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Let $X$ be a $2n$-dimensional symplectic manifold or a smooth $n$-dimensional algebraic variety over $\mathbb{C}$, which is not necessarily compact, but as part of the requirement the small quantum cohomology $\mathit{QH}^\ast(X)$ should be well-defined.

Now consider the $k$-th quantum power of the first Chern class $c_1(X)^{\star k}$, which necessarily lies in $\mathit{QH}^{2k}(X)$, where the grading on $\mathit{QH}^\ast(X)$ is understood in a reduced way, by moding $2N$, with $N$ being the minimal Chern number.

My question is when will there be a $k$ with $\lceil n/2\rceil\leq k\leq n-1$ such that $c_1(X)^{\star k}$ lies in the summand $H^{2k}(X;\mathbb{C})\subset\mathit{QH}^{2k}(X)$? Is there a convenient criterion which ensures that this holds?

As a concrete example, take $X$ to be the total space of the tautological line bundle $\mathcal{O}(-1)$ over $\mathbb{CP}^{n-1}$, then for any $k$ with $1\leq k\leq n-1$, one can check that the quantum power $c_1(X)^{\star k}$ equals the exterior power $c_1(M)^k$.

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  • $\begingroup$ sorry for being naive, could you explain to me in which way $H^{2k}(X, \mathbb{C})$ is a summand of $QH^{2k}(X)$? In sources I have seen it was stated that quantum cohomology is isomorphic to usual cohomology as vector space, only ring structures are different $\endgroup$
    – user74900
    Dec 8, 2017 at 18:50
  • $\begingroup$ @AknazarKazhymurat QH usually doesn't have a $\mathbb{Z}$-grading. It's explained in Mcduff-Salamon for example. $\endgroup$
    – YHBKJ
    Dec 9, 2017 at 2:11

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