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

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Wang Z.

Annales Polonici Mathematici

2016

tom 117

nr 1

41-58

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Wang Z.

1 2016

Wyniki wyszukiwania

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