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Oberseminar WiSe 2010/11 Zusammenfassung des Vortrags von Orsola Tommasi

Orbifold cohomology of the moduli space of curves of genus 3

Orbifold cohomology, also known as Chen-Ruan cohomology, has been introduced by Chen and Ruan in 2001 as part of the project of extending Gromov-Witten invariants to orbifolds. As a vector space, the orbifold cohomology with rational coefficients of an orbifold X coincides with the ordinary cohomology of the inertia stack of X, which is (loosely speaking) the stack parametrizing pairs (x,g) where x is an object of X and g an automorphism of x. The grading of the orbifold cohomology is obtained by shifting the grading of the ordinary cohomology of each connected component of the inertia stack by a rational number, called age or fermionic twist.

In this seminar, I will report on a joint work with Nicola Pagani (KTH) on the orbifold cohomology of the moduli space of curves. I will explain how one can determine all connected components of the inertia stack in this case and compute their age. In the special case of moduli of genus 3 curves, this yields to a complete description of the additive structure of orbifold cohomology.