Smooth asymptotics for collapsing Calabi–Yau metrics

Abstract

We prove that Calabi–Yau metrics on compact Calabi–Yau manifolds whose Kähler 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 $k\text{th}$-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.