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 ‐order remainders that satisfy uniform ‐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 known from previous work of the second‐named author. For , 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.

中文翻译：

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用于折叠 Calabi-Yau 度量的平滑渐近


我们证明了紧凑 Calabi-Yau 流形上的 Calabi-Yau 度量，其 Kähler 类收缩了全态纤维的纤维，具有远离奇异纤维的所有阶数的先验估计。为此，我们证明了这些度量在纤维直径的幂方面的渐近扩展，其中 ‐ 阶余数满足关于折叠的背景度量族的统一估计。这些估计中的常数是统一的，不仅因为它们独立于纤维直径，而且它们仅取决于第二位作者以前的工作中已知的估计中的常数。对于 ，新的估计值通过爆炸和矛盾来证明，并且展开的每个附加项都作为证明余数的一个附加导数的均匀边界的障碍而出现。