Birational Calabi--Yau n-folds have equal Betti numbers

TL;DR

This work proves that birational smooth projective Calabi–Yau $n$-folds over $\mathbb C$ share identical Betti numbers. The authors develop and apply Weil $p$-adic measures associated with gauge forms on regular models, and compare $p$-adic integrals of reductions modulo primes to deduce equal Weil zeta functions for corresponding reductions. By invoking the Weil conjectures and comparison with singular cohomology, they deduce $\dim H^i(X,\mathbb C)=\dim H^i(Y,\mathbb C)$ for all $i$, establishing cohomological invariance under birational maps that preserve the canonical class. The approach extends to generalizations where the canonical class is preserved and yields applications to the McKay correspondence, with deeper motivic implications suggested via Kontsevich’s ideas on motivic integration.

Abstract

Let X and Y be two smooth projective n-dimensional algebraic varieties X and Y over C with trivial canonical line bundles. We use methods of p-adic analysis on algebraic varieties over local number fields to prove that if X and Y are birational, they have the same Betti numbers.

Theorems & Definitions (15)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Theorem 2.5: weil, Theorem 2.2.5
• Definition 2.6
• Theorem 2.7
• Theorem 2.8
• Proposition 3.1