Showing papers on "Calabi conjecture published in 2004"

Weil–petersson geometry on moduli space of polarized calabi–yau manifolds

Abstract: In this paper, we define and study the Weil–Petersson geometry. Under the framework of the Weil–Petersson geometry, we study the Weil–Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil–Petersson metric and the Ricci curvature of the Weil–Petersson metric for Calabi–Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.

Limiting behavior of local Calabi-Yau metrics

Abstract: We use a generalization of the Gibbons-Hawking ansatz to study the behavior of certain non-compact Calabi-Yau manifolds in the large complex structure limit. This analysis provides an intermediate step toward proving the metric collapse conjecture for toric hypersurfaces and complete intersections.

Nef Reduction and Anticanonical Bundles

Abstract: Projective manifolds X with nef anticanonical bundles (i.e. - K X ∙ C = det T X ∙ C ≥ 0 for all curves C ⊂ X ) can be regarded as an interpolation between Fano manifolds (ample anticanonical bundle) and Calabi-Yau manifolds resp. tori and symplectic manifolds (trivial canonical bundle). A differential-geometric analogue are varieties with semi-positive Ricci curvature although this class is strictly smaller -- to get the correct picture one has to consider sequences of metrics and make the negative part smaller and smaller. However we will work completely in the context of algebraic geometry. Our aim is twofold: classification and, as a consequence, boundedness in case of dimension 3. We shall not consider threefolds with trivial canonical bundles, the eventual boundedness of Calabi-Yau threefolds still being unknown. Fano threefolds have been classified a long time ago and threefolds with big and nef anticanonical bundle are very much related with Q-Fano threefolds; therefore we will concentrate here on projective threefolds X with - KX nef and K3X = 0, but KX ≢ 0. The essential problem is to distinguish the positive and flat directions in X.

Quotients of E^n by A_{n+1} and Calabi-Yau manifolds

Abstract: We give a simple construction, starting with any elliptic curve E, of an n-dimensional Calabi-Yau variety of Kummer type (for any n>1), by considering the quotient Y of the n-fold self-product of E by a natural action of the alternating group A_{n+1} (in n+1 variables). The vanishing of H^m(Y, O_Y) for 0