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The ADHM variety and perverse coherent sheaves. (English) Zbl 1229.14010

The authors study the set of solutions of the ADHM equation, the ADHM variety \(\mathcal V\). For given vector spaces \(V,W\) of dimension \(r\) and \(c\), respectively, \(\mathcal V\) is the set of data \((A,B,I,J)\) with \(A,B \in\mathrm{End}(V)\), \(I \in\mathrm{Hom}(W,V)\) and \(J \in\mathrm{Hom}(V,W)\) satisfying the ADHM equation \([A,B]+IJ=0\). The main result is on the disjoint decomposition \(\mathcal V^{(s)}\) of \(\mathcal V\) given by the dimension \(s\) of the stabilizing subspace and on the corresponding filtration \(\mathcal V^{[s]}\): \(\mathcal V^{(s)}\) is an irreducible quasi-affine variety of dimension \(2rc+c^2-(r-1)(c-s)\), which is nonsingular if and only if \(s=c\) or \(s=c-1\), and \(\mathcal V^{[s]}\) is an affine variety, which is irreducible if and only if \(r\geq 2\). Additional results are given for \(r=1\).
A solution of the ADHM equation corresponds to a representation \(R\) of the ADHM quiver with ADHM equation as relation. The authors construct an associated complex \(E^\bullet_R\). This complex is shown to be a perverse coherent sheaf on \(\mathbb P^2\) in the sense of Kashiwara of rank \(r\), charge \(s\) and length \(c-s\), which is trivial at infinity. Here \(s\) is the dimension of the stabilizing subspace of the given ADHM solution. Finally, the authors give a necessary condition on \(E^\bullet_R\) for \(R\) being costable, and a sufficient condition on \(E^\bullet_R\) for \(R\) being regular.

MSC:

14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills)
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