Moduli spaces of framed sheaves and quiver varieties. (English) Zbl 1428.14017
J. Geom. Phys. 118, 20-39 (2017); corrigendum and addendum ibid. 121, 176-179 (2017).
Summary: In the first part of this paper we provide a survey of some fundamental results about moduli spaces of framed sheaves on smooth projective surfaces. In particular, we outline a result by Bruzzo and Markushevich, and discuss a few significant examples. The moduli spaces of framed sheaves on \(\mathbb{P}^2\), on multiple blowup of \(\mathbb{P}^2\) are described in terms of ADHM data and, when this characterization is available, as quiver varieties.
The second part is devoted to a detailed study of framed sheaves on the Hirzebruch surface \(\varSigma_n\) in the case when the invariant expressing the necessary and sufficient condition for the nonemptiness of moduli spaces attains its minimum (what we call the “minimal case”). Our main result is that, under this assumption, the corresponding moduli space is isomorphic to a Grassmannian (when \(n = 1\)), or to the direct sum of \(n - 1\) copies of the cotangent bundle of a Grassmannian (when \(n \geq 2\)). Finally, by slightly generalizing a construction due to Nakajima, we prove that these moduli spaces admit a description as quiver varieties.
The second part is devoted to a detailed study of framed sheaves on the Hirzebruch surface \(\varSigma_n\) in the case when the invariant expressing the necessary and sufficient condition for the nonemptiness of moduli spaces attains its minimum (what we call the “minimal case”). Our main result is that, under this assumption, the corresponding moduli space is isomorphic to a Grassmannian (when \(n = 1\)), or to the direct sum of \(n - 1\) copies of the cotangent bundle of a Grassmannian (when \(n \geq 2\)). Finally, by slightly generalizing a construction due to Nakajima, we prove that these moduli spaces admit a description as quiver varieties.
MSC:
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |
| 16G20 | Representations of quivers and partially ordered sets |
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