arXiv subjects

Arya Yae

Publications and source records attributed to Arya Yae.

3 recordsLinked to original sources

arXiv2026

Star-Shaped Nakajima Quiver Varieties, Parabolic Higgs Bundle Moduli Spaces, and their Holomorphic Symplectic Structures

In this paper, we consider two classes of hyperk\"ahler manifolds: moduli spaces of central-Levi parabolic Higgs bundles on the punctured sphere and star-shaped Nakajima quiver varieties. We produce a map $\mathcal T$ from a given star-shaped quiver variety $\mathcal X$ to a central-Levi parabolic Higgs bundle moduli space $\mathcal M$. We verify that $\mathcal T$ preserves stability and we show that it is a homeomorphism onto the locus of Higgs bundles with trivial underlying holomorphic structure. We then prove our main theorem: that $\mathcal T$ identifies the natural holomorphic symplectic structures on the two spaces. This theorem generalizes work by Biswas, Florentino, Godinho, Mandini from the rank 2, full flag, strongly parabolic case to arbitrary rank, partial flag, and weakly parabolic cases -- namely, those whose Higgs field residues project to the centers of their respective Levi subalgebras.

math.DG↗

arXiv2026

Hyperk\"ahler Degenerations from Parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs Bundles Moduli Spaces on the Punctured Sphere to Hyperpolygon Spaces

Complete hyperk\"ahler 4-manifolds of finite energy are grouped into ALE, ALF, ALG$^{(*)}$, ALH$^{(*)}$, each of these being further classified according to the Dynkin type of their noncompact end. A family of ALG-$D_4$ spaces are modeled by certain moduli spaces of strongly parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles on the Riemann sphere with $n=4$ punctures. Meanwhile, a family of ALE-$D_4$ spaces are modeled by certain Nakajima quiver varieties known as $n=4$ hyperpolygon spaces. There is a map from hyperpolygon space to the moduli space of strong parabolic $\mathrm{SL}(2,\mathbb{C})$-Higgs bundles that is a diffeomorphism onto its open and dense image. We show that under a fine-tuned degenerate limit, the pullback of a family of ALG-$D_4$ metrics parameterized by $R$ converges pointwise to the ALE-$D_4$ metric as $R \to 0$. While the connection to gravitational instantons occurs in the $n=4$ case, we prove our result for any finite $n$.

math.DG↗

arXiv2022

Further Study on Domains and Quasihyperbolic Distances

We establish constructive geometric tools for determining when a domain is $L^s$-averaging and obtain upper and lower bounds for the $L^s$-integrals of the quasihyperbolic distance. We also construct examples which are helpful to understand our geometric tools and the relationship between $p$-Poincar\'{e} domains and $L^s$-averaging domains. Finally, finite unions of $L^s(\mu)$-averaging domains are explored.

math.CA↗