Gromov-Witten invariants of varieties with holomorphic 2-forms

We show that a holomorphic two-form $\theta$ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps $\mgn(X,\beta)$ to the locus where $\theta$ degenerates; it then enables us to define the localized GW-invariant, an algebro-geometric analogue of the local invariant of Lee and Parker in symplectic geometry, which coincides with the ordinary GW-invariant when X is proper. It is deformation invariant. Using this, we prove formulas for low degree GW-invariants of minimal general type surfaces with p_g>0 conjectured by Maulik and Pandharipande.