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Kähler manifolds and the curvature operator of the second kind
Li, Xiaolong
Li, Xiaolong
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2023-03-22
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Curvature operator of the second kind,Orthogonal bisectional curvature,Holomorphic sectional curvature,Rigidity theorems
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Citation
Li, X. Kähler manifolds and the curvature operator of the second kind. Math. Z. 303, 101 (2023). https://doi.org/10.1007/s00209-023-03263-0
Abstract
This article aims to investigate the curvature operator of the second kind on Kähler manifolds. The first result states that an m-dimensional Kähler manifold with $$\frac{3}{2}(m^2-1)$$-nonnegative (respectively, $$\frac{3}{2}(m^2-1)$$-nonpositive) curvature operator of the second kind must have constant nonnegative (respectively, nonpositive) holomorphic sectional curvature. The second result asserts that a closed m-dimensional Kähler manifold with $$\left( \frac{3m^3-m+2}{2m}\right) $$-positive curvature operator of the second kind has positive orthogonal bisectional curvature, thus being biholomorphic to $${{\mathbb {C}}}{{\mathbb {P}}}^m$$. We also prove that $$\left( \frac{3m^3+2m^2-3m-2}{2m}\right) $$-positive curvature operator of the second kind implies positive orthogonal Ricci curvature. Our approach is pointwise and algebraic.
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Preprint version available from arXiv.
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Mathematische Zeitschrift
Volume 303, No. 4
Volume 303, No. 4
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1432-1823