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Intuitively, Gromov-Witten theory makes perfect sense. Via Poincare duality, we look at the cohomology classes $\gamma_1, \ldots, \gamma_n$ corresponding to geometric cycles $Z_i$ on a target space $X$, pull them back and then the integral $$\langle \gamma_1 \cdots \gamma_n\rangle=\int_{\overline{\mathcal{M}}_{g,n}(X,\beta)^{vir}}ev_1^*\gamma_1 \smile \cdots \smile ev_n^*\gamma_n$$ should count the number of curves whose intersection with the given cycles is non-empty.

However, we also have the ψ-classes (or "gravitational descendants") arising from the moduli space $\overline{\mathcal{M}}_{g,n}$ which are the chern classes of the $i$-th cotangent line bundle to a given $(C, x_1, \ldots, x_n) \in \overline{\mathcal{M}}_{g,n}$.

So what, geometrically, do these represent? The fact that they arise from $\overline{\mathcal{M}}_{g,n}$ means that the inclusion of a ψ-class places restriction on the geometry of the curves which we count; that much is clear. What is this restriction?

The reason that I am curious is that I am trying to evaluate the GW-invariants corresponding to maps which have components collapsing to an A1 singularity (i.e. a $B\mathbb{Z}/2$), but such that not all of the curve collapses. It has been mentioned in passing that including a ψ-class could help with this, and while the little I understand makes this sound plausible, I don't exactly see why.

So what are ψ-classes? Can I use them to split my curve up into parts so that a fixed component lands on my stacky point, while the rest of it does whatever else curves do?

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The following answer is unfortunately not quite correct, but it may be useful anyway. I will of course be ignoring any virtual fundamental class issues.

Imagine that you are computing a Gromov-Witten invariant where you require the i-th marked point to land at a specific point (i.e. your i-th insertion γi is the class of a point), and now lets add aditionally the i-th psi-class as an insertion. You can restrict to the subspace of maps with $f(x_i) = x$ for some generic choice of $x \in X$. Fixing an arbitrary non-trivial map $\Phi \colon T_x \to k$ gives you by composition a map from the relative tangent bundle of the universal curve over $M_{g, n}(X)$ at the section xi to the trivial line bundle, in other words a section $\phi$ of the relative cotangent bundle of the universal curve. It will vanish on curves which are tangent to a hypersurface through x with tangent direction matching the zero-locus of the map $\Phi$.

So you can think of Gromov-Witten invariants with psi-classes as counting maps which additionally satisfy tangency conditions at the marked points.

Why is this not correct? The zero locus of $\phi$ computes the Chern class of the relative cotangent bundle at $x_i$ over Mg, n(X), which is not the same as the pull-back of the $\psi$-class from Mg, n. Insertions of the former are sometimes called "gravitational ancestors", and the difference to gravitational descendants is described explicitly in alg-geom/9708024.

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Hi Arend, welcome to MO (presumably you are the Arend I know)! –  Hailong Dao Jul 20 '10 at 18:20
    
Welcome to MO, Arend! And thanks for answering this question. I had been wondering about this question ever since it was posted 2 months ago. –  Kevin H. Lin Jul 20 '10 at 18:25
    
This is more a meta issue than anything, but I would like to mark this as best answer, but there isn't a check mark next to the upvote/downvote arrows. –  Simon Rose Jul 24 '10 at 16:24
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