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Moduli spaces of stable sheaves on abelian surfaces. (English) Zbl 1066.14013

In the paper under review, the author considers various problems concerning the moduli spaces of stable sheaves on an abelian surfaces under a particular additional assumption, thereby enhancing some related study on K3 surfaces conducted before by S. Mukai, K.O’Grady, D.Orlov, the author himself, and others. Actually, this paper is an extended version of the author’s previous articles “Albanese map of moduli of stable sheaves one abelian surfaces” (math.AG/9901013) and “Some examples of isomorphisms induced by Fourier-Mukai functors” (math.AG/9902105).
Let \(X\) be an abelian surface defined over \(\mathbb{C}\) and \(H\) an ample line bundle on \(X\) Then, according to S. Mukai’s approach [cf.: On the moduli space of bundles on \(K3\) surfaces. I: Vector bundles on algebraic varieties, Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 341–413 (1987; Zbl 0674.14023)], each so-called Mukai vector \(v\in H^{\text{even}}(X,\mathbb{Z})\) gives rise to the moduli space \(M_H(v)\) of Gieseker-stable sheaves \(E\) with \(ch(E)= v\) and \(\text{det }E= H\). In the present paper, the author’s main assumption is that the chosen Mukai vector \(v\) is a positive primitive element (in some precise sense). Under this special assumption, and using the framework of the Fourier-Mukai transform as a main tool, he studies the moduli spaces \(M_H(v)\) for a general ample line bundle \(H\) in great depth. More precisely, the author determines the deformation types of these moduli spaces, their Albanese maps, their Bogomolov factors, their Hodge structures of weight 2, and other invariants. In this context, he also discusses the deformation types of moduli spaces of stable sheaves on \(K3\) surfaces, thereby completing the picture for that class of complex surfaces.
The exposition of this highly substantial paper is utmost comprehensive, detailed, rigorous, and clear.

MSC:

14D20 Algebraic moduli problems, moduli of vector bundles
14D22 Fine and coarse moduli spaces
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14K10 Algebraic moduli of abelian varieties, classification

Citations:

Zbl 0674.14023
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