Topological Bounds on Hyperkähler Manifolds

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.

KEYWORDS:

• Hyperkähler manifolds
• Chern numbers
• Betti numbers

Acknowledgments

The author would like to thank Simon Salamon for comments on an earlier draft of this paper.

Notes

1 We use the form interface to LiE at wwwmathlabo.univ-poitiers.fr/∼maavl/LiE/form.html.

Additional information

Funding

The author would like to thank the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support, and the NSF for support through grants DMS-1206309, DMS-1555206, and DMS-2152130.