Let $X$ be a smooth projective variety over $\mathbb{C}$. We consider the small quantum cup product $\star$ on the deRham cohomology ring $\displaystyle H^*(X;\mathbb{C})=\bigoplus_{p,q}H^{p,q}(X)$. Let $H^{(i)}$ denote $\displaystyle\bigoplus_{p-q=i}H^{p,q}(X)$.
In ''Gamma Conjecture via Mirror Symmetry'', Galkin and Iritani claimed in the Appendix that $H^{(i)}\star H^{(j)}\subseteq H^{(i+j)}$ in order to draw a conclusion about the spectrum of the operator $c_1(X)\star$ acting on various subspaces of $H^*(X)$. They said this claim follows from the motivic axiom of the Gromov Witten theory. How should one argue following this line?
Thanks.
Remark: If $X$ is convex, then $\overline{\cal M}_{0,3}(X,\beta)$ is a smooth projective variety so the claim should be obvious. But in general we have to use virtual fundamental cycle to define $\star$. I don't know what we should do then.
