Euler Characteristics of Crepant Resolutions of Weierstrass Models

Abstract

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 apply this pushforward formula to derive generating functions for Euler characteristics of crepant resolutions of singular Weierstrass models given by Tate's algorithm. Since these Euler characteristics depend only on the sequence of blowups and not on the Kodaira fiber itself, nor the associated group, several distinct Tate models have the same Euler characteristic. In the case of elliptic Calabi-Yau threefolds, we also compute the Hodge numbers. For elliptically fibered Calabi-Yau fourfolds, our results also prove a conjecture of Blumenhagen-Grimm-Jurke-Weigand based on F-theory/heterotic string duality.

v4:Typos corrected, references added, 30 pages + Appendix (10 pages), 11 Tables, accepted for publication in Communications in Mathematical Physics

References in corpus (3)

• D-brane Deconstructions in IIB Orientifolds
• Mordell-Weil Torsion and the Global Structure of Gauge Groups in F-theory
• Box Graphs and Singular Fibers

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