Gromov-Witten invariants
count (in a loose sense only) holomorphic maps from genus Riemann
surfaces to a variety which pass through a given collection of
cycles on . In order to define these, we compactify the space of
maps from a variable pointed curve to by allowing the domain
curve to degenerate to a nodal curve so that the corresponding map
always has finite automorphism group. For a fixed genus , image
homology class , and number of marked points , this gives
the moduli space of stable maps
which is
typically a highly singular Deligne-Mumford stack. The Gromov-Witten
invariants of are given by integrals
where
is evaluation at the
marked point and the are elements of
. An important point of the theory is that this
integral is defined via cap product with a distinguished homology
class known as the virtual fundamental class of
. The descendent Gromov-Witten Invariants
are obtained by inserting monomials in the Witten classes
into the integral.

Jeffrey Herschel Giansiracusa
2005-06-27