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Moduli spaces of framed $G$--Higgs bundles and symplectic geometry (1902.06490v2)

Published 18 Feb 2019 in math.AG, math.DG, and math.SG

Abstract: 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 $G$--bundle is a pair $(E_G,\, \phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $\phi$ assigns to each $x\, \in\, D$ a point of the quotient space $(E_G)x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,\, \phi)$ together with a section $\theta\, \in\, H0(X,\, \text{ad}(E_G)\otimes K_X\otimes{\mathcal O}_X(D))$ such that $\theta(x)$ is compatible with the framing $\phi$ for every $x\, \in\, D$. We construct a holomorphic symplectic structure on the moduli space $\mathcal{M}{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $\mathcal{M}{FH}(G)$ to the moduli space $\mathcal{M}{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,\, \theta)$ that forgets the framing, is Poisson. These results generalize \cite{BLP} where $(G,\, {H_x}{x\in D})$ is taken to be $(\text{GL}(r,{\mathbb C}),\, {\text{I}{r\times r}}{x\in D})$. We also investigate the Hitchin system for $\mathcal{M}{FH}(G)$ and its relationship with that for $\mathcal{M}_{H}(G)$.

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