# Euler Characteristics of Crepant Resolutions of Weierstrass Models

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## 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 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.

## Notes

### Acknowledgements

The authors are grateful to Paolo Aluffi, Jim Halverson, Remke Kloosterman, Cody Long, Kenji Matsuki, Julian Salazar, Shu-Heng Shao, and Shing-Tung Yau for helpful discussions. The authors would like in particular to acknowledge Andrea Cattaneo for many useful comments and suggestions. The authors are thankful to all the participants of the workshop “A Three-Workshop Series on the Mathematics and Physics of F-theory” supported by the National Science Foundation (NSF) Grant DMS-1603247. M.E. is supported in part by the National Science Foundation (NSF) Grant DMS-1701635 “Elliptic Fibrations and String Theory”. P.J. is supported by NSF Grant PHY-1067976. P.J. would like to extend his gratitude to Cumrun Vafa for his tutelage and continued support. M.J.K. would like to acknowledge partial support from NSF Grant PHY-1352084. M.J.K. is thankful to Daniel Jafferis for his guidance and constant support.

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