Fujita decomposition over higher dimensional base

Massey Products and Fujita decompositions on fibrations of curves

Let $f:S\to B$ be a fibration of curves and let $f_\astω_{S/B}=\cU\oplus \cA$ be the second Fujita decomposition of $f.$ In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $\cU$. The main result states that their vanishing on a general fibre of $f$ implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we p…

奇异埃尔米特度量与Catanese、Fujita和Kawamata分解定理

Singular hermitian metrics and the decomposition theorem of Catanese, Fujita, and Kawamata

本文证明带半正曲率奇异埃尔米特度量且满足最小延拓性质的无挠层可分解为埃尔米特平坦丛与泛型丰沛层的直和，并将其应用于相对多重典范丛直接像，推广了Fujita、Catanese–Kawamata和Iwai的结果。

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. A…

Fujita-type freeness for quasi-log canonical curves and surfaces

We prove Fujita-type basepoint-freeness for projective quasi-log canonical curves and surfaces.

相对 Fujita-Zariski 定理

The relative Fujita-Zariski theorem

本文证明了 Fujita-Zariski 定理的相对版本，指出真簇上可逆层在基轨迹丰沛时其充分高幂整体生成，推广了 Fujita (1983) 与 Zariski (1962) 的经典结果。

On Fujita's freeness conjecture in dimension 5

Let $X$ be a smooth projective variety of dimension $5$ and $L$ be an ample line bundle on $X$. We show that $|K_X + 6L|$ is base-point free.

On the volume of a line bundle

Using a result of Fujita on approximate Zariski decompositions and the singular version of Demailly's holomorphic Morse inequalities as obtained by Bonavero, we express the volume of a line bundle in terms of the absolutely continuous parts of all the positive curvature currents on it, with a way to pick an element among them which is most homogeneous with respect to the volume. This enables us to introduce the volume of any pseudoeffective class on a compact Kaehler manifold, and Fujita's theorem is then extended to this context.

关于对数多典范丛的正像层与非消失猜想的评述

Remarks on the Direct Image Sheaf of Logarithmic Pluricanonical Bundles and the Non-Vanishing Conjecture

本文利用 Chen-Jiang 分解证明不规则簇上 lc 对的非消失猜想可由低维情形推出，并将 Catanese-Fujita-Kawamata 分解推广至 klt 情形以建立截面存在性。

Answer to a question by Fujita on Variation of Hodge Structures

We first provide details for the proof of Fujita's second theorem for Kähler fibre spaces over a curve, asserting that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. Our main result then answers in the negative the question posed by Fujita whether $V$ is semiample. In fact, $V$ is semiample if and only if $Q$ is associated to a representation of the fundamental group of $B$ having finite image. Our examples are based on hypergeometric integrals.

Effective freeness of adjoint line bundles

In this note we establish a new Fujita-type effective bound for the base point freeness of adjoint line bundles on a compact complex projective manifold of complex dimension $n$. The bound we obtain (approximately) differs from the linear bound conjectured by Fujita only by a factor of the cube root of $n$. As an application, a new effective statement for pluricanonical embeddings is derived.

A characterization of semiampleness and contractions of relative curves

A line bundle with a base-point-free multiple is called semiample. I give a cohomological characterization of semiample line bundles. The result is a common generalization of the Fujita-Zariski criterion for semiampleness and the Grothendieck-Serre characterization of ampleness. Applying the Fujita-Zariski Theorem, I characterize contractible curves in 1-dimensional families.

Big arithmetic divisors on the projective spaces over Z

This paper is an enhancement of the previous note "Explicit computations of Zariski decompositions on P_Z^1". In this paper, we observe several properties of a certain kind of an arithmetic divisor D on the n-dimensional projective space over Z and give the exact form of the Zariski decomposition of D on the projective line over Z. Further, we show that, if n>=2 and D is big and non-nef, then, for any birational morphism f: X --> P^n_Z of projective, generically smooth and normal arithmetic varieties, we can not expect a suitable Zariski decomposition of f^*(D). We also give a concrete constru…

Hodge-Arakelov inequalities for family of surfaces fibered by curves

The Hodge numerical invariants of a variation of Hodge structure over a smooth quas--projective variety are a measure of complexity for the global twisting of the limit mixed Hodge structure when it degenerates. These invariants appear in inequalities which they may have correction terms, called Arakelov inequalities. One may investigate the correction term to make them into equalities, also called Arakelov equalities. We investigate numerical Arakelov type (in)equilities for a family of surfaces fibered by curves. Our method uses Arakelov identities in a weight 1 and also in a weight 2 variat…

通过带解析伴随理想层的扩张定理对藤田自由性猜想的解决

A solution to Fujita's freeness conjecture via an extension theorem with analytic adjoint ideal sheaves

本文利用伴随理想层对乘子理想层不可积轨迹的精细控制，并结合基于内射性定理技术的扩张定理，直接处理 lc 奇点及其极小 lc 中心，从而在 Angehrn-Siu 思路下解决了藤田有效自由性猜想。

Positivity on subvarieties and vanishing of higher cohomology

We study the relationship between positivity of line bundles restricted to complete intersection subvarieties and the vanishing of higher cohomology groups. Based on this connection we prove generalizations of the vanishing theorems of Kawamata-Viehweg and Fujita that are valid for not necessarily big line bundles.

Stable Base Locus Decompositions of Kontsevich Moduli Spaces

In this paper, we determine the stable base locus decomposition of the Kontsevich moduli spaces of degree two and three stable maps to Grassmannians. This gives new examples of the decomposition for varieties with Picard rank three. We also discuss the birational models that correspond to the chambers in the decomposition.

On Mori chamber and stable base locus decompositions

The effective cone of a Mori dream space admits two wall-and-chamber decompositions called Mori chamber and stable base locus decompositions. In general the former is a non trivial refinement of the latter. We investigate, from both the geometrical and the combinatorial viewpoints, the differences between these decompositions. Furthermore, we provide a criterion to establish whether the two decompositions coincide for a Mori dream space of Picard rank two, and we construct an explicit example of a Mori dream space of Picard rank two for which the decompositions are different, showing that our …

Generic Base Change, Artin's Comparison Theorem, and the Decomposition Theorem for Complex Artin stacks

We prove the generic base change theorem for stacks, and give an exposition on the lisse-analytic topos of complex analytic stacks, proving some comparison theorems between various derived categories of complex analytic stacks. This enables us to deduce the decomposition theorem for perverse sheaves on complex Artin stacks with affine stabilizers from the case over finite fields.

Prime decomposition for the index of a Brauer class

We prove that the index of a Brauer class satisfies prime decomposition over a general base scheme. This contrasts with our previous result that there is no general prime decomposition of Azumaya algebras.

广义Severi-Brauer概型的半正交分解

Semiorthogonal decompositions for generalized Severi-Brauer schemes

本文利用保守下降方法为广义Severi-Brauer概型构造半正交分解，并推广到任意基概型上的旗簇，从而推广了Kapranov的经典结果。

Contractions on a manifold polarized by an ample vector bundle

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case $r := rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of paper by Ye-Zhang, Fujita, Andreatta-Ballico-Wisniewski. If $K_X+detE$ is nef, then by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map $π:X \longrightarrow W$ from $X$ onto a normal…

Base change for semiorthogonal decompositions

Consider an algebraic variety $X$ over a base scheme $S$ and a faithful base change $T \to S$. Given an admissible subcategory $\CA$ in the bounded derived category of coherent sheaves on $X$, we construct an admissible subcategory in the bounded derived category of coherent sheaves on the fiber product $X\times_S T$, called the base change of $\CA$, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of the bounded derived category of $X$ is given then the base changes of its components form a semiorthogonal decomposition of the bounded derived catego…

Speeding up Cylindrical Algebraic Decomposition by Gröbner Bases

Gröbner Bases and Cylindrical Algebraic Decomposition are generally thought of as two, rather different, methods of looking at systems of equations and, in the case of Cylindrical Algebraic Decomposition, inequalities. However, even for a mixed system of equalities and inequalities, it is possible to apply Gröbner bases to the (conjoined) equalities before invoking CAD. We see that this is, quite often but not always, a beneficial preconditioning of the CAD problem. It is also possible to precondition the (conjoined) inequalities with respect to the equalities, and this can also be useful in m…

Zero Decomposition with Multiplicity of Zero-Dimensional Polynomial Systems

We present a zero decomposition theorem and an algorithm based on Wu's method, which computes a zero decomposition with multiplicity for a given zero-dimensional polynomial system. If the system satisfies some condition, the zero decomposition is of triangular form.