On the volume of a pseudo-effective class and semi-positive properties of the Harder–Narasimhan filtration on a compact Hermitian manifold

Zhiwei Wang¹

¹ School of Mathematical Sciences Peking University Beijing 100871, China

Annales Polonici Mathematici, Tome 117 (2016) no. 1, pp. 41-58

Résumé

This paper divides into two parts. Let $(X,\omega )$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $\omega $ satisfies the assumption that $\partial \overline {\partial }\omega ^k=0$ for all $k$, we generalize the volume of the cohomology class in the Kähler setting to the Hermitian setting, and prove that the volume is always finite and the Grauert–Riemenschneider type criterion holds true, which is a partial answer to a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical bundle $K^{-1}_X$ is nef, then for any $\varepsilon \gt 0$, there is a smooth function $\phi _\varepsilon $ on $X$ such that $\omega _\varepsilon :=\omega +i\partial \overline {\partial }\phi _\varepsilon \gt 0$ and Ricci$(\omega _\varepsilon )\geq -\varepsilon \omega _\varepsilon $. Furthermore, if $\omega $ satisfies the assumption as above, we prove that for a Harder–Narasimhan filtration of $T_X$ with respect to $\omega $, the slopes $\mu _\omega (\mathcal {F}_i/\mathcal {F}_{i-1})$ are nonnegative for all $i$; this generalizes a result of Cao which plays an important role in his study of the structures of Kähler manifolds.

Détail

DOI : 10.4064/ap3780-11-2015

Keywords: paper divides parts omega compact hermitian manifold firstly hermitian metric omega satisfies assumption partial overline partial omega nbsp generalize volume cohomology class hler setting hermitian setting prove volume always finite grauert riemenschneider type criterion holds which partial answer conjecture posed boucksom secondly observe anticanonical bundle nef varepsilon there smooth function phi varepsilon omega varepsilon omega partial overline partial phi varepsilon ricci omega varepsilon geq varepsilon omega varepsilon furthermore omega satisfies assumption above prove harder narasimhan filtration respect omega slopes omega mathcal mathcal i nonnegative generalizes result cao which plays important role his study structures hler manifolds