[Paper Review] Bridgeland's stabilities on abelian surfaces

Shintarou Yanagida, Kōta Yoshioka|arXiv (Cornell University)|Mar 5, 2012

Algebraic Geometry and Number Theory15 references4 citations

TL;DR

This paper investigates wall and chamber structures for Bridgeland stability conditions on abelian surfaces with Picard number 1, focusing on rank 1 complexes. Using Fourier-Mukai transforms and Mukai lattice techniques, it classifies codimension 0 walls and shows that for primitive Mukai vectors with ⟨v²⟩/2 ≤ 4, the moduli spaces MH(v) are isomorphic to Hilbert schemes or products with X, generalizing Mukai's earlier results. The action of Fourier-Mukai transforms on stability parameters is shown to correspond to the natural SL(2,R)-action on the upper half-plane.

ABSTRACT

In this paper, we shall study the structure of walls for Bridgeland's stability conditions on abelian surfaces. In particular, we shall study the structure of walls for the moduli spaces of rank 1 complexes on an abelian surface with the Picard number 1.

Motivation & Objective

• To understand the wall and chamber structure for Bridgeland stability conditions on abelian surfaces with Picard number 1.
• To study the behavior of these walls under Fourier-Mukai transforms and dual functors.
• To classify moduli spaces MH(v) for primitive Mukai vectors with small self-intersection ⟨v²⟩/2 ≤ 4.
• To relate numerical solutions to codimension 0 walls and describe their geometric structure.

Proposed method

• Uses the Mukai lattice (H∗(X,Z)alg, ⟨·,·⟩) to express the stability function Z(β,ω)(E) = ⟨e^{β+iω}, v(E)⟩.
• Characterizes walls via numerical solutions of semi-homogeneous presentations of complexes V−1 → V0.
• Applies Fourier-Mukai transforms to relate moduli spaces and show that the action on the parameter space corresponds to SL(2,R) acting on the upper half-plane.
• Uses the cohomological action of Fourier-Mukai functors to define a group Gn,ℓ ⊂ GL(2,R) acting on walls.
• Analyzes the (s,t)-plane for stability conditions σ(β+sH,tH) and shows walls form non-intersecting circles or lines.
• Applies results from [14] on quadratic forms to classify equivalence classes of numerical solutions.

Experimental results

Research questions

• RQ1 How do codimension 0 walls for Bridgeland stability on abelian surfaces with Picard number 1 relate to numerical solutions of semi-homogeneous presentations?
• RQ2 What is the structure of the wall and chamber decomposition for rank 1 complexes on abelian surfaces with NS(X) = ZH?
• RQ3 How do Fourier-Mukai transforms act on the space of stability conditions and on the moduli spaces of stable objects?
• RQ4 For which primitive Mukai vectors v with ⟨v²⟩/2 ≤ 4 is MH(v) isomorphic to Hilbert schemes or products with X?
• RQ5 What is the orbit structure of codimension 0 walls under the action of the group Gn,ℓ?

Key findings

• Codimension 0 walls are in bijection with numerical solutions; if √(ℓ/n) ∈ Q, there is a unique such wall, otherwise infinitely many.
• For v = 1 − ℓ̺X with ℓ ≤ 4, the moduli space MH(v) is isomorphic to Hilb⟨v²⟩/2(X) × X or MH(0,2H,−1), generalizing Mukai’s result.
• The moduli space MH(1,0,−4) is not isomorphic to MH(0,2H,−1), as they admit different types of contractions.
• For s = −2 and v = (1,0,−4), the moduli space M(−2H,t2H)(1,0,−4) is isomorphic to MH(0,2H,−1) via a Fourier-Mukai transform.
• The action of Fourier-Mukai transforms on the parameter space corresponds to the standard SL(2,R)-action on the upper half-plane.
• The set of codimension 0 walls forms a single orbit under the action of Gn,ℓ, with two accumulation points at (±√(ℓ/n), 0).