Showing papers on "Calabi conjecture published in 1999"

Kähler–Einstein metrics on Kummer threefold and special Lagrangian tori

Abstract: The concept of special Lagrangian submanifolds is introduced by Harvey and Lawson in the seminal paper [HL]. In [SYZ], Strominger, Yau and Zaslow propose a construction of the mirror threefold Z of a given Calabi-Yau threefold X using special Lagrangian tori. Roughly speaking, the mirror threefold Z should be realized as the compactified moduli space of special Lagrangian 3-torus T on X together with flat ?7(l)-connections on T. In particular, if the mirror threefold of X exists, then X should admit a fibration with general fiber being a special Lagrangian 3-torus. Recall that a Calabi-Yau manifold is a compact connected Kahler manifold with vanishing first Chern class. By Yau's theorem ([Y]), given a Calabi-Yau manifold X there exists an unique Ricci-flat metric UJ in the Kahler class (Here we have identified a Kahler metric with its associated Kahler form). Note that a Ricci-flat Kahler metric is equivalent to a Riemannian metric whose holonomy group is a subgroup of SU(n). This metric is the so-called Calabi-Yau metric. Now we recall the notion of special Lagrangian submanifolds.

Asymptotically Locally Euclidean metrics with holonomy SU(m)

Abstract: Let G be a nontrivial finite subgroup of U(m) acting freely on C^m - 0 Then C^m/G has an isolated quotient singularity at 0 Let X be a resolution of C^m/G, and g a Kahler metric on X We say that g is Asymptotically Locally Euclidean (ALE) if it is asymptotic in a certain way to the Euclidean metric on C^m/G In this paper we study Ricci-flat ALE Kahler metrics on X We show that if G is a subgroup of SU(m) acting freely on C^m - 0, and X is a crepant resolution of C^m/G, then there is a unique Ricci-flat ALE Kahler metric in each Kahler class This is proved using a version of the Calabi conjecture for ALE manifolds We also show the metrics have holonomy SU(m) These results will be applied in the author's book ("Compact manifolds with special holonomy", to be published by OUP, 2000) to construct new examples of compact 7- and 8-manifolds with exceptional holonomy They can also be used to describe the Calabi-Yau metrics on resolutions of a Calabi-Yau orbifold The paper has a sequel, "Quasi-ALE metrics with holonomy SU(m) and Sp(m)", mathAG/9905043, which studies Kahler metrics on resolutions of non-isolated singularities C^m/G

Quasi-ALE metrics with holonomy SU(m) and Sp(m)

Abstract: This is the sequel to "Asymptotically Locally Euclidean metrics with holonomy SU(m)", math.AG/9905041. Let G be a subgroup of U(m), and X a resolution of C^m/G. We define a special class of Kahler metrics g on X called Quasi Asymptotically Locally Euclidean (QALE) metrics. These satisfy a complicated asymptotic condition, implying that g is asymptotic to the Euclidean metric on C^m/G away from its singular set. When C^m/G has an isolated singularity, QALE metrics are just ALE metrics. Our main interest is in Ricci-flat QALE Kahler metrics on X. We prove an existence result for Ricci-flat QALE Kahler metrics: if G is a subgroup of SU(m) and X a crepant resolution of C^m/G, then there is a unique Ricci-flat QALE Kahler metric on X in each Kahler class. This is proved using a version of the Calabi conjecture for QALE manifolds. We also determine the holonomy group of the metrics in terms of G. These results will be applied in the author's book ("Compact manifolds with special holonomy", to be published by OUP, 2000) to construct new examples of compact 7- and 8-manifolds with exceptional holonomy. They can also be used to describe the Calabi-Yau metrics on resolutions of a Calabi-Yau orbifold.