## Non-Archimedean integrals and stringy Euler numbers of log-terminal pairs.(English)Zbl 0943.14004

Let $$V$$ 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.

### 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)

Zbl 0963.14015
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