Automorphisms of Moduli Spaces of Symplectic Bundles

Abstract

Let X be an irreducible smooth complex projective curve of genus g ≥ 4. Fix a line bundle L on X. Let M Sp (L) be the moduli space of semistable symplectic bundles (E, ϕ : E ⊗ E → L) on X, with the symplectic form taking values in L. We show that the automorphism group of M Sp (L) is generated by the automorphisms of the form E → E ⊗ M , where M 2 ∼ = O X , together with the automorphisms induced by automorphisms of X.

A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve

“…The nonabelian Torelli theorem has already been proven in [5] for the moduli of principal bundles in general, and the automorphism group for the moduli of symplectic bundles has been computed in [4]. The approach in [5] was to examine the strictly semistable locus and reduce to the classical Torelli theorem for the principally polarized Jacobians.…”

“…The approach in [5] was to examine the strictly semistable locus and reduce to the classical Torelli theorem for the principally polarized Jacobians. On the other hand, we get the result from the geometric properties on the stable locus as in [4]. Both approaches in [4] and ours are based on the Hecke correspondence, but we use the Hecke curves covering the moduli space to show the result in a more direct way.…”

“…On the other hand, we get the result from the geometric properties on the stable locus as in [4]. Both approaches in [4] and ours are based on the Hecke correspondence, but we use the Hecke curves covering the moduli space to show the result in a more direct way. This paper is organized as follows.…”

The group of automorphisms of the moduli space of principal bundles with structure group $F_4$ and $E_6$

“…Observe that, if (E, ϕ) is an F 4 -Higgs bundle with H(E, ϕ) = a, then the previous equation is a factor of the characteristic polynomial of ϕ. Similarly, if G = E 6 , the element a is of the form a = (a 2 , a 5 , a 6 , a 8 , a 9 , a 12 ) and the spectral curve X a is the curve in |K| defined by the equation 5,6,8,9,12 a r (x)y 12−r = 0.…”

“…where Tr denotes the trace map and A is seen as an element of SL(27, C) via the fundamental 27-dimensional representation of E 6 . For any element A ∈ e 6 , the characteristic polynomial of A is det(yI − A) = y 27 + r=2, 5,6,8,9,12 p r (A)y 27−r .…”