Adiabatic limits of Ricci-flat Kähler metrics

By Valentino Tosatti. 2009

137 citations recorded by the source corpus.

We study adiabatic limits of Ricci-flat Kahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Ampere equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metric has Ricci curvature equal to a Weil-Petersson metric that measures the variation of complex structure of the Calabi-Yau fibers. This generalizes results of Gross-Wilson for K3 surfaces to higher dimensions.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d5c11b42761ab0733f268e079e2a0a2773377412

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Interior derivative estimates for the Kähler–Ricci flow (2011, unknown): “There exist now many generalizations of Calabi’s estimate [Cherrier 1987; Tosatti 2010; Tosatti et al. 2008; Zhang and Zhang 2011].”
The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits (2014, background): “The following estimates are already known by the work of the first-named author [54].”
KAWA lecture notes on the Kähler-Ricci flow (2015, background): “[75]), but note that this convergence falls short of the one obtained in Theorem 5.”
Continuity of Extremal Transitions and Flops for Calabi-Yau Manifolds (2010, unknown): “[2], [8], [9], [13], [30], [37], [41], [56], [57], [58], [49], [60], [64]).”
Remarks on the collapsing of torus fibered Calabi–Yau manifolds (2014, unknown): “2] (which relies on some estimates from [12]) shows that there exists a constant CB such that C−1 B p ∗(ω0 + ωSF) λt p∗ω̃t CBp∗(ω0 + ωSF) holds on B × C, uniformly in t ∈ (0, 1].”
Collapsing of abelian fibred Calabi-Yau manifolds (2011, background): “We need to show that as t goes to zero we have ut → (f ◦ p)∗φ in C∞ loc(B×C, δ), where f∗φ is the C1,α limit of φt from [38].”
Gromov-Hausdorff collapsing of Calabi-Yau manifolds (2013, background): “By [35] and [12], ω̃t converges smoothly to f ∗ω on f−1(K) for any compact K ⊂ N0 as t → 0, and on N0, where ω is the Kähler metric on N0 with Ric(ω) = ωWP obtained in [35] and [29] (see also [28]), and ωWP is a Weil-Petersson semipositive form on N0 coming from the variation of the complex structur”
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):”
Infinite time singularities of the Kähler-Ricci flow (2014, methodology): “In the first setup (which is the same as in Tosatti [24], Gross, Tosatti and”