Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs.

*(English)*Zbl 0943.14004Let \(V\) be a smooth complex algebraic variety having a regular action of a finite group \(G\). The main result of this paper is that the physicists’ orbifold Euler number \(e(V,G)\) defined by the Dixon-Harvey-Vafa-Witten formula is equal to a “stringy” invariant of the orbifold \(V/G\). Let \(X\) be a normal irreducible algebraic variety over \(\mathbb C\), and \(\Delta_X\) a \(\mathbb Q\)-Weil divisor on \(X\). After recalling the notion of a Kawamata log-terminal pair, Batyrev associates a stringy \(E\)-function \(E_{\text{st}}(X,\Delta_X;u,v)\) and a stringy Euler number \(e_{\text{st}}(X,\Delta_X)\) with an arbitrary such pair. Integration over the space of arcs of a nonsingular algebraic variety is needed to prove that the above definitions, given in terms of a log-resolution of the pair, make sense. This technique was introduced by Kontsevich in connection with a previous result of the author [V. V.Batyrev, “Birational Calabi-Yau \(n\)-folds have equal Betti numbers” in: New trends in algebraic geometry, Sel. Pap. Euro Conf., Warwick 1996, Lond. Math., Soc. Lect. Note Ser. 264, 1-11 (1999)], and further developed by Denef-Loeser. Sections 1 to 3, where the above material is settled, and section 4 where an explicit formula for the stringy \(E\)-function of a Kawamata log-terminal pair \((X,\Delta_X)\), with \(X\) toric and \(\Delta_X\) \(T\)-invariant is given, are an abridged version of another paper [V. V. Batyrev, in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30–July 4, 1997, and in Kyoto, Japan, July 7–11 1997. Singapore: World Scientific. 1–32 (1998; Zbl 0963.14015)].

In section 7, a canonical Kawamata log-terminal pair \((X,\Delta_X)\) with \(X:=V/G\) is attached with \((V,G)\) as above, and the equality \(e_{\text{st}}(X,\Delta_X)=e(V,G)\) is proved. The proof is based on a canonical abelianization process leading to a partial desingularization of \(X\) whose singularities are quotient singularities by abelian groups, hence toroidal. It also uses explicit computations of the orbifold \(E\)-function \(E_{\text{orb}}(V,G;u,v)\) defined in section 6, and of the stringy \(E\)-function of its associated Kawamata log-terminal pair, for \(V={\mathbb C}^n\) and \(G\) a finite abelian subgroup of \(\text{GL}(n,{\mathbb C})\) acting by diagonal matrices.

Finally, as an application, the author proves a cohomological version of the McKay correspondence for the quotient singularity \(X:={\mathbb C}^n/G\) by a finite subgroup \(G\) of \(\text{ SL}(n,{\mathbb C})\), assuming the existence of a crepant desingularization of \(X\). An alternative proof of this result may be found in a preprint by J. Denef and F. Loeser, “Motivic integration, quotient singularities and the McKay correspondence”, 21 p., available at math.AG/9903187.

In section 7, a canonical Kawamata log-terminal pair \((X,\Delta_X)\) with \(X:=V/G\) is attached with \((V,G)\) as above, and the equality \(e_{\text{st}}(X,\Delta_X)=e(V,G)\) is proved. The proof is based on a canonical abelianization process leading to a partial desingularization of \(X\) whose singularities are quotient singularities by abelian groups, hence toroidal. It also uses explicit computations of the orbifold \(E\)-function \(E_{\text{orb}}(V,G;u,v)\) defined in section 6, and of the stringy \(E\)-function of its associated Kawamata log-terminal pair, for \(V={\mathbb C}^n\) and \(G\) a finite abelian subgroup of \(\text{GL}(n,{\mathbb C})\) acting by diagonal matrices.

Finally, as an application, the author proves a cohomological version of the McKay correspondence for the quotient singularity \(X:={\mathbb C}^n/G\) by a finite subgroup \(G\) of \(\text{ SL}(n,{\mathbb C})\), assuming the existence of a crepant desingularization of \(X\). An alternative proof of this result may be found in a preprint by J. Denef and F. Loeser, “Motivic integration, quotient singularities and the McKay correspondence”, 21 p., available at math.AG/9903187.

Reviewer: M.Lejeune-Jalabert (Grenoble)

##### MSC:

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

14J81 | Relationships between surfaces, higher-dimensional varieties, and physics |

32S35 | Mixed Hodge theory of singular varieties (complex-analytic aspects) |

32P05 | Non-Archimedean analysis |

58D20 | Measures (Gaussian, cylindrical, etc.) on manifolds of maps |

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

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