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Euler Characteristics of Crepant Resolutions of Weierstrass Models

Mboyo Esole, Patrick Jefferson, Monica Jinwoo Kang

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.

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.

Paper Structure

This paper contains 26 sections, 29 theorems, 106 equations, 1 figure, 11 tables.

Table of Contents

  1. Introduction
  2. Conventions
  3. $G$-models
  4. The pushforward theorem and Jacobi's identity
  5. Strategy
  6. Organization of the paper
  7. Euler Characteristic of Elliptic Fibrations
  8. Crepant resolutions and flops
  9. Batyrev's theorem and the Chern class of a crepant resolution
  10. Pushforwards and Computing the Euler Characteristic
  11. The pushforward theorem
  12. Classes of the blowup centers of crepant resolutions
  13. Hodge Numbers of Elliptically Fibered Calabi-Yau Threefolds
  14. An Illustrative Example: SU(2)-Models
  15. Tables of Results
  16. ...and 11 more sections

Key Result

Theorem 1.8

Let the nonsingular variety $Z\subset X$ be a complete intersection of $d$ nonsingular hypersurfaces $Z_1$, …, $Z_d$ meeting transversally in $X$. Let $E$ be the class of the exceptional divisor of the blowup $f:\widetilde{X}\longrightarrow X$ centered at $Z$. Let $\widetilde{Q}(t)=\sum_a f^* Q_a t^

Figures (1)

  • Figure 1.1: Twisted affine Lie algebras vs affine Lie algebras for non-simply laced cases. Only those on the left appears in the theory of elliptic fibrations as dual graphs of the fiber over the generic point of an irreducible component of the discriminant locus.

Theorems & Definitions (78)

  • Example 1.1
  • Definition 1.2: $\mathcal{K}$-model
  • Definition 1.3: $G$-model
  • Example 1.4
  • Example 1.5: See MP
  • Example 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 1.9: Jacobi
  • ...and 68 more