L'accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ : 277-Indecomposable Cohen-Macaulay modules and irreducible maps DORIN POPESCU1 & MARKO ROCZEN2
The string-theoretic E-functions Estr (X; u, v) of normal complex varieties X having at most log-terminal singularities are defined by means of sncresolutions. We give a direct computation of them in the case in which X is the underlying space of the three-dimensional A-D-E singularities by making use of a canonical resolution process. Moreover, we compute the string-theoretic Euler number for several compact complex threefolds with prescribed A-D-E singularities.
IntroductionThe string-theoretic (or stringy) Hodge numbers h p,q str (X) of normal, projective complex varieties X with at most Gorenstein quotient or toroidal singularities were introduced in [7] in an attempt to determine a suitable mathematical formulation (and generalization) for the numbers which are encoded into the Poincaré polynomial of the chiral and antichiral rings of the physical "integer charge orbifold theory", due to the LG/CY-correspondence of Vafa, Witten, Zaslow and others. (See [47], [49, §3-5], [50, §4]). These numbers are generated by the so-called E strpolynomials and, as it was shown in [7] and [6], they are the right quantities to establish several mirror-symmetry identities for Calabi-Yau varieties. In fact, as long as a stratification (separating singularity types) for such an X is available, the key-point is how one defines the E str -polynomial locally at these special Gorenstein singular points (by "measuring", in a sense, how far they are from admitting of crepant resolutions).Recently Batyrev [4] generalized this definition and made it work also for the case in which one allows X to have at most log-terminal singularities. In this general framework, ones has to introduce appropriate E str -functions E str (X; u, v) instead which may be not even rational. The treatment of varieties X with e str (X) = lim u,v→1 E str (X; u, v) / ∈ Z is therefore unavoidable. Nevertheless, as it turned out, this new language is a very important tool as it unifies the considerations of certain invariants associated to a wide palette of "MMP-singularities" and leads to the use of more flexible manipulations, as for example in the study of the behaviour *
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