Smooth asymptotics for collapsing Calabi-Yau metrics

TL;DR

This paper establishes uniform high-order estimates for collapsing Calabi-Yau metrics on fibered manifolds, providing an asymptotic expansion in terms of fiber diameter with controlled remainders.

Contribution

It introduces a detailed asymptotic expansion of collapsing Calabi-Yau metrics with uniform estimates, extending previous results to all derivative orders.

Findings

Proves uniform C^k-estimates for collapsing metrics

Derives an asymptotic expansion in fiber diameter

Identifies obstructions to higher derivative bounds

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.