Euler Characteristics of Crepant Resolutions of Weierstrass Models

Based on an identity of Jacobi, we prove a simple formula that computes the pushforward of analytic functions of the exceptional divisor of a blowup of a projective variety along a smooth complete intersection with normal crossing. We use this pushforward formula to derive generating functions for Euler characteristics of crepant resolutions of singular Weierstrass models given by Tate’s algorithm. Since the Euler characteristic depends only on the sequence of blowups and not on the Kodaira fiber itself, several distinct Tate models have the same Euler characteristic. In the case of elliptic Calabi–Yau threefolds, using the Shioda–Tate–Wazir theorem, we also compute the Hodge numbers. For elliptically fibered Calabi–Yau fourfolds, our results also prove a conjecture of Blumenhagen, Grimm, Jurke, and Weigand based on F-theory/heterotic string duality.