Abstract

We conjecture that certain curvature invariants of compact hyperkähler manifolds are positive/negative. We prove the conjecture in complex dimension four, give an “experimental proof” in higher dimensions, and verify it for all known hyperkähler manifolds up to dimension eight. As an application, we show that our conjecture leads to a bound on the second Betti number in all dimensions.

中文翻译：

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Hyperkähler 流形上的拓扑界

摘要

我们推测紧致超凯勒流形的某些曲率不变量是正/负的。我们在复数四维中证明了猜想，在更高维中给出了“实验证明”，并针对所有已知的高达八维的超凯勒流形对其进行了验证。作为一个应用，我们证明了我们的猜想导致了所有维度中第二个 Betti 数的界限。