局部正性、乘子理想和阿贝尔簇的合

algebraic varieties的渐近合势

Asymptotic syzygies of algebraic varieties

本文研究了光滑射影簇X的合轨随着嵌入线丛的正性增长的渐近性态。我们证明，至少就分级而言，X的理想的最小分辨率有一个令人惊讶的一致渐近形状：粗略地说，发电机最终出现在几乎所有的度允许的Castelnuovo-Mumford正则性。这特别表明，一个广泛接受的直觉来自曲线的情况下，即合朔变得更简单的嵌入程度增加，可能是误导。对于射影空间的Veronese嵌入，我们给出了一个有效的证明，证明了在某些情况下，Veronese嵌入是最优的，并且证明了Veronese嵌入总是最优的.最后，我们提出了一些问题和开放的问题，关于高维簇的渐近syzygies。

Syzygies of projective varieties of large degree: recent progress and open problems

This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. After a quick overview of results from the 1980s and 1990s concerning the linearity of the first few terms of a resolution, we discuss a non-vanishing theorem to the effect that from an asymptotic viewpoint, essentially all of the syzygy modules that could be non-zero are in fact non-zero. We explain the quick new proof of this result in the case of Veronese varieties due to Erman and authors, and we explore some results a…

On Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves on singular complex spaces and regularity of weak Kähler-Einstein metrics

In this article, we will characterize regular points respectively by the local vanishing, positivity of the Ricci curvature and $L^2$-solvability of the $\overline\partial$-equation together with Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubharmonic (psh) functions on any (reduced) complex space of pure dimension. As a by-product, we show that any weak Kähler-Einstein metric on \emph{singular} $\mathbb{Q}$-Fano/Calabi-Yau/general type varieties cannot be smooth, and that in general there exists no \emph{singular} normal Kähler complex space such that the Kä…

Strong openness of multiplier ideal sheaves and optimal $L^{2}$ extension

In this note, we reveal that our solution of Demailly's strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Kollár and Jonsson-Mustată implies the truth of twisted versions of the strong openness conjecture; our optimal $L^{2}$ extension implies Berndtsson's positivity of vector bundles associated to holomorphic fibrations over a unit disc.