Then Conjectures \[main conj intro\] and \[equivalent\] hold for $X$.

The GIT presentation of a toric variety considered in Theorem \[Main Thm\] is the standard one, coming from its fan, as in [@Cox]. (However, as explained in §\[GIT presentations\] later, the result holds for any other semi-positive GIT presentation $(\CC^{n+r'}, (\CC^*)^{r'}, \theta')$ of $X$; note that the quasimap theories are different for different GIT presentations.)

It is easy to see that semi-positive toric varieties have $I_0(q)=1$. When $X$ is projective and Fano, we have in addition (for the standard GIT presentation) that $I_1(q)=0$. The following Corollary is then an immediate consequence Theorem \[Main Thm\].

If $X$ is a nonsingular projective Fano toric variety, then its quasimap invariants are independent on $\ke$: $$F_g^\ke({\bf t}(\psi))= F^\infty_g({\bf t}(\psi)),$$ for all $\ke\geq 0+$.

More interesting is the case of toric Calabi-Yau targets, for which Theorem \[Main Thm\] is highly relevant to the Physicists’ Mirror Symmetry (see e.g., [@HV]) at genus $g\geq 1$. For $X$ a toric Calabi-Yau $3$-fold, consider the equality for $\ke=0+$ and specialize at ${\bf t}(\psi)=0$. If we use the string and divisor equations, the Gromov-Witten side becomes precisely the $A$-model genus $g$ pre-potential [*after*]{} applying the mirror map. The Mirror Conjecture 