Varieties Of Maximal Albanese Dimension Research Articles

Kodaira Dimension of Fibrations Over Abelian Varieties

Abstract We prove an additivity result for the log Kodaira dimension of algebraic fiber spaces over abelian varieties, a superadditivity result for fiber spaces over varieties of maximal Albanese dimension, as well as a subadditivity result for log pairs.

Relative Severi inequality for fibrations of maximal Albanese dimension over curves

AbstractLet$f: X \to B$be a relatively minimal fibration of maximal Albanese dimension from a varietyXof dimension$n \ge 2$to a curveBdefined over an algebraically closed field of characteristic zero. We prove that$K_{X/B}^n \ge 2n! \chi _f$. It verifies a conjectural formulation of Barja in [2]. Via the strategy outlined in [4], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and$\chi _f> 0$, we prove that the general fibreFoffhas to satisfy the Severi equality that$K_F^{n-1} = 2(n-1)! \chi (F, \omega _F)$. We also prove some sharper results of the same type under extra assumptions.

Moving Seshadri constants, and coverings of varieties of maximal Albanese dimension

Let $X$ be a smooth projective complex variety of maximal Albanese dimension, and let $L \to X$ be a big line bundle. We prove that the moving Seshadri constants of the pull-backs of $L$ to suitable finite abelian etale covers of $X$ are arbitrarily large. As an application, given any integer $k\geq 1$, there exists an abelian etale cover $p\colon X' \to X$ such that the adjoint system $\big|K_{X'} + p^*L \big|$ separates $k$-jets away from the augmented base locus of $p^*L$, and the exceptional locus of the pull-back of the Albanese map of $X$ under $p$.

The eventual paracanonical map of a variety of maximal Albanese dimension

Let $X$ be a smooth complex projective variety such that the Albanese map of $X$ is generically finite onto its image. Here we study the so-called eventual $m$-paracanonical map of $X$ (when $m=1$ we also assume $\chi(K_X)>0$). We show that for $m=1$ this map behaves in a similar way to the canonical map of a surface of general type, while it is birational for $m>1$. We also describe it explicitly in several examples.

Positivity in varieties of maximal Albanese dimension

Abstract Given a generically finite morphism f from a smooth projective variety X to an abelian variety A, we show that f * ⁢ ω X {f_{*}\omega_{X}} is “sufficiently positive” on A. As an application, we prove that when X is of general type, the global sections of ω X 2 {\omega_{X}^{2}} define a generically finite map of X. We also study the structure of X when X is of general type and satisfies χ ⁢ ( X , ω X ) = 0 {\chi(X,\omega_{X})=0} . We formulate a conjectural characterization of such X and prove the conjecture when A has exactly three simple factors.

Holomorphic curves in algebraic varieties of maximal albanese dimension

We prove a second main theorem type estimate in Nevanlinna theory for holomorphic curves f : Y → X from finite covering spaces Y → ℂ of the complex plane ℂ into complex projective manifolds X of maximal albanese dimension. If X is moreover of general type, then this implies that the special set of X is a proper subset of X. For a projective curve C in such X, our estimate also yields an upper bound of the ratio of the degree of C to the geometric genus of C, provided that C is not contained in a proper exceptional subset in X.

Characterization of products of theta divisors

AbstractWe study products of irreducible theta divisors from two points of view. On the one hand, we characterize them as normal subvarieties of abelian varieties such that a desingularization has holomorphic Euler characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$. On the other hand, we identify them up to birational equivalence among all varieties of maximal Albanese dimension. We also describe the structure of varieties $X$ of maximal Albanese dimension, with holomorphic Euler characteristic $1$ and irregularity $2\dim X-1$.

Pluri-canonical maps of varieties of maximal Albanese dimension in positive characteristic

We show that if X is a nonsingular projective variety of general type over an algebraically closed field k of positive characteristic and X has maximal Albanese dimension and the Albanese map is separable, then |4KX| induces a birational map.

Severi inequality for varieties of maximal Albanese dimension

In this paper, we prove the general Severi inequality for varieties of maximal Albanese dimension. Suppose that \(X\) is an \(n\)-dimensional projective, normal, minimal and \(\mathbb {Q}\)-Gorenstein variety of general type in characteristic zero. If \(X\) is of maximal Albanese dimension, then \(K^n_X \ge 2 n! \chi (\omega _X)\).

Continuous families of divisors, paracanonical systems and a new inequality for varieties of maximal Albanese dimension

Given a smooth complex projective variety X, a line bundle L of X an element v of H^1(O_X) and a section s in H^0(L) that deforms to first order in the direction v, we give a sufficient condition on v in terms of Koszul cohomology for this first order deformation to extend to an analytic deformation. We apply this result to improve known results on the paracanonical system of a variety of maximal Albanese dimension, due to Beauville in the case of surfaces and to Lazarsfeld-Popa in higher dimension. In particular, we prove the inequality p_g(X)>=\chi(K_X)+q(X)-1 for a variety X of maximal Albanese dimension without irregular fibrations of Albanese general type.

On varieties of maximal Albanese dimension

We study the birational geometry of varieties of maximal Albanese dimension in this article. Given a non-birational, generically finite, and surjective morphism f: X → Y between varieties of maximal Albanese dimension, we show that the plurigenera P m (X) and P m (Y) for some m ≥ 2 could be equal only in very restrictive situations. We also prove that the 5-th pluricanonical map of a variety of maximal Albanese dimension always induces the Iitaka model.

On the bicanonical map of irregular varieties

From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus 2 2 : varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

Birational characterization of products of curves of genus 2

Let X be a variety of maximal Albanese dimension. In this paper we prove that if �(!X) = 1 then q(X) ≤ 2dimX and if q(X) = 2dimX, then X is birational to a product of curves of genus 2. 2000 Mathematics Subject Classification: 14J40, 14E99.

On the birational geometry of varieties of maximal Albanese dimension

We study the birational geometry of varieties of maximal Albanese dimension. In particular we discuss criteria for a generically finite morphism of varieties of maximal Albanese dimension to be birational; we give a new characterization of Theta divisors; we study the Albanese map and refine some of the results of Koll\'ar; finally we use these results to birationally classify varieties with $P_3(X)=2$ and $q(X)=dim (X)$. Our method combines the generic vanishing theorems of Green and Lazarsfeld, the theory of Fourier Mukai transforms and the results of Koll\`ar on higher direct images of dualizing sheaves.

Pluricanonical maps of varieties of maximal Albanese dimension

Let X be a smooth complex projective algebraic variety of maximal Albanese dimension. We give a characterization of $\kappa (X)$ in terms of the set $V^0(X,\omega_{X} ):=\{ P\in{\mbox{\rm Pic}}^0(X)|h^0(X, \omega_X \otimes P) \ne 0\}$ . An immediate consequence of this is that the Kodaira dimension $\kappa (X)$ is invariant under smooth deformations. We then study the pluricanonical maps $\varphi _m:X \dashrightarrow \mathbb{P} (H^0(X,mK_X))$ . We prove that if X is of general type, $\varphi _m$ is generically finite for $m\geq 5$ and birational for $m\geq 5 \mbox{\rm dim} (X) +1$ . More generally, we show that for $m\geq 6$ the image of $\varphi _m$ is of dimension equal to $\kappa (X)$ and for $m\geq 6\kappa (X)+2$ , $\varphi _m$ is the stable canonical map.