Splitting of the virtual class for genus one stable quasimaps

Abstract

We analyze the local structure of moduli space of genus one stable quasimaps. Combining it with the p-fields theory developed in [8], we prove the hyperplane property for genus one invariants of stable quasimaps to hypersurface in P n .

Quantum Lefschetz property for genus two stable quasimap invariants

“…-in Section 5 so that we can get a tidy form (1.6). Once we do these for all i, then by using [LL,Corollary 1.3] e ref pV 1,1,d q X rQ red 1,1,d pP n qs " rQ 1,1,d pXqs vir ´c1 pL 2 q 12 rQ 0,2,d pXqs vir together with (1.4), the decomposition (1.5) proves Theorem 2.…”

“…of the obstruction bundle over Ă M is not so obvious, but it is proven in [LL,Equation (3.15)]. From now on for simplicity, we denote the domain of the morphism ιi by Q piq , by Q piq p the fiber product Q piq ˆQpiq Q piq p and by P piq the pullback of P piq .…”

“…Combining these with [LL,Theorem 1.1] e ref pV 1,1,d q X rQ red 1,1,d pP n qs " rQ 1,1,d pXqs vir ´rKs dimX´1 X `rM 1,1 s ˆrQ 0,2,d pXqs vir ˘, (5.10) for i " 1 becomes rQ p1q p s vir "p´1q dp ř i i q`m rKs dimX´1 X `rM 1,1 s ˆrQ 1,1,d pXqs vir (5.11) ´p´1q dp ř i i q`m rK 1 s dimX´1 rK 2 s dimX´1 X `rM 1,1 s ˆrQ 0,2,d pXqs vir ˆrM 1,1 s ˘,…”

“…Its nontrivial contribution to the integration over rM 1,1 s ˆpe ref pV 1,1,d q X rQ red 1,1,d pP n qsq is only ´αψ. Hence rQ p1q p s vir " ´p´1q dp ř i i q`m 24 c 1 pLq X pe ref pV 1,1,d q X rQ red 1,1,d pP n qsq Using [LL,Corollary 1.3] e ref pV 1,1,d q X rQ red 1,1,d pP n qs " rQ 1,1,d pXqs vir ´c pLq 12 rQ 0,2,d pXqs vir , we obtain rQ p1q p s vir " ´p´1q dp ř i i q`m c 1 pLq X rQ 1,1,d pXqs vir `2 p´1q dp ř i i q`m 24 2 c 1 pL 1 qc 1 pL 2 q X rQ 0,2,d pXqs vir .…”