# Algebraic deformation invariance of Gromov-Witten invariants

Let $$\mathcal{X}\to S$$ be a smooth family of projective varieties over a smooth curve $$S$$. Let $$\beta$$ be a curve class supported in some fiber and consider the relative moduli space of stable maps $$\overline{M}_{g,n}(\mathcal{X},\,\beta)\to S$$. Is the virtual fundamental cycle $$[\overline{M}_{g,n}(\mathcal{X},\,\beta)]^{\rm vir}$$ represented by a linear combination of algebraic cycles which are flat over $$S$$? Note that this would imply deformation-invariance of the GW invariant when these cycles have relative dimension $$0$$ over $$S$$. I am also happy to consider the case where the moduli space has dimension zero over the general fiber of $$S$$ and the invariant is enumerative. In this case, I'm simply asking whether the closure of the moduli space of the general fiber represents the virtual fundamental class of a special fiber.

If anyone has a citation, that would be very useful!

• Are you asking for a proof/reference for deformation invariance of the GW invariants? In this case, families of cycles can be avoided: deformation invariance comes for free as a property of the (construction of the) virtual class. Dec 15, 2020 at 17:09
• I mean, I wouldn't say no to a reference which proves this explicitly. But I am asking for a stronger statement: Are the virtual fundamental cycles represented by a flat family (when the base is a curve)? Dec 16, 2020 at 13:39
• I am not competent enough on families of cycles to confirm the stronger statement. In particular, I am not sure about the official definition for cycles of dim > 0 (Kollár's? Rydh's?) However the reference for deformation invariance is in Fulton's "Intersection Theory", where he proves that refined Gysin homomorphisms behave well under pullback (part of bivariant intersection theory). I hope this helps. Dec 18, 2020 at 11:21