On the perverse filtration of the moduli spaces of 1-dimensional sheaves on ℙ² and P=C conjecture

2023/12/28 by Yuan Yao, Yuan, Yao
Mathematics · #14D22 #14J26 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.2312.17035

openalex publication_date 2023/12/28 · openalex created_date 2023/12/30 · openalex updated_date 2026/07/28

Abstract

Let M(d,χ) be the moduli space of semistable 1-dimensional sheaves supported at curves of degree d on ℙ², with Euler characteristic χ. We have the Hilbert-Chow morphism π: M(d,χ)→ |dH| sending each sheaf to its support. We study the perverse filtration on H^*(M(d,χ),ℚ) via map π, especially the P=C conjecture posed by Kononov-Pi-Shen. We show that P=C conjecture holds for H^{*≤ 4}(M(d,χ),ℚ) for any d≥ 4, (d,χ)=1. The main strategy is to relate M(d,χ) to the Hilbert scheme S^[n] of n-points and transfer the problem to some properties on H^*(S^[n],ℚ). We use induction on n to achieve the desired properties. Our proof involves some complicated calculations.

Related

• Higher direct images of the structure sheaf via the Hilbert-Chow morphism
• The geometry of the moduli space of one-dimensional sheaves
• Sheaves on non-reduced curves in a projective surface
• Some Betti numbers of the moduli of 1-dimensional sheaves on ℙ²
• Perverse sheaves on symmetric products of the plane