Moduli of autodual instanton bundles

Abstract

Abstract. We provide a description of the moduli space of framed autodual instanton bundles on projective space, focusing on the particular cases of symplectic and orthogonal instantons. Our description will use the generalized ADHM equations which define framed instanton sheaves.

Orthogonal Bundles and Skew-Hamiltonian Matrices

“…In other words we need two skew-symmetric matrices A and B such that rk(AJB − BJA) = r. It seems interesting and useful to give such an explicit construction for the linear monad. For other explicit examples, see [JMW13].…”

On rank 3 instanton bundles on P3$\mathbb {P}^3$

“…Although the family of instanton bundles has been known for quite some time and besides the fact that there is a vast literature on the subject (see [5,12,19,29,30] to mention a few), it was only in the 2010s that it was proved that the moduli space of stable rank 2 vector instanton bundles of charge 𝑛 is a smooth (see [22]) and irreducible (see [31,32]) open subset of the moduli space of stable rank 2 bundles with Chern classes 𝑐 1 = 0 and 𝑐 2 = 𝑛 on ℙ 3 . Since then, several authors pursued what they considered to be a natural direction for the study of instantons: some studied their definition for other projective varieties (see [3,9,13,26]); some considered their degenerations in ℙ 3 (see [14,20]); and others considered the higher rank case [2,8,21].…”

Irreducibility of the moduli space of orthogonal instanton bundles on $$\mathbb {P}^n$$

“…For example, in [9] Costa, Hoffmann, Miró-Roig and Schmitt proved that the moduli space of all symplectic instanton bundles on P 2n+1 with n ≥ 2 is reducible; in [21] Miró-Roig and Orus-Lacort proved that M P 2n+1 (c) is singular for n ≥ 2 and c ≥ 3; Costa and Ottaviani in [10] proved that M P 2n+1 (c) is affine and introduced an invariant which allowed Farnik, Frapporti and Marchesi to prove in [12] that there are no orthogonal instanton bundles with rank 2n on P 2n+1 . Using the ADHM construction introduced by Henni, Jardim and Martins in [15], Jardim, Marchesi and Wißdorf in [17] consider autodual instantons of arbitrary rank on projective spaces, with focus on symplectic and orthogonal instantons; they described the moduli space of framed autodual instanton bundles and showed that there are no orthogonal instanton bundles with trivial splitting type, arbitrary rank r and charge 2 or odd on P n . While in [1] Abuaf and Boralevi proved that the moduli space of rank r stable orthogonal bundles on P 2 , with Chern classes (c 1 , c 2 ) = (0, c) and trivial splitting type on the general line, is smooth and irreducible for r = c and c ≥ 4, and r = c − 1 and c ≥ 8, the results of Farnik, Frapport and Marchesi in [12] and Jardim, Marchesi and Wißdorf in [17], already mentioned, show us that orthogonal instanton bundles on P n , n ≥ 3 are for some reason hard to find and that it is interesting to establish existence criteria for these bundles.…”