On invariance of plurigenera for foliations on surfaces

Invariance of plurigenera fails in positive and mixed characteristic

We construct smooth families of elliptic surface pairs with terminal singularities over a DVR of positive or mixed characteristic $(X,B)\to \mathrm{Spec}R$, such that $P_m(X_k,B_k)>P_m(X_K,B_K)$ for all sufficiently divisible $m>0$. In particular, this shows that invariance of all sufficiently divisible plurigenera does not follow from the MMP and Abundance Conjectures.

凯勒簇的多重典范类形变不变性的若干方面

Aspects of deformation invariance of plurigenera of Kähler varieties

该研究结合准则与凯勒极小模型纲领，证明了凯勒族多重典范类等的多项形变不变性结果，并将广义极化对的相关不变性结论从射影族推广至莫伊谢佐恩族。

多重亏格的不变性与好的极小模型

Invariance of plurigenera and good minimal models

本文将 Nakayama 关于多重亏格形变不变性的结果推广至正或混合特征 DVR 上的奇异对族，并证明了特定一般型簇好极小模型形变的多重亏格不变性及三维 Gorenstein 典范模型的有界性。

Invariance of Plurigenera and boundedness for Generalized Pairs

In this note, we survey some recent developments in birational geometry concerning the boundedness of algebraic varieties. We delineate a strategy to extend some of these results to the case of generalized pairs, first introduced by Birkar and Zhang, when the associated log canonical divisor is ample, and the volume is fixed. In this context, we show a version of deformation invariance of plurigenera for generalized pairs. We conclude by discussing an application to the boundedness of varieties of Kodaira dimension $κ(X)=\dim(X)-1$.

Invariance of Plurigenera

The following conjecture on the deformation invariance of plurigenera is proved. For a smooth projective holomorphic family of compact complex manifolds over the open unit 1-disk such that all the fibers are of general type, every plurigenus of the fiber is independent of the fiber. The proof uses Nadel's multiplier ideal sheaves, Skoda's result on the generation of ideals with L2 estimates with respect to a plurisubharmonic weight, and the extension theorem of Ohsawa-Takegoshi-Manivel for holomorphic top-degree forms which are L2 with respect to a plurisubharmonic weight.

Moduli of weighted stable elliptic surfaces and invariance of log plurigenera

This is the third paper in a series of work on weighted stable elliptic surfaces - elliptic fibrations with section and marked fibers weighted between zero and one. Motivated by Hassett's weighted pointed stable curves, we use the log minimal model program to construct compact moduli spaces parameterizing these objects. Moreoever, we show that the domain of weights admits a wall and chamber structure, we describe the induced wall crossing morphisms on the moduli spaces as the weight vector varies, and describe the surfaces that appear on the boundary of the moduli space. The main technical res…

Canonical singular hermitian metrics on relative logcanonical bundles

This supersedes 0704.0566. We prove the invariance of logarithmic plurigenera for a projective family of KLT pairs and the adjoint line bundle of KLT line bundles. The proof uses the canonical singular hermitian metrics on relative logcanonical bundles.

关于全纯叶层代数重数的拓扑不变性

On the topological invariance of the algebraic multiplicity of holomorphic foliations

本文研究全纯叶层代数重数的拓扑不变性，证明二维下重数为1是拓扑不变量并推广至高维，验证多类叶层的不变性，还修正了Rosas-Bazan的相关结论。