Hyperpolygon spaces and their cores

Hyperpolygons and Hitchin systems

We study the hyperkähler analogues of moduli spaces of semistable n-gons in complex projective space. We prove that the hyperkähler Kirwan map is surjective and produce a formula that recursively calculates the Betti numbers of these spaces for all ranks. Building on a natural analogy between hyperpolygons and parabolic Higgs bundles, we identify hyperpolygon spaces with certain degenerate Hitchin systems, and use this to establish their complete integrability, for ranks up to and including 3.

Hyperpolygon spaces and moduli spaces of parabolic Higgs bundles

Given an $n$-tuple of positive real numbers $α$ we consider the hyperpolygon space $X(α)$, the hyperkähler quotient analogue to the Kähler moduli space of polygons in $\mathbb{R}^3$. We prove the existence of an isomorphism between hyperpolygon spaces and moduli spaces of stable, rank-$2$, holomorphically trivial parabolic Higgs bundles over $\mathbb{C} \mathbb{P}^1$ with fixed determinant and trace-free Higgs field. This isomorphism allows us to prove that hyperpolygon spaces $X(α)$ undergo an elementary transformation in the sense of Mukai as $α$ crosses a wall in the space of its admissible…

有限商奇点的所有81个Crepant解均为超多边形空间

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

本文证明阶为32的例外子群G的商奇点同构于仿射箭图簇，通过GIT商变动将其所有81个射影crepant解构造为超多边形空间，并推广至n角星形箭图，完整描述了相应超多边形空间的双有理几何。

超多边形空间上的辛形式

Symplectic form on hyperpolygon spaces

本文为$CP^1$上的一族抛物Higgs丛提供了典范替代构造，借此计算出其Higgs辛形式，并证明了其与超多边形模空间的同构是辛同胚。

Quasi-parabolic Higgs bundles and null hyperpolygon spaces

We introduce the moduli space of quasi-parabolic $SL(2,\mathbb{C})$-Higgs bundles over a compact Riemann surface $Σ$ and consider a natural involution, studying its fixed point locus when $Σ$ is $\mathbb{C} \mathbb{P}^1$ and establishing an identification with a moduli space of null polygons in Minkowski $3$-space.

Moduli spaces of generalized hyperpolygons

We introduce the notion of generalized hyperpolygon, which arises as a representation, in the sense of Nakajima, of a comet-shaped quiver. We identify these representations with rigid geometric figures, namely pairs of polygons: one in the Lie algebra of a compact group and the other in its complexification. To such data, we associate an explicit meromorphic Higgs bundle on a genus-$g$ Riemann surface, where $g$ is the number of loops in the comet, thereby embedding the Nakajima quiver variety into a Hitchin system on a punctured genus-$g$ Riemann surface (generally with positive codimension).…