Limits of Calabi–Yau metrics when the Kähler class degenerates

By Valentino Tosatti. 2007

101 citations recorded by the source corpus.

We study the behavior of families of Ricci-flat Kahler metrics on a projective Calabi- Yau manifold when the Kahler classes degenerate to the boundary of the ample cone. We prove that if the limit class is big and nef the Ricci-flat metrics converge smoothly on compact sets outside a subvariety to a limit incomplete Ricci-flat metric. The limit can also be understood from algebraic geometry. Einstein metrics, namely metrics with constant Ricci curvature, have been an important subject of study in the field of differential geometry since the early days. The solution of the Calabi Conjecture given by Yau (Y1) in 1976 provided a very powerful existence theorem for Kahler-Einstein metrics with negative or zero Ricci curvature (the negative case was also done independently by Aubin (Au)). This produced a number of nonho- mogeneous examples of Ricci-flat manifolds. These spaces have been named Calabi-Yau manifolds by the physicists in the eighties, and have been thoroughly studied in several different areas of mathematics and physics. Prompted by the physical intuition of mirror symmetry, mathematicians have studied the ways in which Calabi-Yau manifolds can de- generate when they are moving in families. In general both the complex and symplectic (Kahler) structures are changing, and the behavior is not well understood. In this paper we will consider the case when the complex structure is fixed, and so we will be looking at a single compact projective Calabi-Yau manifold. The Kahler class is then allowed to vary inside the ample cone. As long as the class stays inside the cone, the corresponding Ricci-flat metrics vary smoothly, but they will degenerate when the class approaches the boundary of the cone. We will try to understand this degeneration process and see what the limiting space looks like. To introduce our results, let us fix some notation first. LetX be a compact projective Calabi-Yau manifold, of complex dimensionn. This is by definition a projective manifold

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/86145bb6981cacf337a03a330e86ca2b1a23f7bf

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Citations in the corpus

Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields (2011, background): “Kołodziej [24] (see also [16], [36], [31], [6], [33], [34] for substantial improvements and interesting applications of this result):”
Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds (2010, methodology): “Now following the standard Moser iteration argument in [61] with a trick inspired by [56], we are able to get a uniform C0-estimate of the potential function φt.”
Relations between the Kähler cone and the balanced cone of a Kähler manifold (2012, background): “For reader’s convenience, we present some details on how to prove the above lemma in [26].”
Gromov-Hausdorff collapsing of Calabi-Yau manifolds (2013, background): “Thanks to [34] and [41], the diameter of the metrics ω̃t satisfies diamω̃t(M) C, for a constant C > 0 independent of t.”
Kähler currents and null loci (2013, background): “it is the first Chern class of an R-divisor), this was proved by the second author in [55], except for the statement about Gromov-Hausdorff convergence which is due to Rong-Zhang [40] (see also [41, 47]).”
The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits (2014, methodology): “The Ricci-flat metrics ω have a uniform upper bound on their diameter, thanks to [53] (and independently [64]), and have volume bounded below by C−1e−nt, by (4.”
Collapsing of abelian fibred Calabi-Yau manifolds (2011, background): “If α n is strictly positive, then it was proved by the secondnamed author [37] that the Ricci–flat metrics do not collapse, (i.”
A Priori L∞-Estimates for Degenerate Complex Monge–Ampère Equations (2010, unknown): “When n = m, the cohomology class ω0 is big and semiample and this problem has been addressed by several authors recently (see [4, 6, 14, 15]).”
Smooth Convergence Away from Singular Sets (2012, background): “MR 1055713 (91d:32042) [Tos09] Valentino Tosatti, Limits of Calabi-Yau metrics when the Kähler class degenerates, J. Eur.”