  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.
