ON THE BIRATIONAL GEOMETRY OF IRREGULAR VARIETIES

CHRISTOPHER D. HACON

Abstract. The purpose of this course is to illustrate how the theory of Fourier Mukai transforms can be used to give surprisingly precise results in the birational classification of irregular varieties.

1. Introduction

Let X be a smooth complex projective variety and ω_{X} the canonical line bundle. We will denote by

• P_{m}(X) := h^{0}(ω_{X}^{⊗m}) the m-th plurigenus of X.

• q(X) := h^{0}(Ω_{X}^{1}) the irregularity of X.

These numbers are birational invariants of the variety X. (Recall that two varieties X, X′ are birational if and only if they have some isomorphic open subset U ≅ U′.) They are interesting because when they are positive, X comes equipped with a natural algebraic map.

If P_{m}(X) > 0, then the sections of H^{0}(X, ω_{X}^{⊗m}) define a rational map

φ_{X}^{m} : X −−→ ℙ (H^{0}(X, ω_{X}^{⊗m})) .

Recall that the Kodaira dimension

κ(X) ∈ {−∞, 0, 1, ..., dim(X)},

is defined as follows: if P_{m}(X) = 0 for all m > 0 we set κ(X) = −∞, and otherwise κ(X) is the maximum of the dimensions of the images of φ_{X}^{m} for all m > 0. This is also a birational invariant of X.

If q(X) > 0, then by integrating the holomorphic one forms ω ∈ H^{0}(X, Ω_{X}^{1}), one obtains a morphism

a : X ⟶ A(X) := \frac{H^{0}(X, Ω_{X}^{1})^{∨}}{H_{1}(X, ℤ)} .

The dimension of the image of a is the Albanese dimension of X. The Albanese variety A(X) is an abelian variety i.e. a complex projective torus. Its dual abelian variety

{ \atop A(X)} = Pic^{0}(A) ≅ Pic^{0}(X)

parametrises the topologically trivial line bundles on X and A.

One of the standard questions in Algebraic geometry is to study the geometry of these natural maps and their interaction. For example one may ask:

Question: If X is of general type (i.e. κ(X) = dim(X)), then what is the smallest value of m for which φ_{X}^{m} is a birational map?

If dim(X) = 1, 2 then m = 3, 5 will suffice. However, when dim(X) ≥ 3 any explicit estimate seems extremely hard to obtain. In contrast to this, it is shown in [CH3] that for varieties with maximal Albanese dimension, m = 6 always suffices (regardless of the dimension of X!).

Ideally, one could even hope to characterize (the birational equivalence class of) a projective variety X by its birational invariants. For example recall the following result for surfaces:

Castelnuovo’s Criterion: If P_{2}(X) = q(X) = 0 then X is a rational surface.

Enriques: If P_{2}(X) = 1 and q(X) = 2, then X is birational to an abelian variety.

Classification results of this kind are extremely rare, however there is a famous result of Kawamata that generalizes Enriques’ result to higher dimensions:

Theorem 1.1. [Ka1] If X is a smooth complex projective variety with κ(X) = 0 then its Albanese morphism is surjective (in particular q(X) ≤ dim(X)). If also q(X) = dim(X), then X is birational to an abelian variety.

This is somewhat unsatisfactory as the Kodaira dimension is not a naive invariant, it depends on P_{m}(X) for all m > 0. Kollár has proved an effective version of this result: If X is a smooth complex projective variety with P_{m}(X) = 1 for some m ≥ 3, then its Albanese map is surjective. If P_{m}(X) = 1 for some m ≥ 4 and q(X) = dim(X), then X is birational to an abelian variety. In [CH1], Kollár’s result was refined as follows:

Chen-Hacon: If P_{m}(X) = 1, for some m ≥ 2, then the Albanese map a : X ⟶ A is surjective. If also dim(X) = q(X) then X is birational to an abelian variety.

Recently it has been shown that if P_{3}(X) = 2 (resp. P_{3}(X) = 3) and q(X) = dim(X), then X is a double (resp. bi-double) cover of an abelian variety cf. [HP1], [Hac2].

The main techniques used in obtaining these kinds of results are:

• The theory of Fourier-Mukai transforms [Mu] which characterizes coherent sheaves F on an abelian variety A, in terms of the cohomology groups H^{i}(A, F⊗P) with P ∈ Pic^{0}(A). (Actually a sheaf theoretic version of this statement is necessary.)

• The Generic Vanishing Theorems of Green and Lazarsfeld, which for example imply that if X is a smooth projective variety of maximal Albanese dimension, then χ(ω_{X}) ≥ 0.

• The vanishing theorems of Nadel, Kawamata, Viehweg and the results on higher direct images of dualizing sheaves due to Kollár.

2. Sheaves on abelian varieties

Let A be a g-dimensional abelian variety and Â = Pic^{0}(A) its dual abelian variety. We will denote by p_{A}, pÂ the projections of A × Â on to A, Â. Let P be the normalized Poincaré line bundle on A × Â. So for â ∈ Â, the line bundle P|A × â is just the topologically trivial line bundle Pâ corresponding to the point â ∈ Pic^{0}(A) = Â. The main tool in understanding sheaves on an abelian variety is the Theory of Fourier-Mukai transforms. Recall that Mukai defines the functor S_{A} : D_{qc}^{b}(A) ⟶ D_{qc}^{b}(Â) by

S_{A}(F) = Rp_{Â,∗}(P⊗^{L}p_{A}^{∗}M).

Similarly one defines SÂ : D_{qc}^{b}(Â) ⟶ D_{qc}^{b}(A). Here D_{qc}^{b}(A) denotes the derived category of quasi-coherent sheaves on A with bounded cohomology. We will mostly just be interested in the case where F is a sheaf. In this case, for general â ∈ Â, one has that H^{i}(S_{A}(F))⊗ℂ(â) ≅ H^{i}(F ⊗Pâ). By [Mu], one has:

Theorem 2.1. (Mukai) There exists isomorphisms of functors

S_{A} ∘ SÂ ≅ (−1Â)^{∗}[−g]

and

SÂ ∘ S_{A} ≅ (−1_{A})^{∗}[−g].

Here [−g] means: shift the corresponding complex g spaces to the right. Loosely speaking this means that one can recover a sheaf F from its cohomology groups H^{i}(F ⊗P).

Example 2.2. If H^{i}(A, F ⊗P) = 0 for 0 ≤ i ≤ g and all P ∈ Pic^{0}(A), then by S_{A}(F) = 0 and so F = 0. Similarly, if ϕ : F ⟶ G is an inclusion of sheaves inducing isomorphisms H^{i}(F ⊗P) ⟶ H^{i}(G⊗P) for 0 ≤ i ≤ g and all P ∈ Pic^{0}(A), then F ≅ G.

Example 2.3. Let F = ℂ(â) for some point â ∈ Â, then SÂ(F) = Pâ (i.e. the complex given by F^{i} = 0 for i ≠ 0 and F^{0} = Pâ). Therefore, S_{A}(Pâ) = ℂ(â)[−g], in particular R^{i}p_{Â,∗}(p_{A}^{∗}O_{A}⊗P) = 0 for i ≠ g and R^{g}p_{Â,∗}(p_{A}^{∗}O_{A}⊗P) = ℂ(0Â).

Example 2.4. (Mukai) A vector bundle U on A is unipotent if there exists a filtration

0 = U_{0} ⊂ U_{1} ⊂ ... ⊂ U_{n−1} ⊂ U_{n} = U

such that U_{i}/U_{i−1} ≅ O_{A} for all 1 ≤ i ≤ n. One has that RŜ(U) = R^{g}Ŝ(U) and R^{g}Ŝ(U) is supported on ˆ0 ∈ Â. This gives an equivalence between the category of unipotent vector bundles on A and the category of coherent sheaves on Â supported on ˆ0 i.e. the category of Artinian O_{Â,ˆ0}-modules.

Example 2.5. Let F be a coherent sheaf on A such that for all P ∈ Pic^{0}(A) one has

h^{0}(F⊗P) = 1, and h^{i}(F⊗P) = 0 ∀ i > 0.

Then by cohomology and base change L := S_{A}(F) = H^{0}S_{A}(F) is a line bundle on Â. Moreover, one has that SÂ(L) = H^{g}SÂ(L) and hence h^{g}(L⊗P_{x}) ≠ 0 for some x ∈ A. Line bundles on an abelian variety are particularly well understood. One sees that L is negative semidefinite (i.e. L^{∨} is nef). It follows that there exists a (surjective with connected fibers) homomorphism of abelian varieties p : Â ⟶ B and a non-degenerate line bundle M on B such that L⊗P_{x} = p^{∗}M. In particular h^{g}(L⊗P_{x}⊗Q) = 0 for all Q ∉ B̂ and therefore, F is supported on the abelian subvariety B̂ ↪ A.

If the generic rank of F is 1, then h^{g}(L⊗P) = 1 for general P ∈ Pic^{0}(Â) = A and so L is dual to a principal polarization and hence F is a principal polarization (and in particular a line bundle).

3. Generic Vanishing Theorems

For X a smooth complex projective variety (or more generally a Kähler manifold), one can define cohomological support loci

V^{i}(ω_{X}) := {P ∈ Pic^{0}(X)|h^{i}(ω_{X}⊗P) ≠ 0}.

In view of the Fourier-Mukai transform, one would expect the V^{i}(ω_{X}) to “determine” the sheaves a_{∗}(ω_{X}), so it is not surprising that these loci are very important in the study of irregular varieties. Their geometry is governed by the following result of Green and Lazarsfeld [GL]

Theorem 3.1. (Generic Vanishing Theorem)

(1) Any irreducible component of V^{i}(X, ω_{X}) is a translate^{1} of an abelian subvariety of Pic^{0}(X) and is of codimension at least i − (dim(X) − dim(a_{X}(X))).

(2) Let P ∈ T be a general point of an irreducible component T of V^{i}(X, ω_{X}). Suppose that v ∈ H^{1}(X, O_{X}) ≅ T_{P} Pic^{0}(X) is not tangent to T. Then the sequence

H^{i−1}(X, ω_{X} ⊗ P) ⟶^{∪v} H^{i}(X, ω_{X} ⊗ P) ⟶^{∪v} H^{i+1}(X, ω_{X} ⊗ P)

is exact. If v is tangent to T, then the maps in the above sequence vanish. (3) If X is a variety of maximal Albanese dimension, then

Pic^{0}(X) ⊃ V^{0}(X, ω_{X}) ⊃ V^{1}(X, ω_{X}) ⊃ ... ⊃ V^{n}(X, ω_{X}) = {O_{X}}.

The above result is even true for Kähler manifolds and its proof depends on Hodge theory. The idea behind this result is that the cohomology of the above complex measures the first order deformations of cohomology classes in H^{i}(X, ω_{X}), and that if a class deforms to first order, then all higher obstructions automatically vanish.

From (1), one sees that if X has maximal Albanese dimension, then χ(ω_{X}) ≥ 0. If X is a subvariety of general type of an abelian variety, then it has been shown that χ(ω_{X}) > 0. It was conjectured by Kollár that (as in the case of surfaces of general type), if X is of maximal Albanese dimension and general type, then χ(ω_{X}) > 0 and P_{1}(X) ≥ 2. This is however not the case (cf. [EL], [CH4]).

Green and Lazarsfeld also conjecture the following statement which is closely related to their generic vanishing theorems (cf. [GL]):

Conjecture: Let X be a smooth complex projective variety with maximal Albanese dimension, then for the universal family of topologically trivial line bundles L ⟶ X × Pic^{0}(X), one has that

R^{i}π_{Pic^{0}(X),∗}(L) = 0

for all i < dim(X).

In section 5, we will show that this is true, and if X is not of maximal Albanese dimension, an appropriate statement still follows.

1If X is (deformation equivalent to) a projective manifold, then Simpson has shown that: Every irreducible component of V^{i}(X, ω_{X}) is a torsion translate of an abelian subvariety of Pic^{0}(X).

4. A derived category approach to generic vanishing theorems

Let L be an ample vector bundle on Â. By [Mu] Proposition 3.11, L̂ := H^{0}S_{A}(L) ≅ S_{A}(L) is a vector bundle on A of rank h^{0}(L). Moreover,

ϕ_{L}^{∗} \binom{}{L̂^{∨}} ≅ (L)^{⊕h^{0}(L)} .

(Recall that the isogeny ϕ_{L} : Â ⟶ A is defined by ϕ_{L}(â) = t∗ ; âL^{∨}⊗L.)

Theorem 4.1. [Hac4] Let F be a coherent sheaf on an abelian variety A. The following are equivalent:

(1) For any sufficiently ample line bundle L on Â,

H^{i}(A, F⊗L̂^{∨}) = 0 ∀ i > 0;

(2) There is an isomorphism

S_{A}D_{A}(F) ≅ H^{0} (S_{A}D_{A}(F)) .

Recall that for an n-dimensional variety X, the dualizing functor is defined by D_{X}(?) = RHom(?, ω_{X}^{.}[n]) where ω_{X}^{.} is the dualizing complex of X. Therefore, D_{A} = RHom(?, O_{A}[g]).

Proof. By Grothendieck Duality (G.D.)^{2} and the projection formula (P.F.)^{3} it follows that

D_{k}(RΓ(F⊗L̂^{∨})) ≅^{G.D.} RΓ(D_{A}(F⊗L̂^{∨})) ≅

RΓ(D_{A}(F)⊗L̂) ≅ RΓ(D_{A}(F)⊗p_{A,∗}(P⊗p∗ ; ÂL)) ≅^{P.F.}

RΓ(Lp_{A}^{∗}D_{A}(F)⊗P⊗p∗ ; ÂL) ≅^{P.F.} RΓ(S_{A}(D_{A}(F))⊗L).

In particular, D_{k}(RΓ(F⊗L̂^{∨})) is a sheaf if and only if RΓ(S_{A}(D_{A}(F))⊗L) is a sheaf. The theorem now follows from the following remarks:

(1) Notice that there is an isomorphism

H^{0}(A, F⊗L̂^{∨})^{∨} ≅ D_{k}(RΓ(F⊗L̂^{∨}))

if and only if

H^{i}(A, F⊗L̂^{∨}) = 0 for all i > 0.

(2) For L sufficiently ample, the coherent sheaves H^{j}S_{A}(D_{A}(F))⊗L are globally generated and have vanishing higher cohomology. Consider the spectral sequence

E_{2}^{i,j} = R^{i}Γ(H^{j}S_{A}(D_{A}(F))⊗L) ⇒ R^{i+j}Γ(S_{A}(D_{A}(F))⊗L).

Since E_{2}^{i,j} = 0 for all i ≠ 0, this sequence degenerates at the E_{2} level and hence

H^{0}(Â, H^{j}S_{A}(D_{A}(F))⊗L) ≅ R^{j}Γ(S_{A}(D_{A}(F))⊗L).

Therefore,

RΓ(S_{A}(D_{A}(F))⊗L) ≅ R^{0}Γ(S_{A}(D_{A}(F))⊗L)

2(G.D.) Let f : X ⟶ Y be a morphism of projective varieties, F ∈ D_{qc}^{b}(X), then Rf_{∗}D_{X}(F) = D_{Y} (Rf_{∗}(F)). In particular one has RΓ(D_{X}F) ≅ D_{k}(RΓ(F)) where RΓ = Rγ_{∗} and γ : X ⟶ Spec(k) is the structure morphism.

3(P.F.) Let f : X ⟶ Y be a proper morphism of quasi-projective varieties, then there is an isomorphism Rf_{∗}(F)⊗G ⟶ Rf_{∗}(F ⊗Lf^{∗}G).

if and only if

S_{A}(D_{A}(F)) ≅ H^{0}S_{A}(D_{A}(F)).
□

Corollary 4.2. Let F be a coherent sheaf as above. For all i > 0, the support of H^{i}S_{A}(F) has codimension at least i in Â.

Proof. Since DÂ ∘ S_{A} ≅ (−1_{A})^{∗} ∘ SÂ ∘ D_{A}[g] (cf. [Mu] (3.8)), one has that

S_{A}(F) ≅ DÂ ((−1∗ ; Â ∘ S_{A} ∘ D_{A})(F)[g]) ≅ RHom (G, OÂ) ,

where G := −1∗ ; Â(S_{A}(D_{A}(F))) is a sheaf. It suffices therefore to show that for any coherent sheaf G on a smooth affine variety Y = Spec(A), one has that the sheaves R^{i}Hom(G, O_{Y}) are supported in codimension at least i. Let M = Γ(Y, G) and W be an irreducible component of the support of R^{i}Hom(G, O_{Y}) = Ext^{i}(M, A)^{∽}. Let P ∈ Spec(A) such that W = Spec(A/P). Since localization is exact, one has that

0 ≠ Ext^{i}(M, A)⊗A_{P} ≅ Ext^{i}(M_{P} , A_{P}).

Since A is regular, it follows that A_{P} is regular and hence that

i ≤ dim A_{P} = dim(Y) − dim(W)

(cf. [Ha1] §III.6). □

Corollary 4.3. Let F be a coherent sheaf as above, and P ∈ Â, then: (1) If i ≥ 0 and H^{i}(A, F⊗P) = 0, then H^{i+1}(A, F⊗P) = 0.

(2) For all i > 0 any irreducible component of the loci

V^{i}(F) := {P ∈ Pic^{0}(A) | h^{i}(F⊗P) ≠ 0}

has codimension at least i in Â.

(3) If H^{0}(A, F⊗P) = 0 then H^{0}S_{A}(D_{A}(F))⊗k(P^{∨}) = 0.

(4) If H^{0}(A, F⊗P) = 0 for all P ≠ O_{A}, then F is a unipotent vector bundle.

Proof. (1) Since

0 = H^{i}(A, F⊗P)^{∨} ≅ H^{−i}(D_{ℂ}RΓ(F⊗P)) ≅ R^{−i}Γ(D_{A}(F)⊗P^{∨}),

by Cohomology and Base Change, one sees that the natural homomorphism

0 = H^{−i−1}S_{A}(D_{A}(F))⊗k(P^{∨}) ⟶ R^{−i−1}Γ(D_{A}(F)⊗P^{∨})

is surjective and hence that H^{i+1}(A, F⊗P) = 0.

(2) Let W be an irreducible component of V^{i}(F) and P a general point in W. Let j be the largest integer such that W ⊂ V^{j}(F). By Cohomology and Base Change (for sheaves), one has that H^{j}S_{A}(F)⊗k(P) ≠ 0 and by the previous corollary, one has that W has codimension at least j.

(3) This follows immediately from Cohomology and Base Change.

(4) By (3), we have that H^{0}S_{A}(D_{A}(F)) is supported on the closed point ˆ0 ∈ Â corresponding to O_{A}. By Example 2.4, H^{0}S_{A}(D_{A}(F)) is an Artinian module and hence

D_{A}(F) ≅ (−1_{A})^{∗}SÂS_{A}(D_{A}(F))[−g]

is (the shift of) a unipotent vector bundle say V. Finally it is easy to see that

F = D_{A}(D_{A}(F)) = V^{∨}

is also a unipotent vector bundle. □

5. A conjecture of Green and Lazarsfeld

We will need the following results about higher direct images of dualizing sheaves:

Theorem 5.1. ([Ko1], [Ko2]) Let X, Y be complex projective varieties of dimension n, k with X smooth. Let f : X ⟶ Y be a surjective map and L an ample line bundle on Y. Then

(1) R^{i}f_{∗}ω_{X} is torsion free for i ≥ 0,

(2) H^{j}(Y, L⊗R^{i}f_{∗}ω_{X}) = 0 for j > 0,

(3) Rf_{∗}ω_{X} ≅ ∑ R^{i}f_{∗}ω_{X}[−i],

(4) Rf_{∗}O_{X} ≅ ∑^{d}R^{i}f_{∗}O_{X}, where ^{d}R^{i}f_{∗}O_{X} = D_{Y} (R^{n−k−i}f_{∗}ω_{X}[k + i]).

For future reference, we also recall the following vanishing theorem (cf. [Ko3] §10).

Theorem 5.2. Let f: X → Y be a surjective map of projective varieties, X smooth, Y normal. Let M be a line bundle on X such that M ≡ N +f^{∗}L+Δ, where N is a ℚ-divisor on X which is either nef and big or numerically trivial, L is a ℚ−divisor on Y and Δ has normal crossing support with ⌊Δ⌋ = 0. Then

• R^{j}f_{∗}(ω_{X}⊗M) is torsion free for j ≥ 0;

• If L is nef and big, and g : Y ⟶ Z is any morphism with Z projective, then H^{i}(Y, R^{j}f_{∗}(ω_{X}⊗M)) = 0 for i > 0, j ≥ 0 and

H^{i}(Y, R^{j}f_{∗}(K_{X} + M)) = H^{i}(Z, g_{∗}(R^{j}f_{∗}(K_{X} + M))), i, j ≥ 0.

When X is a smooth complex projective variety and a : X ⟶ A is the Albanese morphism, the results of the previous section allow us to recover a generalization of the conjecture of Green and Lazarsfeld mentioned above. To fix the notation, let L = (a, idÂ)^{∗}P and let πÂ denote the projection of X × Â onto Â.

Theorem 5.3. [Hac4] Let X be a smooth complex projective variety of dimension n, and Albanese dimension k := dim a(X). Then

Rπ_{Â,∗}(L) ≅ ∑_{i=0}^{n−k} H^{k+i}S_{A}(^{d}R^{i}a_{∗}O_{X})[−k − i].

In particular, when X is of maximal Albanese dimension (i.e. k = n), one has that Rπ_{Â,∗}(L) ≅ H^{n}S_{A}(^{d}R^{0}a_{∗}O_{X}) is a sheaf.

Proof. Let Y := a(X). We have that

Rπ_{Â,∗}(L) ≅ Rπ_{Â,∗}((a × idÂ)^{∗}P) ≅^{P.F.} Rp_{Â,∗}(Lp_{A}^{∗}(Ra_{∗}O_{X})⊗P) ≅

Rp_{Â,∗}(Lp_{A}^{∗}(∑_{i=0}^{n−k} ^{d}R^{i}a_{∗}O_{X})⊗P) ≅ ∑_{i=0}^{n−k} S_{A}(^{d}R^{i}a_{∗}O_{X}).

Since

dR^{i}a_{∗}O_{X} ≅ D_{Y} (R^{n−k−i}a_{∗}ω_{X}[k + i]) ≅ D_{A}(R^{n−k−i}a_{∗}ω_{X}[k + i])

it suffices to show that the sheaves F^{n−k−i} := R^{n−k−i}a_{∗}ω_{X} satisfy the hypothesis of Theorem 4.1. This is however an immediate consequence of Theorem 5.1 and the observations that follow:

Let F = F^{n−k−i}. As mentioned in the section 4, L̂ := H^{0}S_{A}(L) ≅ S_{A}(L) is a vector bundle on A of rank h^{0}(L) and

ϕ_{L}^{∗} \binom{}{L̂^{∨}} ≅ (L)^{⊕h^{0}(L)} .

We have that H^{i}(A, F⊗L̂^{∨}) is a direct summand of

H^{i}(A, ϕ_{L,∗}(OÂ)⊗F⊗L̂^{∨}) ≅^{P.F.} H^{i}(Â, ϕ_{L}^{∗}(F⊗L̂^{∨})) ≅ ⨁_{i=1}^{h^{0}(L)} H^{i}(Â, ϕ_{L}^{∗}(F)⊗L).

So it suffices to show that H^{i}(Â, ϕ_{L}^{∗}(F)⊗L) = 0 for all i > 0. Let X′ := X ×_{A} Â and a′ : X′ ⟶ Â be the induced morphism. By flat base change, ϕ_{L}^{∗}(F) ≅ R^{n−k−i}a′_{∗}ω_{X′}, and so by Theorem 5.1, we have the required vanishing:

h^{i}(Â, R^{n−k−i}a′_{∗}ω_{X′}⊗L) = 0 for all i > 0.
□

Corollary 5.4. Let a : X ⟶ A be as in the preceding theorem, then

(1) Every irreducible component of the loci V^{i}(R^{j}a_{∗}ω_{X}) has codimension at least i in Â.

(2) For all i < dim a(X) and general P ∈ Pic^{0}(X), one has that H^{i}(X, P) = 0.

(3) If H^{j}(R^{i}a_{∗}ω_{X}⊗P) = 0, then H^{l}(R^{i}a_{∗}ω_{X}⊗P) = 0 for all l > j ≥ 0 and 0 ≤ i ≤ k.

(4) If H^{0}(R^{i}a_{∗}ω_{X}⊗P) = 0 for all P ≠ O_{A}, then R^{i}a_{∗}ω_{X} is a unipotent vector bundle.

Proof. This is analogous to Corollary 4.3. □

Corollary 5.5. If κ(X) = 0, then the Albanese map a : X ⟶ A is surjective.

Proof. By a result of Fujita, there is a generically finite cover X̃ ⟶ X with κ(X̃) = 0 and P_{1}(X̃) = 1. We claim that since κ(X̃) = 0, then h^{0}(KX̃⊗P) = 0 for all P ≠ OX̃. If this where not the case, then by Simpson’s resulf (cf. Theorem 3.1), one can assume that h^{0}(KX̃⊗P) ≠ 0 for some torsion element P ∈ Pic^{0}(X). Pick a divisor D ∈ |K_{X} + P |, then for sufficiently divisible t, one has that tD ∈ |t(K_{X} + P)| = |tK_{X}| and hence h^{0}(2K_{X}) > 1 which is impossible. By the above corollary, a_{X̃,∗}ωX̃ is a unipotent vector bundle on A(X̃) and hence aX̃ : X̃ ⟶ A(X̃) is surjective. It follows easily that X ⟶ A(X) is also surjective. □

6. Effective criteria for birational morphisms

Throughout this section, f : X ⟶ Y denotes a surjective morphism of smooth complex projective varieties, with dim(X) = dim(Y) = n. Recall that X is of general type if κ(X) = dim(X).

Theorem 6.1. (Hacon-Pardini) [HP1] If Y is of general type and maximal Albanese dimension, then f : X ⟶ Y is birational if and only if P_{m}(X) = P_{m}(Y) for some m ≥ 2.

Proof. Assume for simplicity that m = 2. The Albanese morphism a_{Y} : Y ⟶ A(Y) is generically finite onto its image. We would like to compare a_{Y,∗}(f_{∗}ω_{X}^{⊗2}) and a_{Y,∗}(ω_{Y}^{⊗2}). If these are generically isomorphic, then the generic degree of f is 1. The first step is to extract the positive part of ω_{Y} , ω_{X}.

Let H be any ample divisor on A(Y), for some t ≫ 0, one has that |tK_{Y} − a_{Y}^{∗} H| is nonempty. Fix a general divisor

B_{Y} ∈ |tK_{Y} − a_{Y}^{∗} H|

and let B_{X} := f^{∗}B_{Y} + tK_{X/Y}. After replacing X, Y by appropriate birational models, we may assume that B_{X}, B_{Y} have simple normal crossings support. Let

L_{Y} := K_{Y} − ⌊\frac{B_{Y}}{t} ⌋ ≅ \frac{1}{t}a_{Y}^{∗} H + \{\frac{B_{Y}}{t} \},

L_{X} := K_{X} − ⌊\frac{B_{X}}{t} ⌋ ≅ \frac{1}{t}f^{∗}a_{Y}^{∗} H + \{\frac{f^{∗}B_{Y}}{t} \}.

Claim 1. For t ≫ 0, one has that |2K_{Y} | = |K_{Y} + L_{Y} | + ⌊\frac{B_{Y}}{t} ⌋.

Claim 2. The divisor K_{X/Y} + f^{∗}(⌊B_{Y} /t⌋) − ⌊f^{∗}B_{Y} /t⌋ is effective.

Grant these for the time being. By Claim 2, there is an inclusion

f^{∗}(ω_{Y} ⊗L_{Y}) ⟶ ω_{X}⊗L_{X}

and hence an inclusion

i : a_{Y,∗}(ω_{Y} ⊗L_{Y}) ⟶ a_{Y,∗}f_{∗}(ω_{X}⊗L_{X}).

By Theorem 5.2, one has that for all i > 0 and P ∈ Pic^{0}(Y)

H^{i}(A(Y), a_{Y,∗}(ω_{Y} ⊗L_{Y})⊗P) = H^{i}(A(Y), a_{Y,∗}f_{∗}(ω_{X}⊗L_{X})⊗P) = 0.

We also have that

h^{0}(ω_{Y} ⊗L_{Y}) = P_{2}(Y) = P_{2}(X) ≥ h^{0}(ω_{X}⊗L_{X}) ≥ h^{0}(ω_{Y} ⊗L_{Y})

and therefore for all P ∈ Pic^{0}(Y)

h^{0}(a_{Y,∗}(ω_{Y} ⊗L_{Y})⊗P) = χ(a_{Y,∗}(ω_{Y} ⊗L_{Y})⊗P) =
χ(a_{Y,∗}(ω_{Y} ⊗L_{Y})) = h^{0}(a_{Y,∗}(ω_{Y} ⊗L_{Y})) =
h^{0}(a_{Y,∗}f_{∗}(ω_{X}⊗L_{X})) = χ(a_{Y,∗}f_{∗}(ω_{X}⊗L_{X})) =
χ(a_{Y,∗}f_{∗}(ω_{X}⊗L_{X})⊗P) = h^{0}(a_{Y,∗}f_{∗}(ω_{X}⊗L_{X})⊗P).

So the inclusion i induces isomorphisms in cohomology for all i ≥ 0 and all P ∈ Pic^{0}(Y) and hence

a_{Y,∗}(ω_{Y} ⊗L_{Y}) ≅ a_{Y,∗}f_{∗}(ω_{X}⊗L_{X}).

In particular the generic degrees of a_{Y} , a_{Y} ∘ f coincide and hence f is birational. We now turn to the proof of the above claims.

Proof of Claim 1. After replacing Y by an appropriate birational model, we may assume that

|2K_{Y} | = |M| + F

where |M| is free and F + B_{Y} has simple normal crossings support. For l ≫ 0 let G ∈ |lM| be general. Consider

B′_{Y} := B_{Y} + lF + G ∈ |(t + 2l)K_{Y} − a_{Y}^{∗} H|.

We may write B_{Y} = ∑ b_{i}B_{i} and F = ∑ f_{i}B_{i} for distinct reduced irreducible divisors B_{i} on Y. We have that for l ≫ 0, the multiplicity of B′_{Y} along B_{i} satisfies

⌊\frac{mult_{B_{i}}(B′_{Y})}{t + 2l}⌋ = ⌊\frac{b_{i} + lf_{i}}{t + 2l} ⌋ ≤ ⌊\frac{b_{i}}{2l} + \frac{f_{i}}{2} ⌋ ≤ f_{i} = mult_{B_{i}}F.

We now replace t and B_{Y} by t + 2l and B′_{Y}.

Proof of Claim 2. This follows from a local computation (see also Proposition 2.8 of [Ein]).

Let B_{Y} = ∑ b_{i}B_{i} where B_{i} are distinct prime divisors. If the b_{i} are integers, then ⌊(f^{∗}B_{Y})/t⌋ = f^{∗}⌊B_{Y} /t⌋. So we may assume that ⌊B_{Y} /t⌋ = 0 and B_{Y} /t ≠ 0. Let Q ∈ X be any point and P = f(Q) ∈ Y be its image. We may assume that P ∈ B_{i} for 1 ≤ i ≤ s. Let (y_{1}, ..., y_{n}) be local coordinates centered in P such that y_{i} is a local equation for B_{i} for 1 ≤ i ≤ s. Let E be any component of f^{∗}B_{Y} containing Q and choose local coordinates (x_{1}, ..., x_{n}) on X centered in Q such that x_{1} is a local equation of E. One has that f^{∗}y_{i} = x_{1}^{n_{i}}ϵ_{i}, with n_{i} ≥ 0 and ϵ_{i} a regular function not vanishing on E. One sees that a local equation for K_{X/Y} near Q is

given by the determinant of the Jacobian matrix (\frac{∂f^{∗}y_{i}}{∂x_{j}} ), which vanishes on E to order at least (∑ n_{i})−1. So the coefficient of E in K_{X/Y} −⌊(f^{∗}B_{Y})/t⌋+f^{∗}⌊B_{Y} /t⌋ is at least

∑ n_{i} − ⌊∑ n_{i} \frac{b_{i}}{t} ⌋ − 1 ≥ 0.

□

We remark that the above theorem also holds if Y has sufficiently large algebraic fundamental group (see [Hac4] for a discusion of this case).

7. Theta divisors: Singularities and Cohomological characterization

Let (A, Θ) be a principally polarized abelian variety (PPAV), i.e. Θ is an ample divisor with h^{0}(A, O(Θ)) = 1. The singularities of such divisors are closely related to the geometry of the abelian variety A. For example, Kempf has shown that if A is the Jacobian variety of a curve, then the theta divisor has rational singularities. More recently Kollár has shown

Theorem 7.1. (Kollár) [Ko3] Let (A, Θ) be a PPAV, then the pair (A, Θ) is log canonical ^{4}. In particular if

Σ_{k}(Θ) = {x ∈ A|mult_{x}(Θ) ≥ k},

then every component of Σ_{k} has codimension at least k.

Proof. Let ν : A′ ⟶ A be a log resolution of the pair (A, Θ) (i.e. a proper birational morphism such that ν^{−1}(Θ) ∪ {exceptional set of ν} is a divisor with normal crossings support). For any fixed 1 ≫ ϵ > 0 let

I := ν_{∗}(O_{A′}(K_{A′/A} − ⌊(1 − ϵ)ν^{∗}Θ⌋) ⊂ O_{A}

be the multiplier ideal sheaf of the pair (A, Θ). Multiplier ideals are a measure of the singularities of the pair (A, (1 − ϵ)Θ). Their definition does not depend on the choice of the log-resolution ν. By definition, the pair (A, Θ) is log canonical if for all 1 ≫ ϵ > 0, one has I((1 − ϵ)Θ) = O_{A}. We now proceed to show that I = O_{A}. First of all, we have that

ν^{∗}(Θ + P) + K_{A′/A} − ⌊(1 − ϵ)ν^{∗}Θ⌋ ≡ K_{A′} + ϵν^{∗}Θ + {(1 − ϵ)ν^{∗}Θ}

and hence by Theorem 5.2, for all P ∈ Pic^{0}(A),

h^{i}(A, O_{A}(Θ)⊗P ⊗I) = 0 for all i > 0.

If I ≠ O_{A}, then h^{0}(A, O_{A}(Θ)⊗P ⊗I) = 0 for general P ∈ Pic^{0}(A) (as a general translate of Θ will not vanish on the cosupport of I). But then for all P ∈ Pic^{0}(A),

h^{0}(A, O_{A}(Θ)⊗P ⊗I) = χ(O_{A}(Θ)⊗P ⊗I) = χ(O_{A}(Θ)⊗I) = 0

and hence (cf. Example 2.2) O_{A}(Θ)⊗I is trivial, which is impossible. □

The above proof, is due to Ein and Lazarsfeld. The same argument also shows that if D ∈ |mΘ| , then the pair (A, (1/m)Θ) is log canonical.

One can easily construct examples of abelian varieties where the bound on the codimension of Σ_{k}(Θ) is achieved by considering products of abelian varieties of smaller dimension. The following result of Ein and Lazarsfeld shows that this is essentially the only case in which such a phenomenon occurs.

Theorem 7.2. (Ein-Lazarsfeld) [EL] If Θ ⊂ A is an irreducible theta divisor, then Θ is normal and has only rational singularities.

Proof. Let ν : A′ ⟶ A be a log resolution of the pair (A, Θ) and Z be the proper transform of Θ. Pushing forward exact sequence

0 ⟶ ω_{A′} ⟶ ω_{A′}(Z) ⟶ ω_{Z} ⟶ 0

4In this section, we will use the formalism of multiplier ideals and some of their applications.

We refer to [Ein] for an overview of these and related concepts

via ν : A′ ⟶ A, one obtains a short exact sequence^{5}

0 ⟶ O_{A} ⟶ O_{A}(Θ)⊗I ⟶ ν_{∗}ω_{Z} ⟶ 0

where I := ν_{∗}(ω_{A′/A}(Z − ν^{∗}Θ)) is the adjoint ideal of Θ (cf. [Ein] §3). In order to show that Θ has rational singularities, it suffices to show that I = O_{A}^{6}. By the generic vanishing theorem Theorem 3.1, one sees that for general P ∈ Pic^{0}(A)

h^{i}(ν_{∗}(ω_{Z})⊗P) = h^{i}(ω_{Z}⊗ν^{∗}P) = 0 for all i > 0.

If I ≠ O_{A}, then also h^{0}(ν_{∗}(ω_{Z})⊗P) = h^{0}(O_{A}(Θ)⊗I⊗P) = 0 for general P ∈ Pic^{0}(A) and so χ(ν_{∗}ω_{Z}) = 0. This contradicts a result of Kawamata and Viehweg.

□

This leads to the following cohomological characterization of principal polarizations (cf. [Hac1]).

Theorem 7.3. (Hacon) Let Θ ⊂ A be an irreducible reduced divisor of a gdimensional abelian variety, ν : A′ ⟶ A a log resolution of (A, Θ) and Z the proper transform of Θ. Then Θ is a principal polarization iff

h^{i}(ω_{Z}) = \binom{g}{i + 1} for 0 ≤ i ≤ g − 1 and

h^{i}(ω_{Z}⊗ν^{∗}P) = 0 ∀ O_{A} ≠ P ∈ Pic^{0}(A), i > 0

Proof. If Θ is a principal polarization, then for all P ∈ Pic^{0}(A) one has

h^{0}(O_{A}(Θ)⊗P) = 1 and h^{i}(O_{A}(Θ)⊗P) = 0 ∀i > 0.

For all P ≠ O_{A} and 0 ≤ i ≤ g, one has h^{i}(P) = 0. Consider the exact sequence

0 ⟶ O_{A} ⟶ O_{A}(Θ) ⟶ ν_{∗}ω_{Z} ⟶ 0.

Twisting it by P ∈ Pic^{0}(A), one immediately computes h^{i}(ω_{Z}⊗ν^{∗}P) = h^{i}(ν_{∗}(ω_{Z})⊗P).

Suppose now that h^{i}(ω_{Z}⊗ν^{∗}P) are as above. By Example 2.5, it suffices to show that for all P ∈ Pic^{0}(A), one has h^{0}(O_{A}(Θ)⊗P ⊗I) = 1 and h^{i}(O_{A}(Θ)⊗P ⊗I) = 0 for all i > 0 (then O_{A}(Θ)⊗I is a line bundle and hence I = O_{A} and h^{0}(O_{A}(Θ)) = 1). When P ≠ O_{A}, from the above exact sequence, one sees that h^{i}(O_{A}(Θ)⊗P ⊗I) = h^{i}(ν_{∗}ω_{Z}⊗P) as required. (Notice that since h^{i}(ν_{∗}ω_{Z}⊗P) = 0 for all i > 0, one has

h^{0}(ν_{∗}ω_{Z}⊗P) = χ(ν_{∗}ω_{Z}⊗P) = χ(ν_{∗}ω_{Z}) = 1.)

The homomorphism

H^{0}(Ω_{A}^{g−i−1}) ⟶ H^{0}(Ω_{Z}^{g−i−1})

is injective. Otherwise, there would be a morphism Θ ⟶ Θ̄ = Θ/K where K is a subtorous of A. Let F = F_{Z/Θ̄}, for general P ∈ Pic^{0}(A), one has P |K ≠ O_{K} and so

h^{0}((ω_{Z}⊗P)|F) ≅ h^{0}(ω_{F} ⊗P) ≅ h^{0}(ω_{K}⊗P) = 0.

Therefore h^{0}(Z, ω_{Z}⊗P) = 0 which is impossible. By Hodge symmetry and Serre duality, it follows that the homomorphisms

H^{i}(ν_{∗}ω_{Z}) ⟶ H^{i+1}(O_{A})

5Recall that by Theorem 5.1, R^{1}ν_{∗}ω_{A′} = 0.

6Since Θ is Gorenstein and K_{A′/A}(Z − ν^{∗}Θ)|Z = K_{Z} − ν^{∗}K_{Θ}, it follows that I = O_{A} iff ν_{∗}K_{Z} = K_{Θ} i.e. iff Θ has canonical and hence rational singularities cf. [Ein].

is surjective. By dimensional considerations it is an isomorphism and hence

h^{0}(O_{A}(Θ)⊗I) = h^{0}(O_{A}) = 1 and h^{i}(O_{A}(Θ)⊗I) = 0 for i > 0.
□

In general the hypothesis in the above theorem can’t be weakened, however, in [HP1] it is shown that: If X is a smooth projective variety of maximal Albanese dimension, with P_{1}(X) = q(X) = dim(X) + 1, and the Albanese variety of X is simple, then X is birational to a theta divisor.

As an application of the above result, one can also classify surfaces with p_{g} = q = 3 (cf. [HP2], [Pi]).

8. Characterization of abelian varieties

We begin by recalling the following result of Kawamata which is a generalization to higher dimensions of Enriques’ Theorem for surfaces mentioned in the introduction.

Theorem 8.1. (Kawamata) [Ka1] Let X be a smooth complex projective variety with κ(X) = 0, then a : X ⟶ A is surjective (in particular q(X) ≤ dim(X)). If moreover, q(X) = dim(X), then X is birational to an abelian variety.

This gives (a partial) answer to a conjecture of Ueno which in turn is an important test case for the C_{n,m} conjecture of Iitaka. On the other hand, the invariant κ(X) is highly nontrivial as it involves knowing P_{m}(X) for infinitely many values of m. Therefore it would be more satisfactory to give an ”effective” statement such as the following:

Theorem 8.2. (Kollár) [Ko4] Let X be a smooth projective variety. If P_{m}(X) = 1 for some m ≥ 3, then a : X ⟶ A is surjective. If P_{m}(X) = 1 for some m ≥ 4 and q(X) = dim(X), then X is birational to an abelian variety.

We will give an idea of the proof of the following result conjectured by Kollár:

Theorem 8.3. (Chen-Hacon) [CH1], [CH2] Let X be a smooth projective variety. (1) If P_{m}(X) = 1 for some m ≥ 2, then a : X ⟶ A is surjective.

(2) If P_{m}(X) = 1 for some m ≥ 2 and q(X) = dim(X), then X is birational to an abelian variety.

Proof. If κ(X) = 0, both results follow from Kawamata’s Theorem above. We will therefore assume that κ(X) > 0 and derive a contradiction (i.e. that P_{2}(X) ≥ 2).

(1) Let f : X ⟶ V be a birational model of the Iitaka fibration of X^{7}. Let X_{v} be a generic geometric fiber of f, then κ(X_{v}) = 0 and by Kawamata’s Theorem, one sees that a(X_{v}) is an abelian subvariety of A. Since abelian subvarieties of a fixed abelian variety A are rigid, it follows that there is an abelian subvariety K ⊂ A such that for general v, the images a(X_{v}) are translates of K. Let S = A/K. We will denote by Z, W the images of X, V in A, S. After replacing X, V by appropriate birational models, we may assume that there is a commutative diagram:

X ⟶ Z ⟶^{⊂} A
f↓  f̂↓
V ⟶ W ⟶^{⊂} S

Let π : X ⟶ S be the induced morphism. Fix H an ample divisor on S. For r ≫ 0 the linear series |rK_{X} − π^{∗}H| will be non empty. After replacing X by an appropriate birational model, we may assume that

|rK_{X} − π^{∗}H| = F + |M|

with |M| base point free and F with normal crossings support. Fix B a general member in |rK_{X} − π^{∗}H| and let

L := ω_{X}⊗O_{X} (−⌊\frac{B}{r} ⌋) ≡ \frac{π^{∗}H}{r} − \{\frac{B}{r} \}.

7By definition, one has that κ(X_{v}) = 0, f is surjective, the general fiber is irreducible and dim(V) = κ(X). For all sufficiently big and divisible integers m, the morphism f is birational to φ_{m}.

(One may think of L as being the positive part of ω_{X} and ⌊\frac{B}{r} ⌋ as being some negligible divisor. For example if K_{X} is ample, take L = ω_{X} and ⌊\frac{B}{r} ⌋ = 0.) As in the proof of Theorem 6.1, one may arrange that

|2K_{X}| = |K_{X} + L| + ⌊B/r⌋.

Since H/r is ample and {B/r} is KLT (it has simple normal crossings support and ⌊{B/r}⌋ = 0), by Theorem 5.2 one has that

h^{i}(W, π_{∗}(ω_{X}⊗L)⊗P) = 0 for all i > 0 and all P ∈ Pic^{0}(S).

Let

F := π_{∗}(ω_{X}⊗L).

One has

h^{0}(W, F) = h^{0}(W, π_{∗}(ω_{X}⊗L)) = h^{0}(X, ω_{X}⊗L) = h^{0}(X, ω_{X}^{⊗2})

It follows that

h^{0}(W, F⊗P) = χ(W, F⊗P) = χ(W, F) = h^{0}(W, F) = 1.

By Example 2.5, any sheaf F as above must be supported on a translate of an abelian subvariety of S. By Theorem 5.1, the sheaf F is a torsion free sheaf on W. However, W generates S and hence W = S and so Z = A i.e. a : X ⟶ A is surjective.

(2) Here we will only consider the case P_{m}(X) = 1 for some m ≥ 4. By the proof of part (1), one sees that

|K_{X} + L + η| > 0 for all η ∈ Pic^{0}(S).

From the morphism

|K_{X} + L + η| × |K_{X} + L − η| ⟶ |2K_{X} + 2L|

one sees that

P_{4}(X) ≥ h^{0}(ω_{X}^{⊗2}⊗L^{⊗2}) ≥ dim(S) + 1

and so κ(V) = dim(V) ≤ dim(S) = 0 and the result follows from Kawamata’s theorem mentioned above. □

We now give an idea of the proof in the case m = 2. Assume that κ(X) = dim(X). By a result of Ein and Lazarsfeld (see [CH1]), one has that if κ(X) > 0, then

dim V^{0}(X, ω_{X}) > 0.

So fix T a positive dimensional component of V^{0}(X, ω_{X}). By Simpson’s result (cf. Theorem 3.1), we may write

T = P_{0} + T_{0}

where P_{0} is a torsion element of Pic^{0}(X) and T_{0} is a sub-abelian variety of Pic^{0}(X). Consider now for all Q ∈ T_{0} the map

|K_{X} + P_{0} + Q| × |K_{X} + P_{0} − Q| ⟶ |2K_{X} + 2P_{0}|.

Since for all Q ∈ T_{0}, the linear series |K_{X} + P_{0} + Q|, |K_{X} + P_{0} − Q| are non-empty and any effective divisor can be decomposed in to the sum of two effective divisors in finitely many ways, one has that

h^{0}(X, ω_{X}^{⊗2}⊗P_{0}^{⊗2}) ≥ 2.

Proceeding as in the proof of Theorem 6.1, one sees that

h^{0}(ω_{X}^{⊗2}⊗P_{0}^{⊗2}) = P_{2}(X).

This is the required contradiction.

The case when 0 < κ(X) < dim(X) is based on a similar idea, but it is unluckily much more technical. We refer the reader to [CH1] for this part of the proof.

References

[CH1] Chen J. A. and Hacon C D.: Characterization of abelian varieties, Invent. Math. 143, 435-447 (2001)

[CH2] Chen, J. A. and Hacon, C. D.: On algebraic fiber spaces over varieties of maximal albanese dimension, Duke Math. J., Vol. 111, No. 1, 2002

[CH3] Chen, J. A. and Hacon, C. D.: Linear series of irregular varieties, Proceedings of the symposium on Algebraic Geometry in East Asia. World Scientific (2002)

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