Poincaré polynomials of moduli spaces of one-dimensional sheaves on the projective plane
Shuai Guo, Longting Wu, with an appendix by Miguel Moreira
TL;DR
The paper proves that the divisibility property for the Poincaré polynomials P_{β}(y) of moduli spaces M_{β} of stable one-dimensional sheaves on del Pezzo surfaces follows from Bousseau's refined sheaves/Gromov-Witten correspondence, and verifies this in the case S=P^2. It introduces an all-genus local/relative packaging to relate local GW invariants of K_S to maximal-contact invariants of (S,E), and develops a graph-sum framework over labeled rooted trees to compute V^{S/E}_{β} from GW data. Specializing to S=P^2, the authors compute leading Betti numbers for M_d (d≥6) and derive a closed product-form for hatΩ^{P^2}_d, with numerical checks up to d≤16 and conjectures for higher ranges and refinements via perverse/Chern filtrations. The results connect GV invariants of local geometries to the topology of moduli spaces of sheaves, offering new computational tools and framing a set of higher-range conjectures (including Appendix by M. Moreira). Overall, the work advances understanding of cohomological structure in moduli of one-dimensional sheaves and ties geometric representation-theoretic data to topological invariants.
Abstract
Let $M_β$ denote the moduli space of stable one-dimensional sheaves on a del Pezzo surface $S$, supported on curves of class $β$ with Euler characteristic one. We show that the divisibility property of the Poincaré polynomial of $M_β$, proposed by Choi-van Garrel-Katz-Takahashi follows from Bousseau's conjectural refined sheaves/Gromov-Witten correspondence. Since this correspondence is known for $S=\mathbb{P}^2$, our result proves Choi-van Garrel-Katz-Takahashi's conjecture in this case. For $S=\mathbb{P}^2$, our proof also introduces a novel approach to computing the Poincaré polynomials using Gromov-Witten invariants of local $\mathbb{P}^2$ and a local elliptic curve. Specifically, we compute the Poincaré polynomials of $M_{d}$ with degrees $d\leq 16$ and derive a closed formula for the leading Betti numbers $b_i(M_d)$ with $d\geq 6$ and $i\leq 4d-22$. We also propose a conjectural formula for the leading Betti numbers $b_i(M_d)$ with $d\geq 4$ and $i\leq 6d-20$. In the Appendix (by M. Moreira), a more general conjecture concerning the higher range Betti numbers of $M_{d}$ is presented, along with another conjecture that involves refinements from the perverse/Chern filtration.