Orthogonal Bundles and Skew-Hamiltonian Matrices

Abstract

Abstract. Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space M 0 ort (r, n) of stable rank r orthogonal vector bundles on P 2 , with Chern classes (c 1 , c 2 ) = (0, n), and trivial splitting on the general line, is smooth irreducible of dimension (r − 2)n − r 2 for r = n and n ≥ 4, and r = n − 1 and n ≥ 8. We speculate that the result holds in greater generality.

When is a Hamiltonian matrix the commutator of two skew-Hamiltonian matrices?

“…This paper can be considered both preliminary and complementary to [1,. Indeed, it is preliminary because some of our results are amongst the ingredients for the irreducibility theorem proved in [1,Theorem 3.4]. On the other hand, it is complementary as our point of view is different than [1] and we cover some aspects of the problem that are not considered there.…”

“…In fact, the connection with algebraic geometry is the main motivation of our study, that originated after some discussions [3] with the author of [16] and one author of [1]. This paper can be considered both preliminary and complementary to [1,. Indeed, it is preliminary because some of our results are amongst the ingredients for the irreducibility theorem proved in [1,Theorem 3.4].…”

“…The unstructured case of this problem is studied in [12], while its version for symplectic vector bundles is analyzed in [16]. The much more technically involved case of orthogonal vector bundles is addressed in [1]. In fact, the connection with algebraic geometry is the main motivation of our study, that originated after some discussions [3] with the author of [16] and one author of [1].…”

“…The much more technically involved case of orthogonal vector bundles is addressed in [1]. In fact, the connection with algebraic geometry is the main motivation of our study, that originated after some discussions [3] with the author of [16] and one author of [1]. This paper can be considered both preliminary and complementary to [1,.…”

Moduli of autodual instanton bundles

“…We therefore conclude that when E is the bundle defined by (1), its dual E ∨ is the cohomology of the monad (6), which is dual to (1), and this proves the statement.…”