Quantitative extensions of pluricanonical forms and closed positive currents

Positivity of twisted relative pluricanonical bundles and their direct images

Our main goal in this article is to establish a quantitative version of the positivity properties of twisted relative pluricanonical bundles and their direct images. The notion of "singular Hermitian metric" on vector bundles (together with an appropriate definition of positivity of the associated curvature) plays a major role here, and its properties are studied via a version of Ohsawa-Takegoshi extension theorem. Part of this article is based on the joint work of the first named author with Bo Berndtsson, and it can be seen as an expanded and updated version of it.

Boundedness of log-pluricanonical maps for surfaces of log-general types in positive characteristic

In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, Δ)$ satisfying the following properties: (i) $X$ is a projective surface defined over an algebraically closed field, (ii) $(X, Δ)$ is log canonical and the coefficients of $Δ$ are in $I$, and (iii) $K_X+Δ$ is big. Then there is a positive integer $N=N(I)$ depending only on the set $I$ such that the linear system $|\lceil m(K_X+Δ)\rceil|$ defines a birational map onto its image for all $m\geq N$ and $(X, Δ)\in\mathfrak{D}$

通过数值不变量研究复流形的定量双曲性

Quantitative hyperbolicity for complex manifolds via numerical invariants

研究通过引入双曲指标这一数值不变量来研究复流形的双曲性，结合喷流微分和密度流方法，证明了\(\mathbb{P}^{n + 1}\)中一般超曲面\(X_d\)双曲指标随\(d\)至少线性增长趋于无穷，并讨论了小林猜想的解析方法。