Smooth asymptotics for collapsing Calabi-Yau metrics
arXiv:2102.03978 · doi:10.1002/cpa.22228
Abstract
We prove that Calabi-Yau metrics on compact Calabi-Yau manifolds whose Kahler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with k-th order remainders that satisfy uniform C^k-estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for k=0 known from previous work of the second-named author. For k>0 the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.
arXiv comment98 pages; final version to appear in Comm. Pure Appl. Math
References in corpus (15)
- The Kähler-Ricci flow on surfaces of positive Kodaira dimension
- Adiabatic limits of Ricci-flat Kahler metrics
- The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits
- Collapsing of abelian fibred Calabi-Yau manifolds
- Remarks on the collapsing of torus fibered Calabi-Yau manifolds
- Positivity of relative canonical bundles and applications
- Infinite time singularities of the Kähler-Ricci flow
- Higher-order estimates for collapsing Calabi-Yau metrics
- Gravitational instantons with faster than quadratic curvature decay (III)
- Collapsing hyperkähler manifolds
- Triviality of fibered Calabi-Yau manifolds without singular fibers
- Degenerations of Calabi-Yau metrics
- Calabi-Yau manifolds and their degenerations
- Collapsing Calabi-Yau manifolds
- Geometry of twisted Kähler-Einstein metrics and collapsing