The filtration conjecture for quantum cohomology

Let $M$ be the symplectic manifold under consideration, let $\Lambda$ be its Novikov coefficient ring, and let $a_0$ be the grading parameter appearing in the quantum product. For the function $f:M\to\mathbb{R}$ defined by

$$f(x)=\sum_{k:x\in D_k}\lambda_k-2,$$

set $M^j=\{f<j\}$. Denote by $\widetilde{\mathcal{Q}}_{\geq j}H^i(M;\Lambda)$ the induced filtration on quantum cohomology.

The filtration conjecture. One has

$$\widetilde{\mathcal{Q}}_{\geq j}H^i(M;\Lambda)\supset\ker\bigl(H^i(M;\Lambda)\to H^i(M^{ja_0-i};\Lambda)\bigr).$$

When the stated hypothesis on the grading holds, this inclusion is an equality.

This conjecture gives a geometric description of the filtration induced from the completed symplectic cochain complex, in terms of the strata determined by the divisors $D_k$. It is motivated by compatibility with the Novikov-variable shift and by the expected behavior of the logarithmic PSS map; the equality is conditional on the grading hypothesis.