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Negative sectional curvature and the product complex structure

Harish Seshadri
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Abstract

We prove that a product complex manifold cannot admit a complete Kähler metric with sectional curvature $K d$, where $c$ and $d$ are constants. In particular, a product domain in $\C$ cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe that there are complete Kähler metrics with negative sectional curvature on $\C$. Hence the upper sectional curvature bound is necessary.

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Harish Seshadri. 2006-02-14. Negative sectional curvature and the product complex structure. https://arxiv.org/abs/math/0602289

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