Virtual fundamental class

The moduli space of maps $ \overline{\mathcal{M}}_{g,n}(X,\beta)$ is highly singular, and yet it has a homology class which behaves much like a fundamental class, and is hence called the virtual fundamental class $ [\overline{\mathcal{M}}_{g,n}(X,\beta)]^{\mathrm{vir}}$ defined in terms of the deformation theory of stable maps to $ X$. The Gromov-Witten invariants of $ X$ are defined by integrating various geometric cohomology classes over this class.

Jeffrey Herschel Giansiracusa 2005-06-27