Algebraic Fiber Spaces 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.

On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces

In this paper, we study the relative anti-canonical divisor − K X / Y -K_{X/Y} of an algebraic fiber space ϕ : X → Y \phi \colon X\to Y , and we reveal relations among positivity conditions of − K X / Y -K_{X/Y} , certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana–Cao–Matsumura’s equality on Hacon–McKernan’s question, whose original proof depends on analytics methods. The third result proves that algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of Y Y . Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.

On Subadditivity of Okounkov Bodies for Algebraic Fiber Spaces

Abstract The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new approach toward the Iitaka conjecture.

Almost nef regular foliations and Fujita's decomposition of reflexive sheaves

In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally connected fibers such that $\mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. As a by-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sum of a hermitian flat vector bundle and a generically ample reflexive sheaf for any algebraic fiber space $f : X \rightarrow Y$. We also study foliations with nef anti-canonical bundles.

Erratum to “When is the Albanese morphism an algebraic fiber space in positive characteristic?”

Erratum to “When is the Albanese morphism an algebraic fiber space in positive characteristic?”

Subadditivity of generalized Kodaira dimensions and extension theorems

In this paper, we introduce the notion of generalized Kodaira dimension with multiplier ideal sheaves, and prove the subadditivity of these generalized Kodaira dimensions for certain Kähler fibrations, which was originally obtained for Kodaira dimensions of algebraic fiber spaces by Kawamata and Viehweg. Our method is analytic and based on some new results in recent years. The crucial step in our proof is to prove an [Formula: see text] extension theorem for twisted pluricanonical sections on compact Kähler manifolds. Moreover, we also discuss the relation between two previous optimal [Formula: see text] extension theorems with singular weights on weakly pseudoconvex Kähler manifolds.

ALGEBRAIC FIBER SPACES AND CURVATURE OF HIGHER DIRECT IMAGES

AbstractLet$p:X\rightarrow Y$be an algebraic fiber space, and let$L$be a line bundle on$X$. In this article, we obtain a curvature formula for the higher direct images of$\unicode[STIX]{x1D6FA}_{X/Y}^{i}\otimes L$restricted to a suitable Zariski open subset of$X$. Our results are particularly meaningful if$L$is semi-negatively curved on$X$and strictly negative or trivial on smooth fibers of$p$. Several applications are obtained, including a new proof of a result by Viehweg–Zuo in the context of a canonically polarized family of maximal variation and its version for Calabi–Yau families. The main feature of our approach is that the general curvature formulas we obtain allow us to bypass the use of ramified covers – and the complications that are induced by them.

Kodaira dimension of algebraic fiber spaces over surfaces

In this short note we prove the Iitaka C_nm conjecture for algebraic fiber spaces over surfaces

When is the Albanese morphism an algebraic fiber space in positive characteristic?

In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is $F$-pure. Furthermore, we consider a notion of $F$-splitting for morphisms, and investigate it of the Albanese morphisms. We show that an $F$-split variety has $F$-split Albanese morphism, and that the $F$-split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.

Irregular varieties with geometric genus one, theta divisors, and fake tori

We study the Albanese image of a compact Kähler manifold whose geometric genus is one. In particular, we prove that if the Albanese map is not surjective, then the manifold maps surjectively onto an ample divisor in some abelian variety, and in many cases the ample divisor is a theta divisor. With a further natural assumption on the topology of the manifold, we prove that the manifold is an algebraic fiber space over a genus two curve. Finally we apply these results to study the geometry of a compact Kähler manifold which has the same Hodge numbers as those of an abelian variety of the same dimension.

Kodaira dimension of algebraic fiber spaces over abelian varieties

In this article we provide a proof of the Iitaka $$C_{nm}$$ conjecture for algebraic fiber spaces over tori.

Algebraic Fiber Space Whose Generic Fiber and Base Space Are of Almost General Type

We assume that the existence and termination conjecture for flips holds. A complex projective manifold is said to be {\it of almost general type} if the intersection number of the canonical divisor with every very general curve is strictly positive. Let $f$ be an algebraic fiber space from $X$ to $Y$. Then the manifold $X$ is of almost general type if every very general fiber $F$ and the base space $Y$ of $f$ are of almost general type.

Solution algebras of differential equations and quasi-homogeneous varieties: a new differential Galois correspondence

We develop a new connection between Differential Algebra and Geometric Invariant Theory, based on an anti-equivalence of categories between solution algebras associated to a linear differential equation (i.e. differential algebras generated by finitely many polynomials in a fundamental set of solutions), and affine quasi-homogeneous varieties (over the constant field) for the differential Galois group of the equation. Solution algebras can be associated to any connection over a smooth affine variety. It turns out that he spectrum of a solution algebra is an algebraic fiber space over the base variety, with quasi-homogeneous fiber. We discuss the relevance of this result to Transcendental Number Theory.

AN EFFECTIVE VERSION OF A THEOREM OF KAWAMATA ON THE ALBANESE MAP

We study the Albanese map of smooth complex projective varieties with small plurigenera. We provide criteria for the Albanese map to be surjective and to be an algebraic fiber space. These criteria are optimal in some sense.

Kodaira dimension of irregular varieties

Let $f:X\to Y$ be an algebraic fiber space with general fiber $F$. If $Y$ is of maximal Albanese dimension, we show that $\kappa (X)\geq \kappa (Y)+\kappa (F)$.

Semipositivity Theorem for Reducible Algebraic Fiber Spaces

We shall prove an extension of the semipositivity theorem for the case of reducible algebraic fiber spaces.

Varieties fibered by good minimal models

Let f:X->Y be an algebraic fiber space such that the general fiber has a good minimal model. We show that if f is the Iitaka fibration or if f is the Albanese map of relative dimension no more than three, then X has a good minimal model.

Some inequalities for Kodaira–Iitaka dimension on subvarieties

In this paper an investigation is presented of the relationship between the Kodaira–Iitaka dimension of a line bundle on an ambient normal variety and the Kodaira–Iitaka dimension of related line bundles on normal subvarieties. The results that are found extend in some directions those that were presented by Peternell et al. (Int J Math 10:1065–1079, 1999) Those results in turn generalized the well known easy addition theorem for Kodaira–Iitaka dimension on an algebraic fiber space.

Remarks on Projective Algebraic Fiber Spaces Over Curves

Let f:X → C be a fibration over curve C with general fiber F. Some numerical characteristics of h i (𝒪 X ) in terms of h i (𝒪 F ), the genus g(C) and the degree of direct image sheaves R i f * (ω X/C ) are given. If f is the derived fibration induced by m-canonical map, further numerical characterizations of m-plurigenus of F are given.

Remarks on algebraic fiber spaces

In this paper, we collect basic properties of the Albanese dimension and explain how to generalize the main theorem of [F2]: Algebraic fiber spaces whose general fibers are of maximal Albanese dimension. This paper is a supplement and a generalization of [F2]. We also prove an inequality of irregularities for algebraic fiber spaces in the appendix, which is an exposition of Fujita-Kawamata’s semi-positivity theorem.