Birational Calabi-Yau \(n\)-folds have equal Betti numbers.

*(English)*Zbl 0955.14028
Hulek, Klaus (ed.) et al., New trends in algebraic geometry. Selected papers presented at the Euro conference, Warwick, UK, July 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 264, 1-11 (1999).

The author proves that if \(X\) and \(Y\) are birationally equivalent Calabi-Yau manifolds over \(\mathbb{C}\), then \(X\) and \(Y\) have the same Betti numbers. This result has now been generalised in a number of different directions [see for example C.-L. Wang, Differ. Geom. 50, No. 1, 129-146 (1998)], and the ideas of motivic integration of J. Denef and F. Loeser, Invent. Math. 135, No. 1, 201-232 (1999; Zbl 0928.14004)], but this paper gives the very first proof of this result.

The basic idea is as follows. Assume for simplicity that \(X\) is defined over \(\mathbb{Q}\); then one can define a scheme \({\mathcal X}\) over \(\text{Spec} (\mathbb{Z})\) such that \({\mathcal X}(\mathbb{C}) =X\). By the Weil conjectures, the Betti numbers of \(X\) can be computed by knowing the numbers of rational points of \(X\) modulo \(p^n\) for all \(n\) and a fixed prime \(p\). However, the number of rational points, again by an idea of Weil, can also be computed by integrating certain \(p\)-adic measures over \({\mathcal X}(\mathbb{Q}_p)\) where \(\mathbb{Q}_p\), denotes the local field of \(p\)-adic numbers. The point then is that if \(X\) and \(X'\) are birational, then they differ only on a set of measure zero with respect to this \(p\)-adic measure, and thus the numbers of rational points, and hence the Betti numbers are the same for \(X\) and \(X'\), proving the result.

For the entire collection see [Zbl 0913.00032].

The basic idea is as follows. Assume for simplicity that \(X\) is defined over \(\mathbb{Q}\); then one can define a scheme \({\mathcal X}\) over \(\text{Spec} (\mathbb{Z})\) such that \({\mathcal X}(\mathbb{C}) =X\). By the Weil conjectures, the Betti numbers of \(X\) can be computed by knowing the numbers of rational points of \(X\) modulo \(p^n\) for all \(n\) and a fixed prime \(p\). However, the number of rational points, again by an idea of Weil, can also be computed by integrating certain \(p\)-adic measures over \({\mathcal X}(\mathbb{Q}_p)\) where \(\mathbb{Q}_p\), denotes the local field of \(p\)-adic numbers. The point then is that if \(X\) and \(X'\) are birational, then they differ only on a set of measure zero with respect to this \(p\)-adic measure, and thus the numbers of rational points, and hence the Betti numbers are the same for \(X\) and \(X'\), proving the result.

For the entire collection see [Zbl 0913.00032].

Reviewer: Mark Gross (MR 2000i:14059)

##### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14E05 | Rational and birational maps |

14F45 | Topological properties in algebraic geometry |

14F25 | Classical real and complex (co)homology in algebraic geometry |

14E30 | Minimal model program (Mori theory, extremal rays) |

14G20 | Local ground fields in algebraic geometry |