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ADHM construction of perverse instanton sheaves. (English) Zbl 1316.14024
The paper under review studies framed torsion free instanton sheaves on projective varieties. An instanton sheaf on \(\mathbb{P}^{n}\) (\(n\geq 2\)) is given by a torsion free sheaf \(E\) which is the cohomology of a linear monad \[ \mathcal{O}_{\mathbb{P}^{n}}(-1)^{\oplus c}\rightarrow \mathcal{O}_{\mathbb{P}^{n}}^{\oplus a}\rightarrow \mathcal{O}_{\mathbb{P}^{n}}(-1)^{\oplus c}\; , \] where we call \(c\) its charge and its rank is given by \(r=2c-a\). A sheaf \(E\) on a projective space is called of trivial splitting type if there exists an isomorphism \(\phi:E|_{l}\rightarrow \mathcal{O}_{l}^{\oplus r}\) for some line \(l\subset \mathbb{P}^{n}\) and, then, the pair \((E,\phi)\) is a framed sheaf. These definitions can be extended to a projective scheme \(\mathbb{Y}\subset \mathbb{P}^{n}\) which contains a line, and the condition of the composition of the arrows in the monad being zero turns out to be the so called ADHM equation between certain matrices.
These matrices are called an ADHM data, and after introducing an stability condition on them the authors show a correspondence (cf. Theorem 3.8) between isomorphism classes of rank \(r\) and charge \(c\) framed torsion free instanton sheaves on \(\mathbb{Y}\) and the so called globally weak stable ADHM data, which allows to construct a quasi projective moduli space of these instantons as the GIT quotient of the stable data. Theorem 4.2. particularizes this result for \(\mathbb{Y}=\mathbb{P}^{n}\) and proves that the moduli space is fine.
Section 5 is devoted to generalize the theory from torsion free sheaves to perverse sheaves, which can be seen as complexes of sheaves which are in the heart of some particular \(t\)-structure in the derived category, not necessarily the standard one. Theorem 5.9 relates the category of perverse instanton sheaves with the ADHM data. Finally, Theorem 5.13 generalizes this for \(\mathbb{Y}=\mathbb{P}^{n}\) to give a hypercohomological characterization of perverse instanton sheaves, generalizing the known one for torsion free instanton sheaves.

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)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
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