[Paper Review] Perverse coherent sheaves on blow-up. II. wall-crossing and Betti numbers formula

Hiraku Nakajima, Kōta Yoshioka|arXiv (Cornell University)|Jun 3, 2008

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper studies moduli spaces of perverse coherent sheaves on the blow-up of a projective surface at a point, establishing wall-crossing behavior under twist by the exceptional divisor's line bundle and deriving a formula for virtual Hodge numbers. It proves that these moduli spaces are connected via wall-crossings to both the original surface's moduli spaces and the blow-up's usual moduli spaces, with explicit formulas for Betti numbers in terms of partitions and generating functions.

ABSTRACT

This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up of a projective surface at a point. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle associated with the exceptional divisor. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in arXiv:0802.3120: c) The moduli space of stable perverse coherent sheaves is isomorphic to the usual moduli space of stable coherent sheaves on the original surface if the first Chern class is orthogonal to the exceptional divisor. d) The moduli space becomes isomorphic to the usual moduli space of stable coherent sheaves on the blow-up after twisting by sufficiently large negative power of the line bundle associated with the exceptional curve. Therefore usual moduli spaces of stable sheaves on the blow-up and the original surfaces are connected via wall-crossings.

Motivation & Objective

• To describe the wall-crossing behavior in moduli spaces of perverse coherent sheaves on the blow-up of a projective surface at a point.
• To derive a formula for the virtual Hodge numbers (and hence Betti numbers) of these moduli spaces.
• To prove that the moduli space of stable perverse coherent sheaves is isomorphic to the usual moduli space on the original surface when the first Chern class is orthogonal to the exceptional divisor.
• To show that after twisting by $\mathcal{O}(-mC)$ for large $m$, the moduli space becomes isomorphic to the usual moduli space on the blow-up.
• To establish a complete connection between the moduli spaces on the original surface and the blow-up via wall-crossing phenomena.

Proposed method

• Constructing intermediate moduli spaces $\widehat{M}^{m,m+1}(c)$ that connect different $\widehat{M}^{m}(c)$ via birational maps $\xi_m$ and $\xi_m^+$.
• Using torus fixed point techniques and weight space decomposition of the tangent space at fixed points to compute dimensions of negative weight spaces.
• Applying one-parameter subgroup actions $\lambda(t) = (t^{N_1}, t^{N_2}, t^{n_1}, \dots, t^{n_r})$ with $N_2 \gg n_1 > \cdots \gg N_1 > 0$ to analyze the equivariant structure.
• Computing contributions from $\operatorname{Ext}^1(E_\alpha, E_\beta(-\ell_\infty))$ with weights determined by $e_\beta e_\alpha^{-1}$, and summing over fixed point components.
• Deriving a generating function formula for the Poincaré series of moduli spaces using partitions and $q$-Pochhammer symbols.
• Applying the formula to $\widehat{M}^m(c)$ by shifting $m_\alpha \to m_\alpha + k_\alpha$ and adjusting the generating function accordingly.

Experimental results

Research questions

• RQ1 How does the moduli space of perverse coherent sheaves change under twisting by $\mathcal{O}(C)$, and what is the wall-crossing structure?
• RQ2 What is the formula for the virtual Hodge numbers (or Betti numbers) of these moduli spaces?
• RQ3 When is the moduli space of perverse coherent sheaves isomorphic to the moduli space on the original surface $X$?
• RQ4 When does the moduli space become isomorphic to the usual moduli space on the blow-up $\widehat{X}$ after twisting?
• RQ5 Can the moduli spaces on $X$ and $\widehat{X}$ be connected via wall-crossings?

Key findings

• The wall-crossing between $\widehat{M}^m(c)$ and $\widehat{M}^{m+1}(c)$ is described via birational maps $\xi_m$ and $\xi_m^+$, forming a chain of moduli spaces.
• The virtual Hodge numbers of $\widehat{M}^m(c)$ are computed via a generating function involving partitions and $q$-Pochhammer symbols.
• When $(c_1, [C]) = 0$, the moduli space $\widehat{M}^m(c)$ is isomorphic to the moduli space of stable sheaves on the original surface $X$.
• For sufficiently large $m$, twisting by $\mathcal{O}(-mC)$ makes $\widehat{M}^m(c)$ isomorphic to the usual moduli space of stable sheaves on $\widehat{X}$.
• The formula for the Poincaré series $\sum P_t(\widehat{M}^m(c)) \mathfrak{q}^{\Delta(c)}$ is given explicitly in terms of $m_\alpha$, $k_\alpha$, and $t$-weighted inner products.
• In the limit $m \to \infty$, the formula recovers the known result from [23, Cor. 3.10], confirming consistency.