Unstable Kodaira Fibrations

A refined Kodaira dimension and its canonical fibration

Given a (meromorphic) fibration $f:X\to Y$ where $X$ and $Y$ are compact complex manifolds of dimensions $n$ and $m$, we define $L_f$ to be the invertible subsheaf of the sheaf of holomorphic $m$-forms of $X$ given by the saturation of $f^*K_Y$, where $K_Y$ is the canonical sheaf of $Y$. We define the Kodaira dimension of the orbifold base $(Y,f)$ by that of $L_f$ and call the fibration to be of general type if this dimension is the maximal $\dime Y$. We call $X$ special if it does not have a general type fibration with positive dimensional base and call $f$ special if its general fibers are. …

Convergence of scalar curvature of Kahler-Ricci flow on manifolds of positive Kodaira dimension

In this paper, we consider Kahler-Ricci flow on n-dimensional Kahler manifold with semi-ample canonical line bundle and 0< m:= Kod(X)<n. Such manifolds admit a Calabi-Yau fibration over its canonical model. We prove that the scalar curvature of the Kahler metric along the normalized Kahler-Ricci flow converge to -m outside the singular set of this fibration.

On the moduli part of the Kawamata-Kodaira canonical bundle formula

It is conjectured that the moduli b-divisor of the Kawamata-Kodaira canonical bundle formula associated to a klt-trivial fibration $(X,B)\to Z$ is semi-ample. In this paper, we show the semi-ampleness of an arbitrarily small perturbation of the moduli b-divisor by a fixed appropriate divisor which roughly speaking comes from a section of $K_X+B$. We apply the above result to settle a conjecture of Fujino and Gongyo: if $f\colon X\to Z$ is a smooth surjective morphism of smooth projective varieties with $-K_X$ semi-ample, then $-K_Z$ is also semi-ample. We list several counter-examples to show …

K-模空间的准映射与K-稳定的Calabi-Yau纤维模空间的准射影性

K-moduli of quasimaps and on quasi-projectivity of moduli of K-stable Calabi-Yau fibrations over curves

本文研究了K-稳定的Calabi-Yau纤维模空间的准射影性，通过构造具有特定log Fano条件的K-模空间，并探讨了其与准映射K-模空间的关系。