I wonder if there is any situation where one can talk about Gromov-Witten invariants or quantum multiplication for singular varieties. Ideally, I would like have a situation where for a singular variety $X$ one can define quantum multiplication operators by elements of ORDINARY cohomology of $X$ on the INTERSECTION cohomology of $X$ (I have some examples where I know what I want the answer to be, but I don't know how to ask the question).

In fact, I will be ready to start with the following simple example: assume that $X$ just has quotient singularities, i.e. locally it looks like $Y/G$ where $Y$ is smooth and $G$ is a finite group. In this case the intersection cohomology coincides with the ordinary cohomology, so my question is whether in this case one can define quantum multiplication. One warning: I am talking about quantum cohomology of $X$ itself, not about what is called "orbifold quantum cohomology" (which in many cases coincides with the quantum cohomology of a good resolution of $X$).


1 Answer 1


Since no one else has said anything, let me make two naive comments.

  1. The only case I know of where there is a well developed notion of Gromov-Witten invariants for a singular variety is the very special case when the target admits a gluing $X \cup_D Y$ where X and Y are smooth and projective and D is a smooth divisor.

  2. You write that you are not interested in the orbifold Gromov-Witten invariants, but perhaps it is possible to define something similar to what you want in terms of the orbifold invariants. Let $X = Y/G$ and set $\mathscr X = [Y/G]$. The Gromov-Witten invariants of $\mathscr X$ are given by maps $$ I_{g,n,\beta} : H^\ast(\overline I \mathscr X)^{\otimes n} \to H^\ast(\overline M_{g,n}),$$ where $\overline I \mathscr X$ is the rigidified inertia stack of $\mathscr X$. However there is also a natural map $H^\ast(X) \to H^\ast(\overline I \mathscr X)$ induced by: (i) the rigidification morphism $I \mathscr X \to \overline I \mathscr X$, in particular the isomorphism it induces on cohomology; (ii) the forgetful map $I \mathscr X \to \mathscr X$; and (iii) the coarse moduli space map $\mathscr X \to X$. So by precomposition we get a collection of maps $H^\ast(X)^{\otimes n} \to H^\ast(\overline M_{g,n})$, and even though it seems they will not satisfy the axioms required of GW invariants maybe they are what is needed in your situation.

  • $\begingroup$ Thanks. In fact I only care about genus 0 case. I have a question: suppose that $X$ has a crepant resolution $\widetilde{X}$. Then $H^*(X)$ maps to $H^*(\widetilde{X})$ in the obvious way. Question: if one assumes the crepant resolution conjecture, will one get the same construction from this embedding as from yours? $\endgroup$ Apr 6, 2011 at 0:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.