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In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Bea00], and admit a projective symplectic resolution if and only if $g=1$ or $(g, n)=(2,2)$. These results are an application of a recent paper by Bellamy and Schedler [BS16] via the so-called Isosingularity Theorem.
Let $X$ be a compact connected Riemann surface, $D, subset, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x, subsetneq, G_x$ a Zariski closed subgroup for every $x, in, D$. A framed principal $G$--bun
We introduce the notion of a conical symplectic variety, and show that any symplectic resolution of such a variety is isomorphic to the Springer resolution of a nilpotent orbit in a semisimple Lie algebra, composed with a linear projection.
We study moduli spaces of parabolic Higgs bundles on a curve and their dependence on the choice of weights. We describe the chamber structure on the space of weights and show that, when a wall is crossed, the moduli space undergoes an elementary transformation in the sense of Mukai.
In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid o
Let $X$ be a compact connected Riemann surface and $D$ an effective divisor on $X$. Let ${mathcal N}_H(r,d)$ denote the moduli space of $D$-twisted stable Higgs bundles (a special class of Hitchin pairs) on $X$ of rank $r$ and degree $d$. It is known