# Euler Characteristics of Crepant Resolutions of Weierstrass Models

## TL;DR

This paper derives a formula for computing Euler characteristics of crepant resolutions of Weierstrass models, revealing that these characteristics depend only on the sequence of blowups, and confirms a conjecture in string theory contexts.

## Contribution

It introduces a new pushforward formula for analytic functions of exceptional divisors and applies it to compute Euler characteristics and Hodge numbers of crepant resolutions, confirming a string theory conjecture.

## Key findings

- Euler characteristics depend only on blowup sequences
- Derived generating functions for Euler characteristics of Weierstrass models
- Confirmed a conjecture relating to F-theory/heterotic duality

## 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.