代数纤维空间中反典范除子的正性

Albanese morphism of log smooth klt compact Kähler manifold with nef log anticanonical divisor

Let $(X, ω)$ be an n-dimensional compact Kähler manifold. Let $D=\sum (1-β_j) Y_j=\sum (1-β_j) [s_j=0]$ a divisor with simple normal crossings with $β_j \in ]0,1[$ such that $-(K_X+D)$ is nef. We show that its Albanese map is submersion outside an analytic set of codimension larger than two with connected fibres.

Structure of projective varieties with nef anticanonical divisor: the case of log terminal singularities

In this article we study the structure of klt projective varieties with nef anticanonical divisor (and more generally, varieties of semi-Fano type), especially the canonical fibrations associated to them. We show that: 1. the Albanese map for such variety is a locally constant fibration (that is, an analytic fibre bundle with connected fibres that $X$ is equal to the product of the universal cover of the Albanese torus by the fibre of the Albanese map quotient by a diagonal action of the fundamenatl group of the Albanese torus); 2. if the smooth locus is simply connected, the MRC fibration of …

局部常值纤维化与曲率正性

Locally Constant Fibrations and Positivity of Curvature

该数学代数几何领域研究证明了底为K-平凡簇、转移函数局部常值的射影纤维丛具有nef反典范丛，验证了相关结构定理的最优性，并补充了切丛带正曲率奇异埃尔米特度量的簇的结构理论结果。

Examples of Fano manifolds with non-pseudoeffective tangent bundle

Let $X$ be a Fano manifold. While the properties of the anticanonical divisor $-K_X$ and its multiples have been studied by many authors, the positivity of the tangent bundle $T_X$ is much more elusive. We give a complete characterisation of the pseudoeffectivity of $T_X$ for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.

A new method in Fano geometry

We give some bounds on the anticanonical degrees of Fano varieties with Picard number 1 and mild singularities, extending results of Kollár et al. from the early 90's and improving them even in the smooth case. The proof is based on a study of positivity properties of sheaves of differential operators on ample line bundles, and avoids the use of rational curves and bend-and-break. This note is a self-contained exposition of the main ideas of math.AG/9811022

完全交族的新斜率不等式

New slope inequalities for families of complete intersections

本文证明完全交纤维化中 $\mathcal{O}_X(1)$ 的 $f$-正性，给出整体完全交情形下幂次与相对典范层 $f$-正性的数值充要条件，并导出 Chow 不稳定条件。

Calabi-Yau Varieties with Fibre Structures

Motivated by the Strominger-Yau-Zaslow conjecture, we study fibre spaces whose total space has trivial canonical bundle. Especially, we are interest in Calabi-Yau varieties with fibre structure. In this paper, we only consider semi-stable families. We use Hodge theory and the generalized Donaldson-Simpson-Uhlenbeck-Yau correspondence to study the parabolic structure of higher direct images over higher dimensional quasi-projective base, and obtain an important result on parabolic-semi-positivity. We then apply this result to study nonisotrivial Calabi-Yau varieties fibred by Abelian varieties (…

On the Iitaka Conjecture $C_{n,m}$ for Kähler Fibre Spaces

By applying the positivity theorem of direct images and a pluricanonical version of the structure theorem on the cohomology jumping loci à la Green-Lazarsfeld-Simpson, we show that the klt Kähler version of the Iitaka conjecture $C_{n,m}$ (Ueno, 1975) for $f:X\to Y$ (surjective morphism between compact Kähler manifolds with connected general fibre) holds true when the determinant of the direct image of some power of the relative canonical bundle is big on $Y$ or when $Y$ is a complex torus. These generalize the corresponding results of Viehweg (1983) and of Cao-P\u aun (2017) respectively. We …

正特征中的典范丛公式

On the canonical bundle formula in positive characteristic

在正特征域上，研究纤维化dlt对的模部分在f-nef条件下的nef性质，并证明在K-平凡情形下模部分的正性，从而在三维情形无条件得到典范丛公式。

相干概形上的除子与齐性空间

Divisors on coherent schemes and homogeneous spaces

本文研究相干基上齐性空间上除子的正性与可延展性，发展了局部相干概形上的估值除子理论，建立了有效估值除子与秩为一的反射sheaf的精确对应关系，推导出非Noetherian的Ramanujam-Samuel定理。

全纯向量丛与模空间的$m$-正稳定性

$m$-Positive Stability of Holomorphic Vector Bundles and Moduli Spaces

研究提出高阶向量丛的$m$-正性概念，通过关联光滑函数推广芒福德 - 竹本稳定性理论，强调全纯结构等，建立与埃尔米特 - 爱因斯坦几何联系，证明相关丛的性质并研究模空间，比较新旧稳定性概念。