For arbitrary smooth hypersurface X ⊂ P n we construct moduli of quasimaps with P fields. Apply Kiem-Li's cosection localization we obtain a virtual fundamental class. We show the class coincides, up to sign, with that of moduli of quasimaps to X. This generalizes Chang-Li's [CL] numerical identity to the cycle level, and from Gromov Witten invariants to quasimap invariants.

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Using virtual residue, which is a generalization of Grothendieck residue, we generalized Cayley-Bacharach Theorem to the cases with positive dimensions.

We generalize Grothendieck's residues Res ψ s to virtual cases, namely cases when the zero loci of the section s has dimension larger than the expected dimension(zero). We also provide an exponential type integral formalism for the virtual residue, which can be viewed as an analogue of the Mathai-Quillen formalism for localized Euler classes. * Partially supported by Hong Kong GRF grant 16301515 and 16301717. 1 A Landau Ginzburg space is a pair (X, W ) of a complex manifold X and a holomorphic function W : X → C with compact critical locus;

By using the infinitesimally marking point to break the loop in the localization calculation as Kim and Lho, and Zinger's explicit formulas for double J-functions, we obtain a formula for genus one stable quasimaps invariants when the target is a complete intersection Calabi-Yau in projective space, which gives a new proof of Kim and Lho's mirror theorem for elliptic quasimap invariants.