On abelian families and holomorphic normal projective connections

Holomorphic normal projective connections on projective manifolds

We study projective manifolds M admitting a (flat) holomorphic normal projective connection and show that the Iitaka fibration (up to etale coverings) defines a smooth abelian group scheme structure on M.

Growth of balls of holomorphic sections on projective toric varieties

Let $\mathcal{O}(D)$ be an equivariant line bundle which is big and nef on a complex projective nonsingular toric variety $X$. Given a continuous toric metric $\|\cdot\|$ on $\mathcal{O}(D)$, we define the energy at equilibrium of $(X,ϕ_{\bar{D}})$ where $ϕ_{\bar{D}}$ is the weight of the metrized toric divisor $\bar{D}=(D,\|\cdot\|)$. We show that this energy describes the asymptotic behaviour as $k\rightarrow \infty$ of the volume of the $L^2$-norm unit ball induced by $(X,kϕ_{\bar{D}})$ on the space of global holomorphic sections $H^0(X,\mathcal{O}(kD))$.

Holomorphic Cartan geometries and Calabi--Yau manifolds

We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous démontrons que les seules variétés projectives de Calabi--Yau qui possèdent des géométrie holomorphes de Cartan sont les variétés abéliennes.)

Dynamics of Singular Holomorphic Foliations on the Complex Projective Plane

This manuscript is an introduction to the theory of holomorphic foliations on the complex projective plane. Historically the subject has emerged from the theory of ODEs in the complex domain and various attempts to solve Hilbert's 16th Problem, but with the introduction of complex algebraic geometry, foliation theory and dynamical systems, it has now become an interesting subject of its own.

Holomorphic maps from rational homogeneous spaces onto projective manifolds

Answering a problem raised by Lazarsfeld, Hwang and Mok proved that a surjective holomorphic map from a rational homogeneous space of Picard number 1 onto projective manifold different from projective space must be a biholomorphism. THe aim of this paper is to generalized this result to irreducible rational homogeneous space of higher Picard number.

Virtually abelian Kähler and projective groups

We characterise the virtually abelian groups which are fundamental groups of compact Kähler manifolds and of smooth projective varieties. We show that a virtually abelian group is Kähler if and only if it is projective. In particular, this allows to describe the Kähler condition for such groups in terms of integral symplectic representations.

On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a generalization of Yang's solution using RC positivity for Yau's conjecture. As an application, we show that any compact Kähler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.

特殊阿贝尔曲面上的全纯线丛

Holomorphic line bundles on a special abelian surface

本文考虑源自Wang和Lu工作的特殊阿贝尔曲面$A_Ω$，显式研究了该曲面上的全纯线丛。

全纯映射到射影空间的空间的拓扑

The topology of spaces of holomorphic maps to projective space

本文证明了从光滑复射影簇到射影空间的全纯映射与连续映射在依赖次数的范围内同调相同。

Generalized holomorphic analytic torsion

We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective morphisms.

On projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural generalization of the structure theorem proved by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. Specifically, we prove that $X$ admits a locally trivial morphism $ϕ:X \to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite étale cover $A \to Y$ by an abelian variety $A$, by combining the author's previous work with…

Holomorphic line bundles on projective toric manifolds from Lagrangian sections of their mirrors by SYZ transformations

The mirror of a projective toric manifold $X_Σ$ is given by a Landau-Ginzburg model $(Y,W)$. We introduce a class of Lagrangian submanifolds in $(Y,W)$ and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant hermitian metrics on holomorphic line bundles over $X_Σ$. Through this geometric correspondence, we also identify the mirrors of Hermitian-Einstein metrics, which are given by distinguished Lagrangian sections whose potentials satisfy certain Laplace-type equations.

Hyperbolic monopoles and holomorphic spheres

We associate to an SU(2) hyperbolic monopole a holomorphic sphere embedded in projective space and use this to uncover various features of the monopole.

Holomorphic vector fields and rationality

We show that a nonsingular complex projective variety admitting a holomorphic vector field with nonempty isolated zeroes, is rational using a key technique by Harvey-Lawson on finite volume flows. This statement was conjectured by J. Carrell. By the same technique, we obtain a uniform upper bound of Betti numbers of nonsingular complex projective variety admitting a holomorphic vector field with exact one zero point. Such an upper bound depends only on the dimension of the variety, which is a stronger version of a result of Akyildiz and Carrell.

Buser-Sarnak invariant and projective normality of abelian varieties

We show that a general $n$-dimensional polarized abelian variety $(A,L)$ of a given polarization type and satisfying $ h^0(A, L) \geq \dfrac{8^n}{2} \cdot \dfrac{n^n}{n !}$ is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety in a geodesic tubular neighborhood of a subtorus of a compact complex torus.

Abelian fibred holomorphic symplectic manifolds

We study holomorphic symplectic manifolds which are fibred by abelian varieties. This structure is a higher dimensional analogue of an elliptic fibration on a K3 surface. We investigate when a holomorphic symplectic manifold is fibred in this way, and are led to several natural conjectures. We then study the geometry of these fibrations. The expectation is that this point of view will prove useful in understanding holomorphic symplectic manifolds, and possibly lead to a classification.

Abelian integrals in holomorphic foliations

The aim of this paper is to introduce the theory of Abelian integrals for holomorphic foliations in a complex manifold of dimension two. We will show the importance of Picard-Lefschetz theory and the classification of relatively exact 1-forms in this theory. As an application we identify some irreducible components of the space of holomorphic foliations of a fixed degree and with a center singularity in the projective space of dimension two. Also we calculate higher Melnikov functions under some generic conditions.

Rigid geometry on projective varieties

We prove rigidity of various types of holomorphic parabolic geometry on smooth complex projective varieties.

Real projective structures on a real curve

Given a compact connected Riemann surface $X$ equipped with an antiholomorphic involution $τ$, we consider the projective structures on $X$ satisfying a compatibility condition with respect to $τ$. For a projective structure $P$ on $X$, there are holomorphic connections and holomorphic differential operators on $X$ that are constructed using $P$. When the projective structure $P$ is compatible with $τ$, the relationships between $τ$ and the holomorphic connections, or the differential operators, associated to $P$ are investigated. The moduli space of projective structures on a compact oriented…

非标准全纯结构在量子射影线上的线丛上

Non-standard Holomorphic Structures on Line Bundles over the Quantum Projective Line

本文探讨量子射影线上的线丛非标准全纯结构，证明存在无限多个非规范等价结构，反驳了2011年提出的疑问。

Negative Holomorphic curvature and positive canonical bundle

In this note we show that if a projective manifold admits a Kähler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

量子射影线上的线丛的不动点与全纯结构

Fixed points and Holomorphic Structures on Line Bundles over the Quantum Projective Line

研究量子射影线上的线丛全纯结构不唯一性，通过不动点理论分析平坦$\bar\partial$连接，并将可逆解的存在性与特定非线性映射在适当Banach空间中的不动点存在性联系起来。

Codimension one holomorphic distributions on the projective three-space

We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one …

Rational Morita equivalence for holomorphic Poisson modules

We introduce a weak concept of Morita equivalence, in the birational context, for Poisson modules on complex normal Poisson projective varieties. We show that Poisson modules, on projective varieties with mild singularities, are either rationally Morita equivalent to a flat partial holomorphic sheaf, or a sheaf with a meromorphic flat connection or a co-Higgs sheaf. As an application, we study the geometry of rank two meromorphic rank two $\mathfrak{sl}_2$-Poisson modules which can be interpreted as a Poisson analogous to transversally projective structures for codimension one holomorphic foli…

The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties

Let f:C -> A be an entire holomorphic curve into a semi-Abelian variety A. Then the Zariski closure of f(C) is a translate of a semi-Abelian subvariety of A (logarithmic Bloch-Ochiai's theorem). The purpose of the present paper is to establish a quantitative version of the above result for such f i.e., the second main theorem and the defect relation.

Holomorphic Jet Bundles

Chern number formulas for holomorphic jet bundles are computed for projective curves and for projective surfaces. These formulas are used to show that certain minimal surfaces of general type (generic hypersurfaces of degree at least 5 in projective 3-spaces in particular) are hyperbolic in the sense of complex hyperbolic geometry.

Equivariant Holomorphic Morse Inequalities II: Torus and Non-Abelian Group Actions

We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. We apply the results to invariant line bundles over toric manifolds. If the group is non-Abelian, there is in addition an action of the Weyl group on the fixed-point set of its maximal torus. The sum over the fixed points can be rearra…

关于具有非负全纯截面曲率的紧凯勒流形的结构

On the structure of compact Kähler manifolds with nonnegative holomorphic sectional curvature

本文研究具有非负全纯截面曲率的紧凯勒流形，通过建立完整和原理的伪有效版本，证明在无真正平坦切向量等条件下流形必为射影且有理连通，从而回答了 Matsumura-Yang 问题并推广了 Yau 猜想，同时分类了非射影三维情形并推广到正实双截面曲率度量。

Holomorphic symmetric differentials and a birational characterization of Abelian Varieties

A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational characterization of abelian varieties. In particular we prove that, under the conjectures of the Minimal Model Program, a smooth projective variety is birational to an abelian variety if and only if it has Kodaira dimension 0 and some symmetric power of its cotangent sheaf is generically generated by its global sections.

光滑射影簇上全纯一次形式的几何

Geometry of Holomorphic One-forms on Smooth Projective Varieties

本文证明光滑射影簇到简单阿贝尔簇的态射光滑当且仅当存在无零点的拉回全纯一次形式，并研究了有零点的全纯一次形式空间的结构。

Mixed multiplier ideals and the irregularity of abelian coverings of the projective plane

A formula for the irregularity of abelian coverings of the projective plane is established and some applications are presented.

Abelian varieties as automorphism groups of smooth projective varieties

We determine which complex abelian varieties can be realized as the automorphism group of a smooth projective variety.

半阿贝尔簇子簇上全纯映射的布洛赫原理

Bloch's principle for holomorphic maps into subvarieties of semi-abelian varieties

本文将半阿贝尔簇子簇上整曲线的值分布基本定理推广到单位圆盘全纯映射序列，证明了对数一般型子簇的伪小林双曲性，并改进了 Cartan 关于满足 Borel 恒等式的无处消失全纯函数组的经典定理。

Hodge环和CM阿贝尔簇上解析周期之间的二次关系

Hodge cycles and quadratic relations between holomorphic periods on CM abelian varieties

本文研究了CM阿贝尔簇上解析周期的Hodge关系，证明通过单调二次关系可生成所有非平凡关系，并展示了构造的B在特定域上的分裂性质。

Abelian surfaces with anti-holomorphic multiplication

For appropriate $N\ge 3$ and $d<0,$ the moduli space of principally polarized abelian surfaces with level $N$ structure and anti-holomorphic multiplication by $\mathcal O_d$ (the ring of integers in $\mathbb Q(\sqrt{d})$) is shown to consist of the real points of a quasi-projective algebraic variety defined over $\mathbb Q$, and to coincide with finitely many copies of the quotient of hyperbolic 3-space by the principal congruence subgroup of level $N$ in $\mathbf{SL}(2, \mathcal O_d).$

Projective normality of abelian surfaces of type (1,2d)

We show that an abelian surface embedded in P^N by a very ample line bundle L of type (1,2d) is projectively normal if and only if d>=4. This completes the study of the projective normality of abelian surfaces embedded by complete linear systems.

具有CM型的阿贝尔簇的全纯(2,2)定理

A holomorphic (2, 2)-Theorem for Abelian varieties of CM-type

该研究针对CM型阿贝尔簇，证明了第二Milnor K群解析层的第二上同调群到对应Hodge型有理上同调类的映射在不计分母时为满射。

Compact leaves of codimension one holomorphic foliations on projective manifolds

This article studies codimension one foliations on projective man-ifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holo-nomy representation (dynamics of the foliation in the transverse direction). We address in particular the following problems: existence of foliation having as a leaf a given hypersurface with topologically torsion normal bundle, global structure of foliations having a compact leaf whose holonomy is abelian (resp. solvable), and factorization results.

Projective normality of abelian varieties

We show that ample line bundles L on a g-dimensional simple abelian variety, with h^0(L)> 2^g.g!, give projectively normal embeddings, for all g>0.

Some results about the projective normality of abelian varieties

We reduce the problem of the projective normality of polarized abelian varieties to check the rank of very explicit matrices. This allow us to prove some results on normal generation of primitive line bundles on abelian threefolds and fourfolds. We also give two situations where the projective normality always fails. Finally we make some conjecture.

Projective structures on Riemann surface and natural differential operators

We investigate the holomorphic differential operators on a Riemann surface $M$. This is done by endowing $M$ with a projective structure. Let $\mathcal L$ be a theta characteristic on $M$. We explicitly describe the jet bundle $J^k(E\otimes {\mathcal L}^{\otimes n})$, where $E$ is a holomorphic vector bundle on $M$ equipped with a holomorphic connection, for all $k$ and $n$. This provides a description of holomorphic differential operators from $E\otimes {\mathcal L}^{\otimes n}$ to another holomorphic vector bundle $F$ using the natural isomorphism $\text{Diff}^k(E\otimes {\mathcal L}^{\otime…

Smooth finite abelian uniformizations of projective spaces and Calabi-Yau Orbifolds

We give a classification of smooth complex manifolds with a finite abelian group action, such that the quotient is isomorphic to a projective space. The case where the manifold is a Calabi-Yau is studied in detail.

Tropical Geometric Compactification of Moduli, II - $A_g$ case and holomorphic limits -

We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and sho…

Abelian varieties as automorphism groups of smooth projective varieties in arbitrary characteristics

Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This generalizes a result of Lombardo and Maffei for complex abelian varieties.

具有全纯辛形式的簇：LLV代数与导出等价

Varieties admitting a holomorphic symplectic form: LLV algebras and derived equivalences

利用Beauville-Bogomolov分解计算具有全纯辛形式的复光滑射影簇的LLV代数，推广超凯勒簇和阿贝尔簇的已知结果，并证明Orlov猜想对许多此类簇成立，特别对所有四维情形成立。

The Second Main Theorem for Holomorphic Curves into Semi-Abelian Varieties II

We establish the second main theorem with the best truncation level one for an entire holomorphic curve $f:\C \to A$ into a semi-abelian variety $A$ and an arbitrary effective reduced divisor $D$ on $A$; the low truncation level is important for applications. We will actually prove this for the jet lifts of $f$. Finally we give some applications, including the solution of a problem posed by Mark Green.

阿贝尔簇上射影丛的上同调秩函数与合冲

Cohomological rank functions and syzygies of projective bundles on abelian varieties

本文利用Jiang–Pareschi上同调秩函数及Caucci–Ito技术研究阿贝尔簇上射影丛的合冲，从而推广并细化了Chiantipalli关于性质$N_p$的结果。

复射影空间上的四次全纯叶状结构

Holomorphic foliations of degree four on the complex projective space

研究复射影空间上四次全纯叶状结构的结构分类，并对高阶射流的 $d\geq4$ 次叶状结构证明其必属于横截仿射、横截射影或有理拉回三种情形之一。

A bound on the irregularity of abelian scrolls in projective space

We prove that the irregularity of a smooth abelian scroll whose dimension is at least half of that of the surrounding projective space is bounded by 2. In addition we discuss some existence results and open problems.

Projective manifolds modeled after hyperquadrics

We classify projective manifolds with flat holomorphic conformal structures.

Stability conditions on product threefolds of projective spaces and Abelian varieties

In this paper, we prove BG-type inequality conjecture for threefolds in the title. In particular, there exist Bridgeland stability conditions on these threefolds.

The Geometry of the Space of Holomorphic Maps from a Riemann Surface into Complex Projective Space

In this paper we study the topology of the spaces Hol(M,P{n},k) of (basepoint preserving) holomorphic maps of a given degree k from a Riemann surface M of genus g>0 into the n-th complex projective space P{n}, n>0. Using symmetric products of the surface as well as the geometry of the associated Abel-Jacobi maps, we construct an explicit topological compactification of Hol(M,P{n},k) which allow us to construct spectral sequences with known E_1-terms that can be used to determine the homology groups of Hol(M,P{n},k). Complete calculations are given for the elliptic case (g=1). Complete rational…

Projective normality of Abelian threefolds

In this article, we show that very ample line bundles of type (1,1,d) on an abelian threefold A, with NS(A)\simeq \ze, give projectively normal embeddings, if d=22,26 or when d\geq 28.

数值平坦叶状结构与全纯泊松几何

Numerically flat foliations and holomorphic Poisson geometry

本文研究了紧Kähler流形上具有数值平坦切丛的光滑全纯叶状结构，证明了其诱导切丛分解等性质，并由此导出了射影流形上秩为2的Poisson结构的全局Weinstein分裂定理及零点存在性的新Hodge理论条件。