Adhm Construction of Perverse Instanton Sheaves

Abstract

Abstract. We present a construction of framed torsion free instanton sheaves on a projective variety containing a fixed line which further generalizes the one on projective spaces. This is done by generalizing the so called ADHM variety. We show that the moduli space of such objects is a quasi projective variety, which is fine in the case of projective spaces. We also give an ADHM categorical description of perverse instanton sheaves in the general case, along with a hypercohomological characterization of these sheaves in the particular case of projective spaces.

Moduli of autodual instanton bundles

“…This enables us to identify ADHM data that produce isomorphic cohomology bundles. Theorem 1.5 (Theorems 3.8 and 4.2 in [9]). There is a 1-1-correspondence between the following objects:…”

“…Let us recall the ADHM construction for framed instanton bundles, whose details can be found in [9]. Let V and W be K-vector spaces of dimension dim V = c and dim…”

“…Being a special case of autodual instanton bundles, we know from Proposition 2.4, that a framed symplectic instanton bundle gives rise to an extended ADHM datum (A, B, I, G, H), where H satisfies diagram (9).…”

“…Moreover we want the coefficients j 1 , j 8 , j 9 and j 16 not to be zero and the coefficients a 3 , a 7 , b 3 , b 7 to be zero in order to get a regular solution. Hence we want the following relations Finally, we also check that one can get solutions in which the coefficients a 2 , a 6 , b 2 , b 6 are not zero.…”

“…We first describe the moduli space of framed autodual instanton bundles using the ADHM construction introduced by Henni, Jardim and Martins in [9]. This allows us to provide a description of the moduli spaces of symplectic and orthogonal instantons using matrices, a description that is useful in establishing existence and non-existence results for orthogonal instantons.…”