Introduction - Florida State University.pdfVIP

CELESTIAL INTEGRATION, STRINGY INVARIANTS, AND CHERN-SCHWARTZ-MACPHERSON CLASSES PAOLO ALUFFI Abstract. We introduce a formal integral on the system of varieties mapping properly and birationally to a given one, with value in an associated Chow group. Applications include comparisons of Chern numbers of birational varieties, new bi- rational invariants, ‘stringy’ Chern classes, and a ‘celestial’ zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern- Schwartz-MacPherson classes of possibly singular varieties, placing them into a context in which a ‘change of variable’ formula holds. The formalism has points of contact with motivic integration. 1. Introduction 1.1. In this note I review the notion of celestial integration, and sketch a few ap- plications; for proofs and further details, the reader is addressed to [Alu05]. I am very grateful to Jean-Paul and Cidinha for organizing the VIII`eme Rencontre Inter- nationale de S˜ao Carlos sur les singularit´es r´eelles et complexes au CIRM, relocating for the occasion the idyllic surroundings of S˜ao Carlos, Brasil to the idyllic surround- ings of Luminy, France. Perfect weather, exceptionally interesting talks, and spirited conversations made the conference a complete success. What follows was the subject of my lecture at the S˜ao Carlos/Luminy meeting, and preserves (for better or worse) the informal nature of a seminar talk. I thank the Max-Planck-Institut f¨ur Mathematik in Bonn, where much of this material was conceived and where this note was written. 1.2. Summary: to a variety X I will associate a