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- Title
Sectional curvatures of Kähler moduli.
- Authors
P. M. H. Wilson
- Abstract
If X is a compact Kähler manifold of dimension n, we let denote the cone of Kähler classes, and the level set given by classes D with D n=1. This space is naturally a Riemannian manifold and is isometric to the manifold of Kähler forms ? with ? n some fixed volume form, equipped with the Hodge metric, as studied previously by Huybrechts. We study these spaces further, in particular their geodesics and sectional curvatures. Conjecturally, at least for Calabi–Yau manifolds and probably rather more generally, these sectional curvatures should be bounded between and zero. We find simple formulae for the sectional curvatures, and prove both the bounds hold for various classes of varieties, developing along the way a mirror to the Weil–Petersson theory of complex moduli. In the case of threefolds with h 1,1=3, we produce an explicit formula for this curvature in terms of the invariants of the cubic form. This enables us to check the bounds by computer for a wide range of examples. Finally, we explore the implications of the non-positivity of these curvatures.
- Subjects
CALCULUS; COPYING; LETTER services; MATHEMATICS
- Publication
Mathematische Annalen, 2004, Vol 330, Issue 4, p631
- ISSN
0025-5831
- Publication type
Article
- DOI
10.1007/s00208-004-0563-9