SYZYGIES OF ABELIAN VARIETIES Let A be an ample line bundle on an abelian variety X

14 Pages·2000·0.22 MB·English

JOURNALOFTHE
AMERICANMATHEMATICALSOCIETY
Volume13,Number3,Pages651{664
S0894-0347(00)00335-0
ArticleelectronicallypublishedonApril10,2000
SYZYGIES OF ABELIAN VARIETIES
GIUSEPPE PARESCHI
Let A be an ample line bundle on an abelian variety X (over an algebraically
closed (cid:12)eld). A theorem of Koizumi ([Ko], [S]), developing Mumford’s ideas and
results([M1]),statesthatifm(cid:21)3thelinebundleL=A(cid:10)m embedsX inprojective
spaceasaprojectivelynormalvariety. Moreover,acelebratedtheoremofMumford
([M2]),slightlyre(cid:12)nedbyKempf([K4]),assertsthatthehomogeneousidealofX is
generatedbyquadricsassoonasm(cid:21)4. Suchresultsturnouttobeparticularcases
of a statement, conjectured byLRob Lazarsfeld, concerning the minimal resolution
Lof the graded algebra RL = 1h=0H0(X;L(cid:10)h) over the polynomial ring SL =
1 SymhH0(X;L). ThepurposeofthispaperistoproveLazarsfeld’sconjecture.
h=0
Toput suchmattersinto perspective,itis usefulto reviewthe caseofprojective
curves. A classical theorem of Castelnuovo states that a curve X, embedded in
projective space by a complete linear system jLj, is projectively normal as soon as
degL(cid:21)2g(X)+1, and a theorem of Mattuck, Fujita and Saint-Donat states that
if degL (cid:21) 2g(X)+2, then the homogeneous ideal of X is generated by quadrics.
Green ([G1]) uni(cid:12)ed, re-interpreted and generalized these results to a statement
about syzygies. Speci(cid:12)cally, given a (smooth) projective variety X and a very
ample line bundle L on X, a minimal resolution of R as a graded S -module
L L
(notation as above) looks like
(1) 0!(cid:1)(cid:1)(cid:1)!E !(cid:1)(cid:1)(cid:1)!E !E !R !0
p 1 0 L
L
where E =S (cid:8) S ((cid:0)a ) (where, since X is embedded by a complete linear
0 L j L 0j L
system, a (cid:21) 2 for any j), E = S ((cid:0)a ) (where, since the image of X in
0j 1 j L 1j
P(H0(L)_) is notLcontainedin any hyperplane,a1j (cid:21)2 for any j), and, in general,
for p (cid:21) 1, E = S ((cid:0)a ) with a (cid:21) p+1 for any j. Green introduced the
p j L pj pj
followingterminology: Lissaidtosatisfy property N ifE =S . Thismeansthat
0 0 L
the map S ! R is surjective, i.e. that the embedded variety X is projectively
L L
normal. Moreover L is said to satisfy property N if it satis(cid:12)es N and a = 2
1 0 1j
for any j, i.e. the homogeneous ideal of the embedded variety X is generated by
quadrics. Inductively,one saysthat L satis(cid:12)es condition N ifit satis(cid:12)es condition
p
Np(cid:0)1andapj =p+1foranyj. SoN2meansthattherelationsbetweenthequadrics
de(cid:12)ning X are generated by linear ones and, for arbitrary p (cid:21) 2, N means that
p
the (cid:12)rst p(cid:0)1 maps of the resolution of the homogeneous ideal are matrices with
linear entries. In a word N means that, up to the p-th step, the resolution (1) is
p
as \regular" as it could possibly be. The aforementioned theorem of Green states
that if X is a curve and degL (cid:21) 2g(X)+1+p, then L satis(cid:12)es N . This result
p
stimulated many interesting questions (see [G1], [G2], [L] and [EL]). One of these
ReceivedbytheeditorsAugust24,1998and,inrevisedform,March8,2000.
2000 Mathematics Subject Classi(cid:12)cation. Primary14K05;Secondary14F05.
Key words and phrases. Homogeneous ideal,Pontrjaginproduct,vector bundles.
(cid:13)c2000 American Mathematical Society
651
652 GIUSEPPE PARESCHI
is how it extends to higher dimension. For arbitrary varieties there is a general
conjectureofMukaiandresultsofEinandLazarsfeld([EL]). Inthis articlewewill
be concerned with the case of abelian varieties. Lazarsfeld’s conjecture states that
powers of ample line bundles on abelian varieties of arbitrary dimension behave
exactly as in the case of elliptic curves:
Theorem (char(k)=0). Let A be an ample line bundle on an abelian variety X.
If n(cid:21)p+3, then A(cid:10)n satis(cid:12)es condition N .
p
Forelliptic curvesthis amountstoGreen’s theoremand,inarbitrarydimension,
the cases p = 0;1 are the aforementioned results of Koizumi and Mumford. It is
alsoworthwhiletopointoutthat,asitiseasytosee,ifAisaprincipalpolarization,
then the bound of Lazarsfeld’s conjecture is the best one can hope for. We also
prove a generalization of Lazarsfeld’s conjecture (Theorem 4.3), about the (cid:12)rst p
steps of the resolution of RA(cid:10)n when n < p+3 (we refer to Section 4 below for
terminology, statement and proof).
It is worthwhile to recall that, shortly after the appearance of Lazarsfeld’s con-
jecture, Kempf ([K5]) provedit \up to a factor two": his statement, in the present
terminology, is that A(cid:10)n satis(cid:12)es condition N as soon as n (cid:21) maxf3;2p+2g.
p
Thus the aboveTheorem improvesKempf’s result already for abelian surfaces and
p=2.
The main point of the proof will be a criterion { of independent interest { for
the surjectivity of multiplication maps of global sections:
((cid:3)) H0(X;A(cid:10)n)(cid:10)H0(X;E)!H0(X;A(cid:10)n(cid:10)E);
where A and E are respectively a vector bundle and an ample line bundle on an
abelian variety X. Roughly, the criterion (Theorem 3.1 and Corollary 3.3) states
thatthe map ((cid:3))is surjective assoonas certainvanishing conditions onthe higher
cohomology of the Pontrjagin product of A(cid:10)n and E are satis(cid:12)ed. The relation
betweenmapsas((cid:3))andsyzygiesiswellknownandgoesasfollows(incharacteristic
zero): let us denote by M the kernel of the evaluation map H0(L)(cid:10)O !L. If
L X
the multiplication map
((cid:3)(cid:3)) H0(L)(cid:10)H0(M(cid:10)p(cid:10)L(cid:10)h)!H0(M(cid:10)p(cid:10)L(cid:10)h+1)
L L
is surjective for any h (cid:21) 1, then L satis(cid:12)es N . Thus the proof of Lazarsfeld’s
p
conjecture will consist in showing that for any n(cid:21)p+3 and h(cid:21)1 the Pontrjagin
product of A(cid:10)n and E = (MA(cid:10)n)(cid:10)p (cid:10)A(cid:10)nh satis(cid:12)es the required cohomological
conditions.
Letusdescribemorecloselyhowtheaforementionedcriterionforthesurjectivity
of multiplication maps is obtained: in Section 1 it is shown that, given two vector
bundles L and E and a point x on X, under mild hypotheses the multiplication
map H0(T(cid:3)L)(cid:10)H0(E)!H0((T(cid:3)L)(cid:10)E) is identi(cid:12)ed to the evaluation of global
x x
sectionsatxofanothervectorbundle,denotedL^(cid:3)E,the(skew)Pontrjaginproduct
of L and E. This suggests we look for ways of understanding the locus whereL^(cid:3)E
is not globally generated. More generally, it is natural to ask for a cohomological
criterion for the global generation of vector bundles on abelian varieties. This is
the theme of Section 2 (which is independent of the rest of the paper), where the
required cohomological criterion (Theorem 2.1) is supplied. Theorem 2.1 follows
from an important result (Lemma 2.2), implicitly contained in Kempf’s work and
pointed outby R. Lazarsfeld,whose proofwe include. Putting together the results
SYZYGIES OF ABELIAN VARIETIES 653
of Sections 1 and 2 one gets a criterion for the surjectivity of multiplication maps,
intermsofthecohomologyofthe Pontrjaginproducts(Theorem3.1). InSection3
we supply some techniques about how to verify the hypotheses of such a criterion.
If one of the two vector bundles L and E is a line bundle, everything can be put
in a much more explicit form (Theorem 3.8), which, we hope, might have other
applications. In Section 4, we apply the previous results to Lazarsfeld’s conjecture
and to its generalization (Theorem 4.3).
Although we work in the algebraic setting, we have made some simplifying as-
sumptions on the characteristic of the ground (cid:12)eld. Moreover one needs the char-
acteristic zero for condition ((cid:3)(cid:3)) about syzygies.
This work owes much to the reading of George Kempf’s penetrating works on
these and related subjects, especially [K1]{[K6]. My understanding of [K5] in turn
owes a lot to some UCLA lectures and seminar talks given by Rob Lazarsfeld
on the subject, back in ’88-89. More recently, I have also bene(cid:12)tted by some
useful conversations with Lawrence Ein. Part of this work was done while the
author was visiting Colorado State University at Fort Collins and U.N.A.M. at
Morelia (Mexico), supported respectively by a C.N.R.-N.A.T.O. scholarship and
by U.N.A.M. My thanks go to these institutions, and especially to Jeanne Du(cid:13)ot
and Rick Miranda, at Fort Collins, and Alexis Garcia-Zamora and Sevin Recillas,
at Morelia, for really great hospitality.
Throughout this paper X will denote an abelian variety over an algebraically
closed (cid:12)eld k. T :X !X will denote the translation by the point x2 X. Given
x
twointegersmandn,mp +np :X(cid:2)X !X willmeanthemap(x;y)!mx+ny.
1 2
E.g. the group law (x;y)7!x+y will be denoted p +p and not m (which will
1 2 X
denote the multiplication by the integer m).
1. Pontrjagin products and surjectivity of multiplication maps
As mentioned in the introduction, the main theme of the (cid:12)rst part of the paper
will be to (cid:12)nd criteria ensuring the surjectivity of the multiplication map
H0(L)(cid:10)H0(E)!H0(L(cid:10)E);
where L and E are respectively a line bundle and a vector bundle on E. However,
the (crucial) fact that L is a line bundle will be used only later (see Section 3(b)).
So here we will assume that L and E are simply two vector bundles (i.e. locally
free sheaves) on X. We will rather undertake a global study of the family of
multiplication maps
(1) m :H0(T(cid:3)L)(cid:10)H0(E)!H0((T(cid:3)L)(cid:10)E)
x x x
for x varying in X. Let us de(cid:12)ne the \skew Pontrjagin product" of L and E as
L^(cid:3)E :=p ((p +p )(cid:3)L(cid:10)p(cid:3)E)
1(cid:3) 1 2 2
(see Remark 1.2 below for a justi(cid:12)cation of the terminology and some comments).
For the sake of simplicity, we will assume throughout that
(2) Hi((T(cid:3)L)(cid:10)E)=0
x
for any x 2 X and for any i > 0. Then Grauert’s theorem yields that L^(cid:3)E is
locally free and that, at the (cid:12)bers level, the map L^(cid:3)E(x) !H0((T(cid:3)L)(cid:10)E) is an
x
isomorphism. On the other hand, there is a natural isomorphism
(cid:8):H0(L)(cid:10)H0(E)!(cid:24) H0(L^(cid:3)E)
654 GIUSEPPE PARESCHI
obtainedasfollows: viathe automorphism(cid:30)=(p +p ;p )ofX(cid:2)X wehavethat
1 2 2
(p +p )(cid:3)L(cid:10)p(cid:3)E =(cid:30)(cid:3)(p(cid:3)L(cid:10)p(cid:3)E). Therefore we have the isomorphism
1 2 2 1 2
(cid:30)(cid:3) :H0(L)(cid:10)H0(E)!H0((p +p )(cid:3)L(cid:10)p(cid:3)E):
1 2 2
The isomorphism (cid:8) follows by composing with the isomorphism
H0((p1+p2)(cid:3)L(cid:10)p(cid:3)2E)(cid:24)=H0(p1(cid:3)((p1+p2)(cid:3)L(cid:10)p(cid:3)2E)):
It follows that the evaluation map of L^(cid:3)E at x composed with the isomorphism
(cid:8),
H0(L)(cid:10)H0(E)(cid:0)!(cid:8) H0(L^(cid:3)E)(cid:0)e(cid:0)v!x L^(cid:3)E(x);
coincides with the composed map
H0(L)(cid:10)H0(E)(cid:0)T(cid:0)x(cid:3)(cid:0)(cid:2)(cid:0)i!d H0(T(cid:3)L)(cid:10)H0(E)(cid:0)M(cid:0)!x H0((T(cid:3)L)(cid:10)E):
x x
Therefore we have
Proposition 1.1. Assuming hypothesis (2), the locus of points x 2 X such that
the multiplication map m is not surjective coincides with the locus where L^(cid:3)E is
x
not generated by its global sections.
Of course the advantage of this is that the fact that the maps m are surjective
x
for any x2X is equivalentto the globalgenerationofthe vectorbundle L^(cid:3)E and
this last condition can be investigated via global cohomologicalmethods.
Remark 1.2. Given two sheaves F and E on X, the sheaf F^(cid:3)E is isomorphic to
F (cid:3)((cid:0)1)(cid:3) E, were \(cid:3)" means the Pontrjagin product as de(cid:12)ned in [Mu], p.160,
X
i.e. G (cid:3) H = (p1 + p2)(cid:3)(p(cid:3)1G (cid:10) p(cid:3)2H): The isomorphism can be seen as above
using the automorphism (p + p ;(cid:0)p ). Note that ^(cid:3) is \skew-symmetric", i.e.
1 2 2
F^(cid:3)E (cid:24)=((cid:0)1)(cid:3) (E^(cid:3)F).
X
2. A cohomological criterion for the global generation
of vector bundles on abelian varieties
In view of Proposition 1.1, it is natural to look for criteria guaranteeing that
a vector bundle on an abelian variety is generated by its global sections. E.g.
a basic and elementary result on abelian varieties states that the product of two
amplelinebundlesisbase-pointfree. Wegeneralizeittothefollowingresult,which
can be seen as an analogue for vector bundles on abelian varieties of the classical
Castelnuovo-Mumfordcriterion for sheaves on projective spaces.
Theorem 2.1. Let A and F be respectively an ample line bundle and a vector
bundle on an abelian variety X. If hi(F (cid:10)A_(cid:10)(cid:11)) =0 for any i>0 and for any
line bundle (cid:11)2Pic0X, then F is generated by its global sections.
(a) A lemma of Kempf (according to Lazarsfeld). Theorem 2.1 will be a
consequenceofa verysharpargumentimplicitin Kempf’s work([K5], [K6]). Since
the matter is not easy to isolate in Kempf’s paper, we include the proof for the
reader’s bene(cid:12)t. The following formulation is due to Lazarsfeld.
SYZYGIES OF ABELIAN VARIETIES 655
Lemma 2.2 (Kempf). Let E and F be vector bundles on an abelian variety X. If
hi(E (cid:10)(cid:11))=hi(F (cid:10)(cid:11))=0 for any i>0 and for any (cid:11)2Pic0X, then the map
M
H0(E (cid:10)(cid:11))(cid:10)H0(F (cid:10)(cid:11)_)m!(cid:11) H0(E (cid:10)F)
(cid:11)2U
is surjective for any non-empty Zariski open set U (cid:26)Pic0X.
Proof. STEP1. Letusdenotebyp thethreeprojectionsonX(cid:2)X(cid:2)Pic0X andby
i
p the three intermediate projections. Moreover let (cid:1) be the diagonal in X (cid:2)X,
ij
and let P be a Poincar(cid:19)e line bundle on X (cid:2)Pic0X. Finally, let
W =p(cid:3) (p(cid:3)E (cid:10)P)(cid:10)p(cid:3) (p(cid:3)F (cid:10)P_);
13 1 23 2
andlet(cid:8):W !Wj(cid:1)(cid:2)Pic0X betherestrictionmap. Notethat,sincep(cid:3)13P(cid:10)p(cid:3)23P_is
trivialwhenrestrictedto(cid:1)(cid:2)Pic0X,thetargetisisomorphictothesheafp(cid:3) (E(cid:10)F)
X
on X (cid:2)Pic0X (via the isomorphism X !(cid:24) (cid:1)). By the hypothesis on E and F and
Grauert’s theorem, the sheaf p (W) is locally free, with (cid:12)ber at (cid:11) isomorphic to
3(cid:3)
H0(E(cid:10)(cid:11))(cid:10)H0(F(cid:10)(cid:11)_). Moreover,denotingp (cid:8):p W !H0(E(cid:10)F)(cid:10)O ,
3(cid:3) 3(cid:3) Pic0X
we have that p (cid:8)((cid:11)) is identi(cid:12)ed to the multiplication map m : H0(E (cid:10)(cid:11))(cid:10)
3(cid:3) (cid:11)
H0(F (cid:10)(cid:11)_)!H0(E (cid:10)F). L
STEP 2. The surjectivity of the map m of the statement of Lemma 2.2
(cid:11)2U (cid:11)
is equivalent to the surjectivity of the map Hg(p (cid:8)):Hg(p W)!H0(E (cid:10)F)(cid:10)
3(cid:3) 3(cid:3)
Hg(OPic0X) (where g =dimX; of couLrse Hg(OPicL0X)(cid:24)=k).
Proof. The surjectivity of the map m : p W((cid:11))!H0(E(cid:10)F) is
(cid:11)2U (cid:11) (cid:11)2U 3(cid:3)Q
equivalenttotheinjectivityofthedualmapH0(E(cid:10)F)_ ! p W((cid:11))_ which
(cid:11)2U 3(cid:3)
is in turn equivalent to the fact that the map (p (cid:8))_ : H0(E (cid:10)F)(cid:10)O !
3(cid:3) Pic0X
p W_isinjectiveattheglobalsectionslevelandthis,bySerreduality,isequivalent
3(cid:3)
to the surjectivity of Hg(p (cid:8)).
3(cid:3)
STEP 3. The Leray spectral sequence Hi(Rjp W) ) Hi+j(W) degenerates
3(cid:3)
giving an isomorphism Hg(p W)!(cid:24) Hg(W) (byhypothesis,Rip (W)iszerofor
3(cid:3) 3(cid:3)
i>0).
STEP4. There is a canonical isomorphism Hg(W)!(cid:24) H0(E(cid:10)F)(cid:10)Hg(O )
Pic0X
such that the map Hg(p (cid:8)) of Step 2 is the composition of this isomorphism with
3(cid:3)
the one supplied by Step 3. Therefore Hg(p (cid:8)) is itself an isomorphism. This
3(cid:3)
proves Lemma 2.2.
Proof. In the (cid:12)rst place let us note that, by the projection formula,
(1) Rip12(cid:3)(W)(cid:24)=p(cid:3)1E (cid:10)p(cid:3)2F (cid:10)Rip12(cid:3)(p(cid:3)13P (cid:10)p(cid:3)23P_):
On the other hand, it is well known (see [K1] or [K6], p.53) that
(
(2) Rip12(cid:3)(p(cid:3)13P (cid:10)p(cid:3)23P_)(cid:24)= O0 (cid:10)Hg(O ) iiff ii=<gg:;
(cid:1) Pic0X
Let us recall the proof of (2) for the sake of self-containedness: one considers the
di(cid:11)erencemapd=p (cid:0)p :X(cid:2)X !X. Bythesee-sawprincipleitfollowsimme-
1 2
diatelythatp(cid:3) P(cid:10)p(cid:3) P_ (cid:24)=(d;id )(cid:3)P. Thereforewearereducedtocomputing
theRip12(cid:3)’so1f3theri2g3hthandsideP.icB0Xy(cid:13)atbaseextensionRip12(cid:3)((d;idPic0X)(cid:3)P)(cid:24)=
d(cid:3)(Rip (P)). At this point one invokes Mumford’s duality result ([M1], Ch.13,
X(cid:3)
[K2], Th.3.15) which states that Rip (P) = Hg(O )(cid:10)O if i = g and it is
X(cid:3) Pic0X e
656 GIUSEPPE PARESCHI
zerootherwise(eistheidentitypointofX). Thisproves(2). Thenby(1)itfollows
that
(
(3) Rip12(cid:3)(W)(cid:24)= p(cid:3)1E (cid:10)p(cid:3)2F (cid:10)Hg(OPic0X)(cid:10)O(cid:1) if i=g;
0 if i<g:
Finally, to get the isomorphism of Step 4 let us consider the other natural spec-
tral sequence, that is, Hi(Rjp (W)) ) Hi+j(W). By (3) it degenerates to the
12(cid:3)
isomorphism Hg(W) (cid:24)= H0(Rgp12(cid:3)(W)) (cid:24)= H0(E (cid:10)F)(cid:10)Hg(OPic0X). The second
part of the statement of Step 4 follows easily.
(b) Proof of Theorem 2.1. We startwith a couple ofeasy corollariesof Lemma
2.2, whose proof is immediate.
Corollary 2.3. If E and F satisfy the hypotheses of Lemma 2.2, then there is a
positive integer N such that given N general line bundles ((cid:11) ;:::;(cid:11) )2(Pic0X)N
L 1 N
the map N H0(E (cid:10)(cid:11) )(cid:10)H0(F (cid:10)(cid:11)_)m!(cid:11)i H0(E (cid:10)F) is surjective.
i=1 i i
Corollary 2.4. Let E be a vector bundle on an abelian variety X such that
Hi(E (cid:10)(cid:11)) = 0 for any i > 0 and for any (cid:11) 2 Pic0X. Then there is a positive
integer N such that given N general line bundles ((cid:11) ;:::;(cid:11) ) 2 (Pic0X)N the
1 N
L
map N H0(E (cid:10)(cid:11) )(cid:10)(cid:11)_ (cid:30)!(cid:11)i E is surjective (where (cid:30) denotes the evaluation
map of ig=lo1bal sectionsi of Ei(cid:10)(cid:11) tensored with (cid:11) _). (cid:11)i
i i
L
Proof. LetAbeanamplelinebundleonX andL=A(cid:10)n. Ifthemap ((cid:30) (cid:10)L):
L (cid:11)i
N H0(E (cid:10)(cid:11) )(cid:10)L(cid:10)(cid:11)_ !E (cid:10)L is not surjective and n is big enough, then it
i=1 i i
is not surjective at the H0 level and this is in contrast with Corollary 2.3.
Theorem 2.1 is now an easy consequence of Corollary 2.4:
Proof of Theorem 2.1. ThehypothesisallowsustoapplyCorollary2.4tothevector
bundle E = F (cid:10) A_: given a tuple (cid:11)(cid:22) = ((cid:11) ;:::;(cid:11) ) 2 (Pic0X)N we have the
1 N
commutative diagram:
LN
H0(F (cid:10)A_(cid:10)(cid:11) )(cid:10)H0(A(cid:10)(cid:11)_)(cid:10)O (cid:0)! H0(F)(cid:10)O
i i X X
(cid:11)i=1 ??L ??
(4) y (id(cid:10)evi) yev
LN H0(F (cid:10)A_(cid:10)(cid:11) )(cid:10)A(cid:10)(cid:11)_ L((cid:30)(cid:0)(cid:11)!i(cid:10)A) F
i i
(cid:11)i=1
Letusdenote: B(cid:11) isthebaselocusofthelinebundleA(cid:10)(cid:11)_,B(cid:11)(cid:22) =B(cid:11)1[(cid:1)(cid:1)(cid:1)[B(cid:11)N
andB(F)isthesupportofthecokernelofthemapev(theevaluationmapofglobal
sections of F). The point is that A(cid:10)(cid:11)_ is always e(cid:11)ective (since A is ample), so
thatB(cid:11)(cid:22) isalwaysaproperoremptysubvarietyofX. ByCorollary2.4wehavethat
L
N ((cid:30) (cid:10)A) is surjective for (cid:11)(cid:22) general in (Pic0X)N and therefore the diagram
i=1 (cid:11)i
shows that B(F) is contained in B(cid:11)(cid:22) for (cid:11)(cid:22) varying in a certain non-empty Zariski-
open set V of (Pic0X)N. (Actually it follows that B(F) is containedTin B(cid:11)(cid:22) for
any (cid:11)(cid:22) 2Pic0X.) But such intersection is clearly empty, since already B(cid:11) is
(cid:11)2U
empty for any Zariski open set U of Pic0X (it is the intersection of translates of
the base locus B(A) by points varying in a Zariski open set of X).
SYZYGIES OF ABELIAN VARIETIES 657
3. Calculus with Pontrjagin products
Going back to the surjectivity of the multiplication maps (1) of Section 1, we
have, as a consequence of Proposition 1.1 and Theorem 2.1:
Theorem 3.1. Let L and E be vector bundles on an abelian variety X such that
Hi((T(cid:3)L)(cid:10)E) = 0 for any i > 0 and for any x 2 X. If there is an ample line
x
bundle A on X such that, for any i>0 and for any (cid:11)2Pic0X,
Hi((L^(cid:3)E)(cid:10)A_(cid:10)(cid:11))=0;
then the multiplication maps m : H0(T(cid:3)L)(cid:10)H0(E) ! H0((T(cid:3)L)(cid:10)E) are sur-
x x x
jective for any x2X.
Givena sheafF,we willsaythatF has the vanishing property if Hi(F(cid:10)(cid:11))=0
for any i > 0 and for any (cid:11) 2 Pic0X. Theorem 3.1 leaves the problem of how to
check in practice that a \mixed product" such as (L^(cid:3)E)(cid:10)A_ has the vanishing
property. The goal of this section is to develop some tools to this purpose, under
the further hypothesis, used in (b) below, that L is a line bundle.
(a) Exchanging tensor and Pontrjagin product under cohomology.
Lemma 3.2. Let L, E and M be vector bundles on X such that hi((T(cid:3)L)(cid:10)E)=
x
hi((T(cid:3)L)(cid:10)M)=0 for any i>0 and for any x2X. Then, for any i(cid:21)0;
x
hi((L^(cid:3)E)(cid:10)M)=hi((L^(cid:3)M)(cid:10)E):
Proof. By the projection formula we have that
(L^(cid:3)E)(cid:10)M (cid:24)=p1(cid:3)(p(cid:3)1(M)(cid:10)(p1+p2)(cid:3)(L)(cid:10)p(cid:3)2(E));
(L^(cid:3)M)(cid:10)E (cid:24)=p1(cid:3)(p(cid:3)1(E)(cid:10)(p1+p2)(cid:3)(L)(cid:10)p(cid:3)2(M)):
To simplify the notation let us denote by respectively F and G the right hand
sides of the (cid:12)rst and of the second isomorphism above. Of course, since F and G
are exchanged by the automorphism (p ;p ) of X (cid:2)X, Hi(F) (cid:24)= Hi(G) for any i.
2 1
The projectionformula andthe hypothesis ensure that Rip F and Rip G vanish
1(cid:3) 1(cid:3)
for i > 0. Therefore the Leray spectral sequence degenerates to isomorphisms
Hi((L^(cid:3)E)(cid:10)M)(cid:24)=Hi(F) and Hi((L^(cid:3)M)(cid:10)E)(cid:24)=Hi(G).
As a corollary of Theorem 3.1 and Lemma 3.2 we get
Corollary 3.3. Let A and E be respectively an ample line bundle and a vector
bundle on X, and let n > 1 be an integer such that E (cid:10)A(cid:10)n has the vanishing
property. If
(1) Hi((A(cid:10)n^(cid:3)(A_(cid:10)(cid:11)))(cid:10)E)=0
for any i > 0 and (cid:11) 2 Pic0X, then the multiplication maps m : H0(T(cid:3)A(cid:10)n)(cid:10)
x x
H0(E)!H0((T(cid:3)A(cid:10)n)(cid:10)E) are surjective for any x2X.
x
(b)Pullingback via the multiplicationby an integer. Sofarwehavemerely
assumed that L and E, the factors of the Pontrjagin product, are vector bundles.
From this point we will make the (essential) further assumption that one of them,
namely L, has rank one.
Let n be a positive integer and let us denote by n :X !X the map x7!nx:
X
If L is a line bundle, we have the following standard formula, which will be a
fundamental tool in the proof of Lazarsfeld’s conjecture.
658 GIUSEPPE PARESCHI
Proposition 3.4. n(cid:3) (L^(cid:3)E)(cid:24)=(L(cid:10)n^(cid:3)(E(cid:10)L(cid:10)(cid:0)n+1))(cid:10)n(cid:3) L(cid:10)L(cid:10)(cid:0)n .
X X
Letuspointoutwhyoneshouldexpectaformulalikethis. Bythetheoremofthe
square,(T(cid:3)L)(cid:10)n (cid:24)=T(cid:3)(L(cid:10)n)(cid:24)=T(cid:3) L(cid:10)L(cid:10)n(cid:0)1. ThereforeT(cid:3)L(cid:10)E =T(cid:3)L(cid:10)L(cid:10)n(cid:0)1(cid:10)
E(cid:10)L(cid:10)(cid:0)nx+1 (cid:24)= (T(cid:3) xL(cid:10)n)(cid:10)E(cid:10)nxL(cid:10)(cid:0)n+1, where x=n meaxns any z 2xX such that
x=n
nz =x. It is therefore naturalto expect that such an isomorphism at the H0 level
globalizes to an isomorphism n(cid:3) (L^(cid:3)E)(cid:24)=(L(cid:10)n^(cid:3)(E(cid:10)L(cid:10)(cid:0)n+1))(cid:10)(a line bundle).
X
Proof of Proposition 3.4. By (cid:13)at base extension we have
n(cid:3)Xp1(cid:3)((p1+p2)(cid:3)L(cid:10)p(cid:3)2E)(cid:24)=p1(cid:3)((nX;1X)(cid:3)(p1+p2)(cid:3)L(cid:10)p(cid:3)2E)
(2) =p ((np +p )(cid:3)L(cid:10)p(cid:3)E):
1(cid:3) 1 2 2
On the other hand,
(3) (np +p )(cid:3)L(cid:24)=(p +p )(cid:3)L(cid:10)n(cid:10)p(cid:3)(n(cid:3) L(cid:10)L(cid:10)(cid:0)n)(cid:10)p(cid:3)L(cid:10)(cid:0)n+1:
1 2 1 2 1 X 2
For n = 2, (3) follows by plugging f = g = p and h = p in [M3], Cor.2, p.58
1 2
(Theorem of the Cube). For any n, (3) follows by induction in the same way.
Finally, plugging (3) into the last member of (2) and using the projection formula
we get the statement.
Let us remark that one could also deduce a (conceptually slightly more compli-
cated)proofofProposition3.4fromMukai’ssetting(seeRemark1.2). Infact,given
asheafG,letusconsiderits\Fouriertransform"G^ =q (q(cid:3)G(cid:10)P)(whereq arethe
2(cid:3) 1 i
projections on X(cid:2)Pic0X and P is a Poincar(cid:19)esheaf). Then one can prove the fol-
lowingformula(whichcanbededucede.g. from[Mu],3.10): L^(cid:3)E (cid:24)=(cid:30)(cid:3)(L\(cid:10)E)(cid:10)L
L
(where we denote by (cid:30) : X ! Pic0X the polarization associated to L). Then,
L
since (cid:30)L(cid:10)n = n(cid:30)L, we have n(cid:3)X(L^(cid:3)E) (cid:24)= (cid:30)L(cid:10)n(L\(cid:10)E)(cid:10)n(cid:3)XL. Thus Proposition
3.4 follows from the formula above applied to L(cid:10)n^(cid:3)(E(cid:10)L(cid:10)(cid:0)n+1).
The main reason why it is convenient to consider pullbacks of type n(cid:3) (L^(cid:3)E)
X
ratherthan L^(cid:3)E itself is that if L and E are both line bundles algebraicallyequiv-
alent to powers of the same ample line bundle A, then there is always a positive
integernsuchthatthevectorbundlen(cid:3) (L^(cid:3)E)istrivialuptotwistbyalinebundle
X
algebraically equivalent to a power of A. To see this we need some basic remarks.
Remarks 3.5. (a)Let L, E and (cid:11) be respectivelytwo coherentsheaveson X anda
line bundle in Pic0X. Then
(4) (F (cid:10)(cid:11))^(cid:3)E (cid:24)=(F^(cid:3)(E(cid:10)(cid:11)))(cid:10)(cid:11):
Indeedif(cid:11)2Pic0X,then(p +p )(cid:3)(cid:11)(cid:24)=p(cid:3)(cid:11)(cid:10)p(cid:3)(cid:11). Pluggingthisintothede(cid:12)nition
1 2 1 2
of (F (cid:10)(cid:11))^(cid:3)E one gets (4) by the projection formula.
(b) Let F be any coherent sheaf on X. There is a natural isomorphism
(cid:9):F^(cid:3)O !(cid:24) H0(F)(cid:10)O :
X X
Indeedlet betheautomorphism =(p ;p (cid:0)p )ofX(cid:2)X. Then (cid:3)(p +p )(cid:3)F =
1 2 1 1 2
p(cid:3)F and we have the isomorphism (cid:3) : p ((p +p )(cid:3)F) ! p (p(cid:3)F): Then (cid:9) is
ob2tainedbycomposing (cid:3)withtheKu(cid:127)nnet1h(cid:3)iso1morp2hismp1(cid:3)(p1(cid:3)2(cid:3)F)2(cid:24)=H0(F)(cid:10)OX.
(a) and (b) yield
(c) Let F be a coherent sheaf on X and (cid:11)2Pic0X. Then F^(cid:3)(cid:11)(cid:24)=H0(F (cid:10)(cid:11))(cid:10)
(cid:11)_.
SYZYGIES OF ABELIAN VARIETIES 659
Let us assume e.g. that E is a line bundle algebraically equivalent to a positive
multiple of L. If (cid:11) 2 Pic0X and n (cid:21) 2, by Proposition 3.4 and Remark 3.5(c) we
get that
(5) n(cid:3) (L^(cid:3)(L(cid:10)n(cid:0)1(cid:10)(cid:11)))(cid:24)=H0(L(cid:10)n(cid:10)(cid:11))(cid:10)n(cid:3) (L)(cid:10)L(cid:10)(cid:0)n(cid:10)(cid:11)_:
X X
Since n(cid:3) L is algebraically equivalent to L(cid:10)n2, we get that n(cid:3) (L^(cid:3)(L(cid:10)n(cid:0)1(cid:10)(cid:11))) is
X X
isomorphic to a trivial bundle twisted by a line bundle algebraically equivalent to
L(cid:10)n(n(cid:0)1). More generally, we have the following
Proposition 3.6. Let A be a line bundle, and let a and b two integers such that
a and a+b are positive. Then (a+b)(cid:3) (A(cid:10)a^(cid:3)(A(cid:10)b(cid:10)(cid:11))) is isomorphic to a trivial
X
bundle times a line bundle algebraically equivalent to A(cid:10)ab(a+b). Speci(cid:12)cally
(a+b)(cid:3) (A(cid:10)a^(cid:3)(A(cid:10)b(cid:10)(cid:11)))(cid:24)=H0(A(cid:10)a+b(cid:10)(cid:11))(cid:10)(a+b)(cid:3) (A(cid:10)a)
X X
(cid:10)a(cid:3) (A(cid:10)(cid:0)a(cid:0)b)(cid:10)(cid:11)_:
X
Proof. By Proposition 3.4 we know that
(6) (a+b)(cid:3) (A(cid:10)a^(cid:3)(A(cid:10)b(cid:10)(cid:11)))(cid:24)=(A(cid:10)a(a+b)^(cid:3)(A(cid:10)b(cid:0)a(a+b(cid:0)1)(cid:10)(cid:11)))
X
(cid:10)(a+b)(cid:3) (A(cid:10)a)(cid:10)A(cid:10)(cid:0)a(a+b);
X
(7) a(cid:3) (A(cid:10)a+b^(cid:3)(cid:11))(cid:24)=(A(cid:10)a(a+b)^(cid:3)((cid:11)(cid:10)A(cid:10)(cid:0)(a+b)(a(cid:0)1)))
X
(cid:10)a(cid:3) (A(cid:10)a+b)(cid:10)A(cid:10)(cid:0)a(a+b):
X
The statement is obtained by plugging (7) into (6) (note that b(cid:0)a(a+b(cid:0)1) =
(cid:0)(a+b)(a(cid:0)1)), and using Remark 3.5(c).
(c) Examples and applications.
Example 3.7. Let us recover the following statement (see [K6]), which includes
Koizumi’s Theorem: if a;b (cid:21) 2 and a + b (cid:21) 5, then the multiplication maps
H0(T(cid:3)A(cid:10)a) (cid:10) H0(A(cid:10)b) ! H0((T(cid:3)A(cid:10)a) (cid:10) A(cid:10)b) are surjective for any x 2 X.
x x
Proof: by Theorem 3.1 we need to show that (A(cid:10)a^(cid:3)A(cid:10)b)(cid:10)A_ has the vanishing
property. Pullingbackvia(a+b)(cid:3) ,byProposition3.6wegetatrivialbundletimes
X
a line bundle algebraically equivalent to A(a+b)(ab(cid:0)a(cid:0)b), which is positive precisely
when a;b (cid:21) 2 and a+b (cid:21) 5. (By means of a (cid:12)ner analysis it is also possible to
prove along these lines the stronger result that m : H0(T(cid:3)A(cid:10)2)(cid:10)H0(A(cid:10)2) !
x x
H0((T(cid:3)A(cid:10)2)(cid:10) A(cid:10)2) is surjective for general x 2 X. One can also recover the
x
explicit description of the locus where m is not surjective.)
x
The same resultcan be recovered,with a slightly di(cid:11)erent proof, as a particular
case of the following more general statement.
Theorem 3.8. LetX bean abelian variety overan algebraically closed (cid:12)eld k, and
let n (cid:21)2 be an integer (such that char(k) does not divide n (cid:0)1). Also let A and
0 0
E be respectively an ample line bundle and a vector bundle on X such that E(cid:10)A(cid:10)n
has the vanishing property for any n(cid:21)n .
0
If(n0(cid:0)1)(cid:3)X(E)(cid:10)A(cid:10)(cid:0)n0(n0(cid:0)1) has thevanishingproperty, thenthemultiplication
map
(8) m :H0(T(cid:3)An)(cid:10)H0(E)!H0((T(cid:3)A(cid:10)n)(cid:10)E)
x x x
is surjective for any x2X and for any n(cid:21)n .
0
660 GIUSEPPE PARESCHI
Proof of Theorem 3.8. By hypothesis and Corollary 3.3 what we need to show is
that hi((A(cid:10)n0^(cid:3)(A_(cid:10)(cid:11)))(cid:10)E)= 0 for any i >0, (cid:11) 2Pic0X. By Proposition 3.6,
pulling back via the map (n (cid:0)1) we get a trivial bundle times (n (cid:0)1)(cid:3) (E)(cid:10)
0 X 0 X
A(cid:0)n0(n0(cid:0)1)(cid:10)(cid:11).
Taking n = 2 and E = A(cid:10)m with m (cid:21) 3 one recovers Example 3.7. More
0
generally, when n = 2 one recovers the following criterion, which is (implicitly)
0
the tool used by Kempf to prove his result on syzygies ([K5]): if E (cid:10) Ak = 0
has the vanishing property for any k (cid:21) (cid:0)2, then for n (cid:21) 2 the multiplication map
H0(T(cid:3)A(cid:10)n)(cid:10)H0(E)!H0((T(cid:3)A(cid:10)n)(cid:10)E) is surjective for any x2X. Thisresult
x x
is clearly optimal but its weakness lies in the fact that in many applications (e.g.
syzygies) one deals with higher powers of A, say n (cid:21) n , and what is needed is a
0
criterionfor the surjectivity of multiplication maps suchas (8) whichis e(cid:11)ective in
function of n . Theorem 3.8 showsthat, if n >2, to get suchan optimal criterion
0 0
one needs to pullback E via a suitable map (the multiplication by n (cid:0)1). This of
0
course makes the criterion more di(cid:14)cult to apply. As we will see in the sequel, for
thebundlesappearinginsyzygyproblemsthisdi(cid:14)cultycanbeeasilycircumvented.
4. Proof of Lazarsfeld’s conjecture
Let us begin by rewiewing some background material { well-known and largely
due to Green { about the relation between conditions about syzygies, like N , and
p
the Koszul complex resolving the residue (cid:12)eld k as a module over the symmetric
algebra. The main fact (see [G1], [G2], Thm 1.2, [L], p.511, for details) is that
condition N is equivalent to the exactness in the middle of the complex
p
p^+1 ^p p^(cid:0)1
(1) H0(L)(cid:10)H0(L(cid:10)h)! H0(L)(cid:10)H0(L(cid:10)h+1)! H0(L)(cid:10)H0(L(cid:10)h+2)
for any h (cid:21) 1. Vector bundles come into play as follows: let M be the kernel of
L
the evaluation map of L,
(2) 0!M !H0(L)(cid:10)O !L!0:
L X
Taking wedge products of (2) one gets, for any p, exact sequences
p^+1 p^+1 ^p
(3) 0! M ! H0(L)(cid:10)O ! M (cid:10)L!0:
L X L
It follows (see e.g. [L]) that the exactness of the complex (1) is equivalent to the
surjectivity of the map
p^+1 ^p
(4) H0(L)(cid:10)H0(L(cid:10)h)!H0( M (cid:10)L(cid:10)h+1)
L
obtained by twisting (3) with L(cid:10)h and taking H0 of the third arrow. Therefore, if
p^+1
(5) H1( M (cid:10)L(cid:10)h)=0
L
for any h (cid:21) 1, then condition N holds (the converse is also true as soon as
p
H1(L(cid:10)h)=0, as for abelian vareties). This leads to the following lemma.