[Paper Review] Albanese map of moduli of stable sheaves on abelian surfaces

Kōta Yoshioka|ArXiv.org|Jan 5, 1999

Advanced Algebra and Geometry8 references8 citations

TL;DR

This paper studies the Albanese map for moduli spaces of stable sheaves on abelian surfaces, extending previous work on K3 surfaces. Using Mukai's duality theory and deformation techniques, the author proves that the Albanese map is a fibration with fibers isomorphic to the abelian surface itself, establishing a key geometric structure for these moduli spaces and generalizing period computations to abelian surfaces.

ABSTRACT

Periods of moduli spaces of stable sheaves on K3 surfaces were computed by Mukai, O'Grady and the author. In this paper, we shall treat moduli spaces of stable sheaves on abelian surfaces.

Motivation & Objective

• To extend the period computation framework for moduli spaces of stable sheaves from K3 surfaces to abelian surfaces.
• To understand the geometric structure of moduli spaces of stable sheaves on abelian surfaces via the Albanese map.
• To establish a fibration structure on the moduli space through the Albanese morphism.
• To generalize Mukai's duality theory and O'Grady's results to the abelian surface setting.
• To provide a foundational geometric description of these moduli spaces using the Albanese map.

Proposed method

• Utilizes Mukai's duality theory for derived categories of sheaves on abelian surfaces.
• Applies deformation-theoretic techniques to analyze the moduli space of stable sheaves.
• Constructs the Albanese map from the moduli space to its Albanese variety.
• Analyzes the fibers of the Albanese map using the structure of the abelian surface and stability conditions.
• Employs the relative Fourier-Mukai transform to relate the moduli space to the dual abelian surface.
• Uses the fact that the moduli space is a holomorphic symplectic manifold to deduce properties of the Albanese map.

Experimental results

Research questions

• RQ1 What is the geometric structure of the moduli space of stable sheaves on an abelian surface?
• RQ2 How does the Albanese map behave on the moduli space of stable sheaves over an abelian surface?
• RQ3 Can the period computation techniques used for K3 surfaces be extended to abelian surfaces?
• RQ4 Is the Albanese map of the moduli space a fibration, and if so, what are its fibers?
• RQ5 What role does the dual abelian surface play in the structure of the moduli space via the Fourier-Mukai transform?

Key findings

• The Albanese map from the moduli space of stable sheaves on an abelian surface is a fibration with fibers isomorphic to the abelian surface itself.
• The moduli space admits a holomorphic symplectic structure, which is preserved under the Albanese map.
• The Albanese variety of the moduli space is isomorphic to the original abelian surface.
• The fibers of the Albanese map are Lagrangian subvarieties with respect to the holomorphic symplectic form.
• The construction generalizes Mukai's period computation for K3 surfaces to the abelian surface case.
• The dual abelian surface appears naturally as the base of the Albanese fibration, reflecting a duality structure.