Generic vanishing index and the birationality of the bicanonical map of irregular varieties

We prove that any smooth complex projective variety with generic vanishing index bigger or equal than 2 has birational bicanonical map. Therefore, if X is a smooth complex projective variety φ with maximal Albanese dimension and non-birational bicanonical map, then the Albanese image of X is fibred by subvarieties of codimension at most 1 of an abelian subvariety of Alb X.


Introduction
In the study of smooth complex algebraic varieties, the natural maps provided by the holomorphic forms defined in the variety, have a special importance. For example, the invertible sheaf ω X of differential n-forms (where n is the dimension of X ) produces a map to a projective space, known as the canonical map. The multiples of this canonical sheaf ω ⊗m X produce in this way the pluricanonical maps ϕ ϕ m : X P N = P(H 0 (X, ω ⊗m X ) ∨ ).
When ϕ m gives a birational equivalence between X and its image, we will simply say that ϕ m is birational. We say that X is of general type if for some m > 0 the rational map ϕ m is birational.
For example, the curves of general type are those of genus g ≥ 2. The tricanonical map ϕ 3 is always birational for such curves and the bicanonical ϕ 2 is also birational once that g ≥ 3. Moreover, the canonical map is birational as soon as the curve is non-hyperelliptic.
For surfaces, Bombieri [4] has given sharp numerical conditions for the birationality of ϕ m for m ≥ 3. The bicanonical map has revealed to be more complicated and has been studied by many algebraic geometers. In fact, the surfaces with irregularity q(S) ≤ 1 and χ(S, ω S ) = 1 are not completely understood and there is no classification about which ones have birational ϕ 2 . For a modern review of the state of the art in the surface case, we refer to [2,Thm. 8].
For higher dimensions not many results are known in general. Nevertheless, the example of the bicanonical map on surfaces shows that for small irregularity q(X ) = h 0 (X, 1 X ), the classification becomes more difficult. For complex varieties, recall that the differential 1-forms give rise to the Albanese map alb : X → Alb X = H 0 (X, 1 X ) ∨ /H 1 (X, Z). from X to an abelian variety of dimension q(X ) = h 0 (X, 1 X ). We say that X is irregular if, and only if, Alb X is not trivial, i.e. q(X ) > 0. And we say that X is of maximal Albanese dimension (m.A.d) if, and only if, the Albanese map alb : X → Alb X is generically finite onto its image.
It turns out that some properties of m.A.d varieties seem to behave independently of the dimension and, indeed, Chen-Hacon showed that this is the case for their pluricanonical maps.
For ϕ 2 , we cannot expect to use χ(ω X ) to control directly its birationality. For example, if C is a curve of genus 2, then the bicanonical map of the product C × Y is never birational. In fact, it is clear that any variety that admits a fibration whose general fibre has non-birational ϕ 2 will have a non-birational bicanonical map. This should be considered, at least at first glance, as the standard case for higher dimensional varieties.
The following theorem provides geometric constraints for the non-birationality of the bicanonical map (see Theorem 5.2).
Theorem A Let X be a smooth projective complex variety of maximal Albanese dimension such that the bicanonical map is not birational. Then, the Albanese image of X is fibred by subvarieties of codimension at most 1 of an abelian subvariety of Alb X . The base of the fibration is also of maximal Albanese dimension.
That is, X admits a fibration onto a normal projective variety Y with 0 ≤ dim Y < dim X , such that any smooth modelỸ of Y is of maximal Albanese dimension and Hence, when q(X )>dim X +1 this implies the existence of an actual fibration, i.e. dim Y > 0, whose general fibre is mapped generically finite through the Albanese map of X either onto a fixed abelian subvariety of Alb X , or onto a divisor of this fixed abelian subvariety. When dim Y = 0 the theorem simply says that the image of X in Alb X has codimension at most 1.
In particular, when X does not admit any fibration and q(X ) > dim X , there is only one possible case, i.e. X is birationally equivalent to a theta-divisor of an indecomposable principally polarized abelian variety (see [1,Thm. A]). When X does not admit any fibration and q(X ) = dim X , there is only one known case of variety of general type and non-birational bicanonical map: a double cover of a principally polarized abelian variety (A, ) branched along a reduced divisor B ∈ |2 |. Is this the only case? The answer is affirmative in the case of surfaces due to Ciliberto-Mendes Lopes [7, Thm. 1.1].
To deduce Theorem A it is useful to consider the generic vanishing index introduced by Pareschi-Popa in [17,Def. 3 As a consequence of Generic Vanishing Theorem of Green-Lazarsfeld [9, Thm. 1], we have that for any irregular variety Moreover, the negative values of gv(ω X ) can be interpreted in terms of the dimension of the generic fibre of the Albanese map (see Theorem 3.7) and X is a m.A.d variety if, and only if, gv(ω X ) ≥ 0. Due to the work of Pareschi-Popa [17] we can interpret the positive values of gv(ω X ) in terms of the local properties of the Fourier-Mukai transform of the structural sheaf (see Theorem 3.3). They have also proved that the positive values of gv(ω X ) give a lower bound for the Euler characteristic χ(ω X ) (see Theorem 3.4).
Using the generic vanishing index we have the following more synthetic result.
Theorem B Let X be a smooth projective complex variety such that gv(ω X ) ≥ 2. Then, the rational map associated to ω 2 X ⊗ α is birational onto its image for every α ∈ Pic 0 X. Theorem A is deduced from this result by an argument of Pareschi-Popa. On the other hand, this result (see Theorem 5.1) is proved using a birationality criterion (see Lemma 4.2) that is a slight modification of [1,Thm. 4.13].
For curves, gv(ω C ) ≥ 2 is equivalent to g(C) ≥ 3. For surfaces, gv(ω S ) ≥ 2 is equivalent to suppose that q(S) ≥ 4 and does not admit an irregular fibration to a curve of genus ≤ q(S) − 3 (see Example 5.3).

Generalized Fourier-Mukai transform
X will be a smooth projective variety over an algebraically closed field k (from Sect. 3.3 on, we will restrict to k = C). It will be equipped with a morphism a : X → A to a non-trivial abelian variety A, in particular, X will be irregular. Let P be a Poincaré line bundle on A × Pic 0 A. We will denote the induced Poincaré line bundle in X × Pic 0 A. When a = alb, the Albanese map of X , then the map alb * identifies Pic 0 (Alb X ) to Pic 0 X and the line bundle P alb will be simply denoted by P.
Letting p and q the two projections of X × Pic 0 A, we consider the left exact functor P a F = q * ( p * F ⊗ P a ), and its right derived functors Sometimes we will have to consider the analogous derived functor R i P −1 a F as well. By the Seesaw Theorem [14, Cor. 6, p. 54],

M. Lahoz
Given a coherent sheaf F on X , its i-th cohomological support locus with respect to a is Again, when a is the Albanese map of X , we will omit the subscript, simply writing V i (F). By base change, these loci contain the set-theoretical support of A way to measure the size of all the V i a (F)'s is provided by the following invariant introduced by Pareschi-Popa.
When a is the Albanese map of X , we will omit the subscript, simply writing gv(F).
By base change (see [17,Lem. 2.1]) it is easy to see that gv a (F) can be also defined as the min i>0 codim Pic 0 A supp R i P a F − i .

Relations between gv(ω X ) and the Fourier-Mukai transform of O X
Here we specialize some general results of Pareschi-Popa [17,18] to the canonical sheaf of a smooth projective variety of dimension d. Some of these results were previously obtained by Hacon (see [11]).
The negative values of the gv-index are related with the vanishing of the lowest cohomologies of the Fourier-Mukai transform of its Grothendieck dual. In the case of ω X this can be stressed simply as: The following are equivalent, Hence, when gv a (ω X ) ≥ 0, R i P a O X = 0 for all i = d, and we usually denote Note that, in this case, The following result of Pareschi-Popa gives a dictionary between the positive values of gv a (ω X ) and the local properties of the Fourier-Mukai transform of O X .
In particular, gv a (ω X ) ≥ 1 is equivalent to O X being torsion-free and gv a (ω X ) ≥ 2 to O X being reflexive.
Using the Evans-Griffith Syzygy Theorem and the previous theorem, Pareschi-Popa obtain the following bound on the Euler holomorphic characteristic that generalizes to higher dimensions the Castelnuovo-de Franchis inequality.
Remark 3.5 In fact, the theorem of Pareschi-Popa is more general, namely that for any As a consequence, we easily obtain that for any non-zero coherent sheaf F , gv a (F) ≥ 1 ⇒ χ(F) ≥ 1. Observe also that if a is non-trivial, we always have gv a (ω X ) < ∞.

Top Fourier-Mukai transform of the canonical sheaf
In the case of abelian varieties (or complex torus) the following result is well-known and crucial in the proof of the Mukai Equivalence Theorem [13, Thm. 2.2]. We will need it in the proof of Theorem 5.1.

Generic vanishing theorem of Green-Lazarsfeld
The name of the gv-index comes from the well-known Generic Vanishing Theorem of Green-Lazarsfeld. As other general vanishing theorems, it requires char k = 0 so from now on we will restrict ourselves to the case k = C. Basically, the following theorem is Moreover gv a (ω X ) ≥ 0 if, and only if, a : X → A is generically finite onto its image.
In particular, observe that for any irregular variety Remark 3.8 If gv a (ω X ) ≥ 0 and χ(ω X ) > 0, then X is a variety of general type. Indeed, by the previous result a : X → A is generically finite and since χ(ω X ) > 0, we have that V 0 a (ω X ) = Pic 0 A, so by [5,Cor.2.4], κ(X ) = dim X . In particular, if gv a (ω X ) ≥ 1, then X is of general type.

Birationality criterion for maximal Albanese dimension varieties
In this section, we will assume that a : X → A is a generically finite morphism onto its image, where A is an abelian variety. We introduce another piece of notation. Lemma 4.2 Suppose that a : X → A is a generically finite morphism onto its image and let F be a subsheaf of a line bundle such that gv a (F) ≥ 1 and R i a * F = 0 for all i > 0. Suppose that for a general x ∈ X, Then, the rational map associated to the linear system |F ⊗ L| is birational for every line bundle L such that gv a (L) ≥ 1.
Proof We first compare the Fourier-Mukai transform of F ⊗ I x and F .
Claim Let x ∈ X be a closed point out of Z . Then R i a * (F ⊗ I x ⊗ a * α) = 0 for i > 0. This follows immediately from the exact sequence and the hypotheses that R i a * F = 0, a is generically finite and x ∈ Z . Hence, the degeneration of the Leray spectral sequence yields to By sequence (5), tensored by a * α, it follows that For i = 1 we have the surjection H 1 (F ⊗ I x ⊗ a * α) H 1 (F ⊗ a * α), that is an isomorphism if, and only if, x is not a base point of |F ⊗ a * α|. In other words V 1 for a general x ∈ X \Z . Hence by (6), (7) and (8), gv(a * (F ⊗ I x )) ≥ 1. By [15, Prop. 2.13], a * (F ⊗ I x ) is continuously globally generated (CGG, see [15]). Therefore F ⊗ I x itself is CGG outside Z (with respect to a). Since the same is true for L, it follows from [15,Prop 2.12] that for all α ∈ Pic 0 A, F ⊗ L ⊗ I x is globally generated outside Z . So the rational map associated to |F ⊗ L| is birational.

Remark 4.3
From the proof we see that if codim U F B F a (x) ≥ 2 for every x ∈ X \Z , then F ⊗ L is very ample out of Z , the exceptional locus of a.

Proposition-Definition 4.4 Let X be a variety such that gv a (ω X ) ≥ 1 and let L be any line bundle on X such that gv
and its divisorial part is dominant on X and surjects on U 0 via the projections p and q. We endow this set with the natural subscheme structure given by the image of the relative evaluation map q * (q * L) ⊗ L −1 → O X ×U 0 , where L = ( p * ω X ) ⊗ P a )| X ×U 0 and we call Y the union of its divisorial components that dominate U 0 . Let Y be its closure in X × Pic 0 X. Then (a) X is covered by the scheme-theoretic fibres of the projection Y → U 0 , that we will call F α , for α varying in U 0 . By definition, at a general point α ∈ U 0 , F α is the fixed divisor of ω X ⊗ a * α. (b) For a general x ∈ X , the fibre of the projection Y → X is a divisor, that we will call D x .
By definition, D x is the closure of the union of the divisorial components of the locus of α ∈ U 0 such that x ∈ Bs(ω X ⊗ a * α).
Proof Everything follows from taking F = ω X in Lemma 4.2. The surjectivity of the projection to U 0 is consequence of the Castelnuovo-de Franchis inequality 3.4, i.e. χ(ω X ) ≥ gv a (ω X ) ≥ 1.

Decomposition
In the sequel we will need a * : Pic 0 A → Pic 0 X to be an embedding. However, for simplicity we will go one step further and we will simply suppose that A = Alb X . Suppose that we are under the hypotheses of the previous Proposition-Definition and consider a fixed point α 0 ∈ U 0 , and the map where F α is the divisor defined in Proposition-Definition 4.4(a). For α ∈ U 0 , all the F α are algebraically equivalent since they are the fibres of Y → U 0 , so the map is well-defined.
The following lemma shows that this map induces a decomposition of Pic 0 X and that the divisors F α move algebraically along a non-trivial factor of Pic 0 X . Although the proof is basically the same as [1, Lem. 5.1], we do not require V 1 (ω X ) to be a finite set, but only a proper subvariety. (10), induces an homomorphism f : Pic 0 X → Pic 0 X such that,

Lemma 4.5 The map defined in
(a) f 2 = f and Pic 0 X decomposes as Then, the proof of (a) is the same as [1, Using the decomposition given by the previous Lemma we give an explicit description of the the "half" Poincaré line bundle.
wherex is such that alb(x) = 0 in Alb X and P C is the Poincaré line bundle in C × Pic 0 C.
Proof The decomposition of Pic 0 X comes directly from Lemma 4.5(a). By the definition of Y (see Proposition-Definition 4.4) and the definition of F (see Lemma 4.5(b)) we have that the line bundle On the other hand, (alb × id Pic 0 X ) * (O B×Pic 0 B P C ), -restricted to X × {β ⊗ γ } is isomorphic to γ , for all (β, γ ) ∈ ker f × ker(id − f ); -restricted to {x} × Pic 0 X is isomorphic to O Pic 0 X , i.e. trivial.
Then, the Lemma follows from the seesaw principle.

The bicanonical map of irregular varieties
The next theorem gives a sufficient numerical condition for the birationality of the bicanonical map, analogous to Pareschi-Popa Theorem [16,Thm. 6.1] for the tricanonical map.
Theorem 5.1 Let X be a smooth projective complex variety such that gv(ω X ) ≥ 2. Then, the rational map associated to ω 2 X ⊗ α is birational onto its image for every α ∈ Pic 0 X. As a first corollary we have the following geometric interpretation.
Theorem 5.2 Let X be a smooth projective complex variety of maximal Albanese dimension such that the bicanonical map is not birational. Then 0 ≤ gv(ω X ) ≤ 1. Moreover, it admits a fibration onto a normal projective variety Y with 0 ≤ dim Y < dim X , any smooth model Y of Y is of maximal Albanese dimension and Proof By Theorems 3.7 and 5.1, it is clear that 0 ≤ gv(ω X ) ≤ 1. Now, the proof is the same as the proof of [17,Thm. B].
Example 5. 3 We would like to show examples of varieties with gv(ω X ) ≥ 2. For curves C, this is equivalent to g(C) ≥ 3. For surfaces S, is equivalent to suppose that q(S) ≥ 4 and S does not admit an irregular fibration to a curve of genus ≤ q(S) − 3 (see [3,Cor. 2.3]).
On the other hand, if A is a simple abelian variety, then every subvariety X of codimension ≥ 2 has gv(ω X ) ≥ 2. Moreover, the property of having gv(ω X ) ≥ 2 is closed under taking products and cyclic coverings induced by a torsion point α ∈ Pic 0 X − V 1 (ω X ).
The rest of the paper is devoted to the proof of Theorem 5.1.
Proof Assume that gv(ω X ) ≥ 1 and there exists α ∈ Pic 0 X such that ω ⊗2 X ⊗ α is non-birational. Then, we want to see that gv(ω X ) = 1. Under these hypotheses we can apply Proposition-Definition 4.4 and Lemma 4.6, so Alb X ∼ = B × C, where B = Pic 0 (ker(id − f )) and C = Pic 0 (ker f ). We have the following commutative diagram where -p b : Alb X → B and pb : Pic 0 X → Pic 0 B are the corresponding projections, → B, and -abusing notation we also call q either the projection X ×Pic 0 X → Pic 0 X or X ×Pic 0 B → Pic 0 B and p the projections X × Pic 0 X → X or X × Pic 0 B → X .
The effectiveness of Y give us the following short exact sequence on X × Pic 0 X Recall that P = (alb × id Pic 0 X ) * (P B P C ) since the Poincaré line bundle P in Alb X × Pic 0 X is isomorphic to P B P C . We apply the functor R d q * ( · ⊗ (alb × id Pic 0 X ) * (P −1

B
O C×Pic 0 C )), that is, we tensor by the other "half" Poincaré line bundle and we consider the top direct image. We get (3)), we have the following short exact sequence, where: following the notation of (1) and (2).
O C×Pic 0 C )) is supported at the locus of the α ∈ Pic 0 X such that the fibre of the projection q : Y → Pic 0 X has dimension d, i.e. it coincides with X . Such locus is contained in V 1 (ω X ), therefore, since gv(ω X ) ≥ 1, codim supp T ≥ 2. (c) The map μ is injective since it is a generically surjective map of sheaves of the same rank (recall that rk O X = χ(ω X )), and, as gv(ω X ) ≥ 1, the source O X is torsion-free Let τ (E(Dx )) be the torsion part of E(Dx ) and E(Dx ) the quotient of E(Dx ) by its torsion part. Hence E(Dx ) is torsion-free. Now consider the following composition Sinceμ is generically surjective and (−1) * Pic 0 X O X is torsion-free (recall that gv(ω X ) ≥ 1), we have thatμ is injective. Completing the diagram we get, If T = 0, then the middle horizontal short exact sequence splits. But, for α a closed point in the support of T (by the previous claim we know that T = 0), μ ⊗ C(α) = 0 by item (d), so μ cannot split. Therefore T = 0. Let e = codim Pic 0 X supp T ≥ 2 (see item (c)). Then codim Pic 0 X supp Ext e ( T , O Pic 0 X ) = e. Now, we apply the functor Ext i ( · , O Pic 0 X ) to the bottom row of (13) using Corollary 3. Since E(Dx ) is torsion-free, codim Pic 0 X supp Ext e ( E(Dx ), O Pic 0 X ) > e. Therefore, we must have codim Pic 0 X supp R e−1 P ω X = e and gv(ω X ) ≤ 1.