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Let $X$ be some (possibly?) arbitrary compact, complex manifold of dimension $d$, and let $\overline{M}_{g,n}(X, \beta)$ be the compactified moduli space of stable holomorphic maps $f: C \to X$, where $C$ is a connected genus $g$ curve with $n$ marked points, and the image lands in the homology class $\beta$ of $X$.

The (complex) virtual dimension of this moduli space is famously

$$(1-g)(d-3)+n+\int_{C}f^{*}(c_{1}(X)).$$

Now, I'm trying to get my hands on this moduli space, even just in the simplest case of $\beta=0$, which corresponds to $f$ taking the entire curve $C$ to a point in $X$. Intuitively, I would think that the moduli space would simply decompose into a product of $\overline{M}_{g,n}$ and $X$,

$$\overline{M}_{g,n}(X,0) = \overline{M}_{g,n} \times X \,\,\,? $$

After all, it seems like the only moduli in this case, is the choice of a complex structure on the curve, and the choice of a point in $X$ to map the whole curve to! It seems like these bits of data don't interact. Thus, I would expect the dimension to be

$$3g-3 + n + d.$$

But the actual virtual dimension given earlier, seems to indicate that the dimension would be

$$3g-3+d(g-1) + n.$$

Where in my reasoning is this discrepancy arising? Maybe the dimension formula above doesn't hold when $\beta=0$? I can't imagine how my reasoning is flawed in the degree zero case. I understand one also needs to think about stability but I feel like the problem isn't arising from that.

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    $\begingroup$ The product decomposition in the degree $0$ case is correct. I'm not seeing where the discrepancy arises; stability is only an issue in the low genus case, so that's not an issue. The virtual dimension is calculated using the deformation theory of maps from a pointed curve, which may very well break down (numerically speaking) in the trivial case, which could be the cause of the issue. $\endgroup$ – Tabes Bridges Dec 20 '16 at 19:24
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    $\begingroup$ What you're seeing is the fact that the virtual fundamental class is not the usual fundamental class even when the moduli space is smooth. The virtual dimension tells you the degree in the Chow ring (or cohomology) that the virtual fundamental class lies in, not the actual dimension of the moduli space which is why these numbers don't (and shouldn't) agree. $\endgroup$ – Dori Bejleri Dec 21 '16 at 14:20
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There is indeed a discrepancy in the case between the virtual dimension (also called the expected dimension) and the dimension one could expect according to arguments above. Nevertheless, it is the virtual dimension that is the "good" one. I think you might enjoy doing a computation with obstruction bundles, which is explained in Joachim Kock's "Notes on psi classes" here. Precisely the example you are looking at is described there, in the end.

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