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

Zhiwei Wang. "On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold." Annales Polonici Mathematici 117.1 (2016): 41-58. <http://eudml.org/doc/286170>.

@article{ZhiweiWang2016,
abstract = {This paper divides into two parts. Let (X,ω) be a compact Hermitian manifold. Firstly, if the Hermitian metric ω satisfies the assumption that $∂∂̅ω^\{k\} = 0$ 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 ε > 0, there is a smooth function $ϕ_\{ε\}$ on X such that $ω_\{ε\} := ω + i∂∂̅ϕ_\{ε\} > 0$ and Ricci $(ω_\{ε\}) ≥ -εω_\{ε\}$. Furthermore, if ω satisfies the assumption as above, we prove that for a Harder-Narasimhan filtration of $T_\{X\}$ with respect to ω, the slopes $μ_\{ω\}(ℱ_\{i\}/ℱ_\{i-1\})$ 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.},
author = {Zhiwei Wang},
journal = {Annales Polonici Mathematici},
keywords = {cohomology; nef class; pseudo-effective class; big class; closed positive current; Gauduchon metric; Monge-Ampére equation; Harder-Narasimhan filtration; stability},
language = {eng},
number = {1},
pages = {41-58},
title = {On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold},
url = {http://eudml.org/doc/286170},
volume = {117},
year = {2016},
}

TY - JOUR
AU - Zhiwei Wang
TI - On the volume of a pseudo-effective class and semi-positive properties of the Harder-Narasimhan filtration on a compact Hermitian manifold
JO - Annales Polonici Mathematici
PY - 2016
VL - 117
IS - 1
SP - 41
EP - 58
AB - This paper divides into two parts. Let (X,ω) be a compact Hermitian manifold. Firstly, if the Hermitian metric ω satisfies the assumption that $∂∂̅ω^{k} = 0$ 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 ε > 0, there is a smooth function $ϕ_{ε}$ on X such that $ω_{ε} := ω + i∂∂̅ϕ_{ε} > 0$ and Ricci $(ω_{ε}) ≥ -εω_{ε}$. Furthermore, if ω satisfies the assumption as above, we prove that for a Harder-Narasimhan filtration of $T_{X}$ with respect to ω, the slopes $μ_{ω}(ℱ_{i}/ℱ_{i-1})$ 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.
LA - eng
KW - cohomology; nef class; pseudo-effective class; big class; closed positive current; Gauduchon metric; Monge-Ampére equation; Harder-Narasimhan filtration; stability
UR - http://eudml.org/doc/286170
ER -