Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

The Cohomological Crepant Resolution Conjecture for Transverse A-singularities

Let $[X]$ be a Gorenstein orbifold such that its coarse moduli space $X$ has a crepant resolution $\rho: Z \rightarrow X$. In general the orbifold cohomology ring $H^*_{orb}([X])$ of $[X]$ and the cohomology ring of $Z$ are not isomorphic. The Cohomological Crepant Resolution Conjecture (by Y.Ruan) states that the difference between the two rings can be expressed in terms of Gromov-Witten invariants of $Z$ of rational curves which are contracted by the resolution morphism $\rho : Z\rightarrow X$. We study this conjecture in the case where $X$ is a variety with transverse ADE-singularities and $[X]$ is the associated canonical reduced orbifold. In the $A_n$-case we compute both the orbifold cohomology ring and the Gromov-Witten invariants of $Z$. Finally we verify the conjecture in the $A_1$ and $A_2$-case. In both cases we give an explicit isomorphism between the orbifold cohomology ring of $[X]$ and the quantum corrected cohomology ring of $Z$.