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Originally it is in symplectic geometry. Is it just curosity or any other special reason? Thank you for clarifying.

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The first complete definition of GWI in algebraic geometry is more or less contemporary to the first complete definition in symplectic geometry. In algebraic geometry you can, e.g., use virtual localization techniques which (as far as I know) have no counterpart on the symplectic side. More generally, using both symplectic and algebraic techniques on the same problem is almost always a good idea since the strengths of one approach tend to be complementary to those of the other.

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  • $\begingroup$ Hi,but is there any example that we use algebraic GW theory to solve something in symplectic category? For example,how can we use virtual localization technique to solve something in symplectic geometry that we generally can't solve using symplectic geometry techniques? $\endgroup$
    – HYYY
    Commented Jul 6, 2010 at 3:42
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    $\begingroup$ Maybe my answer to your question on functoriality of virtual fundamental classes might provide an example. Usually anything that can be proven using algebraic techniques can also be proven by symplectic techniques, and conversely. In my opinion it's not a bug, but a feature, which allows everyone to choose the language they're more comfortable with. $\endgroup$
    – Barbara
    Commented Jul 6, 2010 at 8:47
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Part of it is that, for the special case of homogeneous spaces and genus 0, it can be shown that GW invariants count the solutions to certain enumerative problems (how many rational curves of degree $d$ are there that intersect general translates of given cohomology classes) and some rather old problems in algebraic geometry were solved this way, for instance, Kontsevich's formula for (stable) rational plane curves of degree $d$ passing through $3d-1$ points in general position.

More generally, they are invariants that let us distinguish different varieties of the same dimension, by generalizing the cohomology ring.

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