>

and M 2n gGOm(U, Tl).

(1.19) A f u n d a m e n t a l

inequality

Theorem 1.19.1 Let X/ k be smooth and geometrically connected, of non-negative dimension, and let M on X/k be perverse, and i-mi xed of weight < 0. Write its weight filtration 0

—>

M < _£ —* M

Gr^(M) —> 0,

for some £ > 0. Let M be an abstract trace function which is an approximate trace function for M. Suppose that Gr^(M) is geometrically irreducible. Suppose that for some real e > 0, some

Basic results on perversity and higher m om ents

63

integer n > 2 and some real num bers X and Agn, we have an inequality X E IM|2n < A 2 n (*E)(1-n)x + 0((*E)(1~n )x - e / 2 ), for E/k a variable finite extension. Then we have the following results. 1) dim(Supp(Gr^(M))) > X. 2) If dim(Supp(Gr^(M))) = A, then M 2 n ^GOm(X, Gr^(M)) < Agn . proof Let us put d := dim(Supp(Gr^(M))). Take a smooth stratification {Za ) of X which is adapted to all three objects M 2) we have d > X. 2) If d = X, we have A2 n (* E)(1" n)d + 0 ((* E )(1~n)d - e/2) > S E I M|2n > S E |M|Za |2n = Z E I9l2n + 0 ( ( * E)(1 _n )d -e/2). Divide through by ( # E ) ^ _ n ^d and take the lim sup over E. QED

Corollary 1.19.2 Let X /k be smooth and geometrically connected of dimension dim X > 0. Let M on X /k be perverse, and i-mixed of weight < 0. Write its weight filtration 0

M 0. Let M be an abstract trace function which

is an

approxim ate trace function for M. Suppose that Gr^(M) is geometrically irreducible. Suppose th at for some real z > 0, some integer n > 2 and some integer A 2 n , we have an inequality Z E IM|2n < A 2n( * E ) (1- n)dimX + 0 ( ( * E ) (1- n)dimX “ e/2), for E/k a variable finite extension. Then the support of Gr®(M) is X itself, and M 2ngeo m (X, Gr°(M)) < A2 n . proof By part 1) of Theorem 1.19.1, dim (Supp(G rt-)(M))) > dim X. As X is geometrically irreducible, we have equality. Then by part 2) of Theorem 1.19.1, we have the asserted inequality M 2 n ge° m ( X ' Gr°(M)) < A2 n .

QED

(1.20) Higher m o m e n t e s tim a te s for T w ist(L ,K ,< 3r,h) (1.20.1) Recall that for an even integer 2n > 2, 2n!! is the product 2n !! := (2n-l)(2n-3)...(l) of the odd integers in the interval [0, 2n]. Higher M o m e n t Theorem 1.20.2 Suppose we are given standard input (m > 1, K, V, h, L, d > 2, (7, t )). Suppose in addition that Gr^(K) is not geometrically constant, and

Basic results on perversity and higher m o m e nts

65

that the following three additional hypotheses hold: 1) The perverse sheaf L is £[dimV] for some constructible (Q^-sheaf A m ,

such tha t the perverse sheaf K on A m is i*K[dimW] for some some constructible (Q^-sheaf DC on W. 3) We have the inequality dim V + d im W > m +1. Denote by M the perverse sheaf M := Twist(L,K,1r,h) on 7 . Denote by M q the central norm alization of its trace function. For each integer n > 1 with 2n < d, there exists a real 8 > 0 such that we have the following results. 1) If G r°(M (d im T 0/2)) nest *s no* geometrically self dual, then S E |M0(dim3r0/2)l2n = (n!)((^E)(1' n ) d i m 7 )(l + 0((*E )" s /2 )). 2) If Gr0(M(dim'3r0/2)) nest *s geometrically self dual, then S E |M0( d im T 0/2)l2n = (2n!!)((^E)(1-n )d i m T )(l + 0 ((* E )“ e/2)). Before giving the proof of the theorem, let us give its m a in consequence. Corollary 1.20.3 In the situation of Theorem 1.20.2, suppose in addition that d > 4. Then we have the following results concerning the perverse sheaf M := Twist(L,K,T,h) on 7 . 1) The support of G r ^ ( M ( d im T Q /2 ) ) ncs-t is all of 7 . 2) For any dense open set U c 7 on which M is lisse, M(dim3rQ/2)|U is of the form Tfl(dimT/2)[dimT], TTl(dimT/2)[dimT], for a lisse (Q^-sheaf (Qa-sheaf Til on U which is t-mixed of weight < 0. The nonconstant part Gr^(Tfl)ncst of the highest weight quotient Gr^(Tfl) of !JTl as lisse sheaf on U is geometrically irreducible, and Gr^(UTL)ncs-(;[dim< 3r] on U is the Tatetwisted restriction ( G r ^ ( M ( d im < 3rQ/2))ncst:|lJ)(-dim^r/2).

3) The necessary and sufficient condition (cf. Le m m a 1.15.5, part 3)) for the equality G r°(M (d im T 0/2)) = G r ° ( M ( d im 7 0/2))ncst of perverse sheaves on 7 , nam ely th at Hcm ((V xA m )®k, p r ^ * L ® p r 2*K) is i-mixed of weight < m - e, for some s > 0 is also a necessary and sufficient condition for the equality Gr°(m) = Gr°(T[l)ncst of lisse sheaves on U. 4) Fix n > 1 w ith 2n < d. We have the m o m e n t estimates

66

Chapter 1

M 2nge° m ( u - Gr°(TTl)ncst) * n !> if G r° ( m )n cst is not geometrically self dual on U, M 2nge° m (U - G r0(Tfl)nCst) * 2n!! if G r°^m )n cst is geometrically self dual on U. 5) Fix n > 1 w ith 2n < d. Suppose th a t rank(Gr^(!fIl))ncst ^ n, and th at Gr^(Tfl) is not geometrically self dual on U. Then we have M 2n georn(U> Gr°(TTL)ncst) = n!. 6) Fix n > 1 w ith 2n < d. Suppose th at rank(Gr^(?TL)ncst ) £ 2n, and that Gr^(TTl) ncst

geometrically self dual on U. Then we have M 2 n ge° m ( U ’ Gr° ( m )ncst) = 2n!!'

7) The geometric Frobenius-Schur indicator of Gr^(JTl)ncst on U is given by the product form ula FSl8eo m (U, Gr°(Tfl)ncst) = (-l)rnFSl8eom (V, Gr°(L))FSl8eom (A m , Gr°(K)). proof of C orollary 1.20.3 1) We have d > 4, so we m a y take n = 2 in Theorem 1.20.1. Now apply Corollary 1.19.3. 2) On any dense open set U C H on which any perverse sheaf N is lisse, N|U is of the form TUdimT/2)[dim< 3r] for a lisse (Q^-sheaf Tl on U. If N is i-mixed of weight < 0, then Tl is L-mixed of weight < 0, and the highest weight quotient Gr^(Tl) of Tl as lisse sheaf on U is related to the highest weight quotient Gr^(N) of N as perverse sheaf by Gr^(N)|U = (Gr^(Tl)[dimcT])(dim< 3r/2). Similarly, the nonconstant part Gr^(Tl)nCst

Gr^(Tl) as lisse sheaf on U is related to the

nonconstant part Gr^(N)nCst

Gr^(N) as perverse sheaf by

Gr0(N)n cstlU = ( G r ° m ) n cst[dim the latter

geometrically isomorphic to (a constant sheaf 9)[dim < f]. Thus on 7 k we have G r°(M (d im T 0/2)) s G r°(M (d im ‘3r0/2))ncst © 9[dimf], So on any dense open set U on which M is lisse, we have a direct sum decomposition of lisse sheaves on U ® k k

Basic results on perversity and higher m om e nts Gr°(m) = Gr°(JTl)ncst ©

9

67

.

Thus we have Gr^(!JR) = Gr®(Tfl)ncs1- if and only if 9 = 0, if and only if we have Gr°(M(dim'3ro/2)) = G r ° ( M ( d im f 0/2))n csf 4) Simply apply Corollary 1.19.3. 5) We will reverse the inequality. For any finite-dimensional representation V of any group G, and any non-negative integers a and b, we have the a priori inequality M a>b(G, V) > M ajb(GL(V), V). In characteristic zero, we have M a,a(GL(V)>V) = a! if dimV * a So if rank(Gr^(JR) ncst) ^ n, we have M 2 n geo m (u, G r°(m )ncst) > n!. 6) We reverse the inequality. For any finite-dimensional symplectic (resp. orthogonal) representation V of any group G, and any n o n ­ negative integers a and b, we have the a priori inequality M a j b (G, V) > M a>b(Sp(V), V),

(resp. M a>b(G, V) > M a>b(0(V), V)). In characteristic zero, we have M a,a(SP(V)> V) = M a,a(0(V)>V) = 2a!! lf dimV * 2aSo if rank(Gr^(TTL)ncs(;) > 2n, and Gr^(JR) is geometrically self dual (and hence either orthogonally or symplectically self dual), we have M 2 n Seom (U, G r°(m )ncst) > 2n !!. 7) This is just a rewriting of the already established (cf. Theorem 1.15.6, part 3)) m ultiplicative form ula for the Frobenius-Schur indicator, na m ely FSIgeom (T, Gr 0 ( M (d im T 0 /2))ncst) = ((-l)d im T 0)*FSIgeo m (Am , Gr°(K))xFSIgeo m (V, Gr°(L)). Since G r^(M (d im 7 Q /2))ncs1; has support all of J , and FSlSeom does not see Tate twists, we have FSIgeom ('3r, G r°(M (d im T 0 /2))ncst) ;= (_ 1 )d i m 7 FSIgeom(Ui G r°(m )ncst(dim372)) = ( . 1 )d i m 7 FSIgeom(Ui Gr ° ( m ) ncst).

QED

(1.21) Proof of the Higher M o m e n t T heorem 1.20.2: c o m b in a to r ia l p r e lim in a r ie s (1.21.1) We begin by recalling [Stan-ECI, 3.7.1] a version of the Moebius inversion formula. Suppose we are given a finite, partially ordered set P. Then there exists a unique assignment of integers |j(p, q), one for each pair (p, q) of elements of P with p > q, w ith the following property:

68

Chapter 1

for any abelian group A, and any m a p g : P —» A, if we define a m a p f :P A by the rule ftp) = 2 q

Q, if Q, is a coarsening of TP in the sense th at each set Q a in the partition Q is a union of sets P ^ in the partition P. In terms of equivalence relations, P > Q means that P-equivalence implies Q-equivalence. (1.21.3) Given a partition (i.e., an equivalence relation) P , we denote by X = X(P) the n u m b e r of subsets (i.e., the n u m b e r of equivalence classes) into which {1, 2,..., 2n} is divided by P. X = x(P) := * P We label these subsets Pj_,..., P ^ by the following convention: Pj_ is the subset containing 1, and, if X > 1, P 2 is the subset containing the least integer not in P ^ , et cetera (i.e., for 1 < i < X-l, Pj+j_ is the subset containing the least integer not in the union of those P j w ith j < i). We denote by v = 'u(P) the n u m b e r of singletons in P , i.e., v is the n u m b e r of one-element equivalence classes, or the n u m b e r of indices i for which * P j = 1. v := ^{singletons am ong the Pj}. Each subset Pj has a "type" (aj, bj) in 2 >q x 2> q , defined by aj := * ( P j f i { l , 2,..., n}), bj := * (P jfl{n + l, n+2,..., 2n}). Thus we have v := #{i such th a t aj + bj = 1}. L e m m a 1.21.4 Let P > P' be partitions of the set (1, 2,..., 2n). Put X = X(P), X' := X(P'), v = v(V), v'= ^(P'). Then we have the following inequalities. 1) 2n > 2X - v, i.e., X - n - v/2 < 0, w ith equality if and only if * Pj < 2 for all i. 2) X > 3) v > 4) X proof

X', w ith equality if and only if P = P'. v' . v/2 > X* - v72, i.e., X' -X -iv' - v)/2 < 0. 1) Since P is a partition of(1, 2,...,2n},we have

Basic results on perversity and higher m om e nts 2n =

= i to X ^ ^ i = v +

69

w ith * P j > 2 *^i-

There are \ - v indices i for which # P j > 2, so we have v + S i with * P j > 2 ^ i

* v + 2{\ - v) = 2 \ - v.

Thus 2n > 2X - v, w ith equality if and only if every i w ith # P j > 2 has * P j = 2. 2), 3), and 4) Since P > P', P' is obtained from P by collapsing together various of the Pj. So either P = P', in which case there is nothing to prove, or we can pass from P to P' through a sequence of intermediate coarsenings where at each step we collapse precisely two sets into one. Thus we m a y reduce to the case where P' is obtained from P by collapsing precisely two sets, say P j and P j. Thus X' is A - 1. If neither Pj nor P j is a singleton, then v = v'. In this case, X ’ - v7 2 = X - 1 - v/2 < X - v/2. If exactly one of Pj or P j is a singleton, then v' = v - 1. In this

case, X' - v / 2 = X -1 - (v-l )/2 = X - 1/2 - v/2 < X - v/2. If both Pj and P j are singletons, then v' = v - 2. In this case, X' - v'/2 = X -1 - (u-2)/2 = X - v/2.

QED

(1.21.5) W h a t do these combinatorics have to do w ith the Higher Mom ent Theorem? To see the relation, we first restate th a t theorem in terms of the sums Zjr |Mq I^n , rather than Zjr |MQ(dim< 3rQ/2)|^n . Higher M o m e n t T heorem bis 1.21.6 Hypotheses and notations as in the Higher M om ent Theorem 1.20.2, denote by M the perverse sheaf M := Twist(L,K,5,h) on T. Denote by Mq its centrally normalized trace function. For each integer n > 1 w ith 2n < d, there exists a real e > 0 such tha t we have 1)

£ e |M0 l2n = (n !)((* E )d im 7 " n m )(l + 0(( * E)_ s / 2 )),

if Gr°(M(dim'3r0/2)) ncs\is not geometrically self dual, 2)

Z E IMo12n = (2n!!)((*E)dim