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Let $X$ be a sympletic manifold and $A\in H_2(X;\mathbb{Q})$. Let $g$ and $k$ be nonnegative integers. Assume that $$\mathcal{M}_{g,k}(X;A)$$

is dense in

$$\overline{\mathcal{M}}_{g,k}(X;A).$$

Are the primary Gromov-Witten invariants corresponding to $X$ and $A$ enumerative?

If not, when does the condition $$\mathcal{M}_{g,k}(X;A)$$

dense in

$$\overline{\mathcal{M}}_{g,k}(X;A)$$

imply that the corresponding primary Gromov-Witten invariants are enumerative?

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1 Answer 1

up vote 3 down vote accepted

I am not sure what "primary" means. However, I believe the answer to your first question is "no". For a sufficiently general quintic hypersurface $X$ in $\mathbb{C}P^4$, for sufficiently small curve classes $A$, all genus $0$ curves in $X$ of class $A$ are pairwise disjoint and smooth with normal bundle $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus \mathcal{O}_{\mathbb{P}^1}(-1)$. This implies that the moduli space $\mathcal{M}_{0,k}(X;A)$ are dense in the compactified moduli spaces $\overline{\mathcal{M}}_{0,k}(X;A)$. Nonetheless, the corresponding Gromov-Witten invariants are fractions, thus not enumerative. The issue is that some of the connected components of $\mathcal{M}_{0,k}(X;A)$ have dimension that is larger than the "expected dimension", essentially because they parameterize multiple covers of embedded, smooth curves. Once one takes this into account via the multiple-covering formula, the corresponding instanton numbers are expected to be integers and enumerative (although, of course, the degree $10$ curves of Vainsencher - Pandharipande contradict enumerativity even of these instanton numbers). All of this is discussed in Cox-Katz, Example 7.4.4.1, pp. 205-206.

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