Singularities of divisors on abelian varieties

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1 Singularities of divisors on abelian varieties Olivier Debarre March 20, 2006 This is joint work with Christopher Hacon. We work over the coplex nubers. Let D be an effective divisor on an abelian variety A of diension g. If D is not aple, there exists a quotient abelian variety A B such that D is the pull-back of a divisor on B. Since we are interested in the singularities of D, we will henceforth assue that D is aple. 1 Singularities of pairs Let D be an effective Q-divisor on a sooth projective variety A. A log resolution of the pair (A, D) is a proper birational orphis µ : A A such that the union of µ 1 (D) and the exceptional locus of µ is a divisor with siple noral crossing support. Write µ (K A + D) = K A + a i D i where the D i are distinct prie divisors on A. We define the ultiplier ideal sheaf I (A, D) O A by I (A, D) = µ ( ωa /A( µ D ) ) = µ ( OA ( a i D i ) ) and, if D is a prie divisor, the adjoint ideal sheaf J (A, D) O A as fitting into an exact sequence 0 ω A ω A (D) J (A, D) f ω X 0 (1) of sheaves on A, for any desingularization f : X D. The pair (A, D) is 1

2 log canonical if a i 1 for all i; this is equivalent to I (A, td) = O A for all t Q (0, 1); log terinal if a i < 1 for all i; this is equivalent to I (A, D) = O A ; (D prie) canonical if a i 0 for all i such that D i is µ-exceptional; this is equivalent to J (A, D) = O A. These properties have consequences for the singularities of D: for any positive integers and k, if (A, 1 D) is log canonical, 1 +1 D = 0 and codi A (Sing k D) k; if (A, 1 D) is log terinal, 1 D = 0 and codi A (Sing k D) > k; (D prie) if (A, D) is canonical, D is noral with rational singularities and codi A (Sing k D) > k for k 2. Kollár was the first to use vanishing theores to prove results on the singularities of divisors in abelian varieties. 2 The results Let A be an abelian variety of diension g. A polarization l on A is an aple nuerical equivalence class. Its degree is given by 1 g! lg = h 0 (A, L) for any line bundle L on A that represents l. A polarization of degree 1 is called principal and a divisor representing it is called a theta divisor. A polarized abelian variety (A, l) is indecoposable if it is not the product of nonzero polarized abelian varieties. Previous results were as follows: if (A, [Θ]) is a principally polarized abelian variety and D Θ, the pair (A, 1 D) is log canonical (Kollár, 1993 for = 1; Ein Lazarsfeld, 1997) and log terinal if 1 D = 0 (Hacon, 1999); if Θ is irreducible, the pair (A, Θ) is canonical (Ein Lazarsfeld, 1997). 2

3 if (A, l) is indecoposable of degree 2 and D l, the pair (A, 1 D) is log canonical (Hacon, 2000). Here is Ein Lazarsfeld s proof of the first point. Let Z be the subschee of D defined by I (A, t D), for soe t Q (0, 1). The Nadel Kawaata Viehweg vanishing theore yields H i (A, O A (Θ a ) I Z ) = 0 for all i > 0 and all a A. If Z is nonepty, we have Z Θ a for a general, hence H 0 (A, O A (Θ a ) I Z ) = 0. It follows that χ(a, O A (Θ a ) I Z ) = H 0 (A, O A (Θ a ) I Z ) = 0 for a A general hence for all a because the Euler characteristic is a nuerical invariant. We conclude that H i (A, O A (Θ) P I Z ) = 0 for all i and all P Pic 0 (A). By the Fourier Mukai theory, this iplies O A (Θ) I Z = 0, which is absurd. The other proofs are ore involved and use generic vanishing results of Green Lazarsfeld. We ll coe back to that. Here are our results. Let (A, l) be an indecoposable polarized abelian variety of degree d and diension g > (d + 1) 2 /4 and let D l. If A is siple, if = 1, the divisor D is prie and the pair (A, D) is canonical; if 2, the pair (A, 1 D) is log terinal unless D = E, with E l. If d = 2, if = 1 and D is prie, the pair (A, D) is canonical; if 2, the pair (A, 1 D) is log canonical and is log terinal if 1 D = This condition holds unless D = E, with E l; 3

4 Rearks 1 1) The optial bound sees to be g > d. 2) If A is not siple, but still indecoposable, l ay very well contain reducible eleents (see footnote below). There are also exaples, in any diension 2, and for any d 3 and d 1, of pairs (A, 1 D) that are not log canonical on an indecoposable polarized abelian variety of degree d. 2 3 The proofs Skipping the proof of log terinality, 3 the point is to show that if = 1, the ideal J (A, D) is trivial; if 2 and t Q (0, 1), the ideal I (A, td) is trivial. In both cases, let Z be the subschee of D defined by the ideal and set h = h 0 (A, L I Z P ) = h 0 (A, L a I Z ) [0, d] for P and a general in Pic 0 (A). If h = d, all sections of L contain all translates of Z, which ust be epty. or there are nonzero principally polarized abelian varieties (B 1, [Θ 1 ]) and (B 2, [Θ 2 ]), and an isogeny p : A B 1 B 2 of degree 2, such that with D 2 Θ 2. D = p (Θ 1 B 2 ) + p (B 1 D 2 ) 2 Let (A 1, [L 1 ]) be a general polarized abelian variety of type (d 1) and let E be an elliptic curve. Pick an isoorphis ψ : K(L 1 ) E[d 1] and consider the quotient A of A 1 E by the subgroup {(x, ψ(x)) x K(L 1 )}. There is a divisor Θ on A that defines a principal polarization and O A (Θ) restricts to L 1 on A 1. The line bundle L = O A (Θ+A 1 ) defines an indecoposable polarization of degree d on A that is indecoposable if d 3. The linear syste (d 1)Θ A 1 is nonepty, hence so is the linear syste Θ ( d )A 1 for d 1 > d 1. If D is in that linear syste, D = D + A 1 is in L and the pair (A, 1 D) is not log canonical since it has a coponent with ultiplicity > 1. 3 The difficulties are first to prove that the Green Lazarsfeld theory still applies in this case (this follows fro previous work of Hacon), then to prove, in the notation below, h > 0 (this follows fro the hypothesis 1 D = 0 and other work of Hacon). 4

5 Assue h = 0. In the case 2, we conclude as in the Ein Lazarsfeld proof O A (Θ) I Z = 0, which is absurd. In the case = 1, this iplies, by work of Ein and Lazarsfeld, 4 that D is fibered by nonzero abelian varieties, which is also absurd. So we assue 0 < h < d (and Z nonepty). Set J = {(s, a) PH 0 (A, L) A s Z+a 0} The fiber of a point a of A for the second projection q : J A is PH 0 (A, L I Z+a ) PH 0 (A, L a I Z ) and a unique irreducible coponent I of J doinates A. It has diension g + h 1. Let p : I PH 0 (A, L) be the first projection. The fibers F s = q ( p 1 (s) ) are either epty or of diension g + h d; they satisfy Z + F s div(s) hence di(f s ) g 1. If h = d 1 (this is the case when d = 2), F s has diension g 1 and p is surjective. The divisor of a general section s being prie, the inclusion z + F s div(s) is an equality for all z in Z. This iplies that Z is finite. If A is siple, the inclusion Z + F s div(s) iplies di Z g 1 di F s d 1 h For a general in A, the subvariety p(q 1 (a)) = PH 0 (A, L I Z+a ) of PH 0 (A, L) is a linear subspace of diension h 1. It ust vary with a, because a nonzero s does not vanish on all translates of Z. It follows that the linear span of p(i) has diension at least h. For s 1,..., s h+1 general eleents in p(i), one has 5 di(f s1 F sh+1 ) g (h + 1)(d h) g (d + 1) 2 /4 For a F s1 F sh+1, the sections s 1,..., s h+1 all vanish on Z + a, hence h 0 (A, L a I Z ) h + 1. Since the Euler characteristic χ(a, L a I Z ) is independent of a, this proves that V >0 = {P Pic 0 (A) H i (A, L I Z P ) 0 for soe i > 0} 4 By the Green Lazarsfeld theory, the cohoological locus V >0 defined below is not Pic 0 (A). If h = 0, we have V 0 Pic 0 (A) as well, so that χ(x, ω X ) = 0 for any desingularization X of D (use (1)) and this iplies that D is fibered by nonzero abelian varieties. 5 As in the projective space, subvarieties of a siple abelian variety that should eet for diensional reasons actually eet. 5

6 has diension g (d + 1) 2 /4 > 0. Now it follows fro the Green Lazarsfeld theory that in our case, the cohoological loci V i = {P Pic 0 (A) H i (A, L I Z P ) 0} have striking properties: every irreducible coponent of V i is an abelian subvariety of codiension i of Pic 0 (A) translated by a torsion point. In our case, the inequality h > 0 eans V 0 = Pic 0 (A). This iediatley finishes the proof when A is siple, since V >0 ust then be finite. When d = 2 (and h = 1), since V >0 Pic 0 (A), we get that for a general, there is an exact sequence 0 H 0 (A, L a I Z ) H 0 (A, L a ) H 0 (Z, L a Z ) 0 hence Z is a single point z. An eleent ϕ L (a) Pic 0 (A) is in V 1 if the restriction H 0 (A, L a ) H 0 (Z, L a Z ) is zero. This happens exactly when z + a is in the base locus of L, hence V 1 = ϕ L (Bs( L ) z) which has codiension 2 in A. A rather technical lea proves that for any i > 0 such that V i is nonepty, di Z i 1 + di V i, which is a contradiction. 4 Base loci of aple linear systes Not uch sees to be known about the singularities, or even the diension, of the base locus of an aple linear syste L on an abelian variety A. 4.1 Diension Let us begin with the following reark: let B be an abelian subvariety of A on which L has degree e and let C be an abelian subvariety of A such that the su orphis f : B C A is an isogeny. The base locus of L contains f(bs( L B ) C) = Bs( L B ) + C 6

7 hence has codiension e if e di(b). Hence L has degree e on an abelian = codi(bs( L )) e (2) subvariety of A of diension e It is classical and easy to show that the converse holds for e = 1: L has degree 1 on a nonzero abelian subvariety of A the polarized abelian variety (A, [L]) is decoposable with a principally polarized factor all eleents of L are reducible codi(bs( L )) = 1 I would like to conjecture that the converse holds for e = 2. 6 Conjecture 1 Let (A, [L]) be a polarized abelian variety. If Bs( L ) has codiension 2 in A, the abelian variety A contains an abelian subvariety of diension 2 on which L has degree 2. The conjecture holds in diension g 3. 7 I can also prove it if soe codiension 2 coponent of the base locus is a Cartier divisor in soe eleent of L. 6 and perhaps for any e! Note that for e 2, the property on the left-hand of (2) side does occur on soe indecoposable polarized abelian varieties. 7 We ay assue that Bs( L ) has codiension 2 and a general eleent of L is irreducible. If g = 2, the fixed part consists of at ost (L) 2 = 2d points. Since it is stable by translation by the subgroup K(L) of order d 2, we get d 2 2d. If g = 3 and d > 2, I showed 20 years ago in y thesis (p. 103) that every coponent of Bs( L ) is a translated elliptic curve in A. More precisely, for any principally polarized quotient p : (A, L) (A 0, L 0 ), either (A 0, L 0 ) is a product E 1 E 2 E 3 of principally polarized elliptic curves and one checks that, after reordering, p(k(l)) ust be contained in E 1 E 2 {0}, so that E 3 ebeds in A and L has degree 1 on it; or A 0 is the Jacobian of a bielliptic curve D that has a orphis D E of degree 2 and p(k(l)) E, so that a copleentary surface S A 0 ebeds in A and L has degree 2 on it. 7

8 One could also include in the conjecture that all codiension 2 coponents of the base locus occur as in the construction above, so that, if d > 2, they are in particular stable by translation by a nonzero abelian subvariety of A. 4.2 Singularities Let (B, [Θ]) be an indecoposable principally polarized abelian variety and let b B be a point of order 2. In all exaples that I know of, the intersection Θ Θ b is reduced. For instance, if C is a hyperelliptic curve with hyperelliptic involution τ, the intersection Θ Θ p τp = 2(W g 2 + p) is not reduced for any p C, but p τp is never of order 2. If one believes the conjecture above in its strong for, we should also have the following. Conjecture 2 Let (A, [L]) be a polarized abelian variety. Any coponent of Bs( L ) that has codiension 2 in A is (generically?) reduced. 4.3 Characterization of hyperelliptic Jacobians The following conjecture has been around for soe years. It was probably first explicitly forulated in 1977 by Beauville, who proved it, using his theory of generalized Pry varieties, in diension g 5. Conjecture 3 Let (B, [Θ]) be a principally polarized abelian variety. If Sing(Θ) has codiension 3 in B, the principally polarized abelian variety (B, [Θ]) is a hyperelliptic Jacobian. Let (A, [L]) be a polarized abelian variety of diension g such that the base locus of L has a coponent Z of codiension 2 that is not generically reduced. Let (A, [L ]) be a polarized abelian variety of the sae type and construct a principally polarized quotient π : (A, L) (A, L ) (B, Θ) (we have to check that it is indecoposable). The equation of π Θ is s i (x)s i(x ) = 0 i If x Z, we have s i (x) = 0 and rank(d j s i (x)) 1. On Z A, the singular locus of π Θ has equations i D j s i (x)s i(x ) = 0 8

9 for j {1,..., g}. For fixed x Z, this deterines a divisor in A. It follows that Sing(Θ) has codiension at ost 3 in B. On the other hand, (B, [Θ]) cannot be a hyperelliptic Jacobian: since Θ contains Z + A, the intersection Θ Θ x is reducible for all x A ; on a Jacobian, this can only happen if A has diension 1 and the curve is bielliptic, but a curve cannot be at the sae tie hyperelliptic and bielliptic. This shows that Conjecture 3 iplies Conjecture 2. In particular, Conjecture 2 holds in diension 4 for polarizations of type (d). 9