Holomorphic Euler number of K$\ddot{a}$hler manifolds with almost nonnegative Ricci curvature

Xiaoyang Chen

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arXiv · 1909.05116

Holomorphic Euler number of K$\ddot{a}$hler manifolds with almost nonnegative Ricci curvature

Xiaoyang Chen

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Abstract

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A crucial step in the proof is to show a vanishing theorem of Dolbeault-Morse-Novikov cohomology.

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Xiaoyang Chen. 2019-09-10. Holomorphic Euler number of K$\ddot{a}$hler manifolds with almost nonnegative Ricci curvature. https://arxiv.org/abs/1909.05116

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