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!

  • $\begingroup$ 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. $\endgroup$ Dec 15, 2020 at 17:09
  • $\begingroup$ 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)? $\endgroup$ Dec 16, 2020 at 13:39
  • $\begingroup$ 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. $\endgroup$ Dec 18, 2020 at 11:21


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