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Michael Lennox Wong

Publications and source records attributed to Michael Lennox Wong.

9 recordsLinked to original sources
arXiv2023

Arithmetic and metric aspects of open de Rham spaces

In this paper we determine the motivic class--in particular, the weight polynomial and conjecturally the Poincaré polynomial--of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank $n$ bundle on $\mathbb{P}^1$. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss--Leclerc--Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the $4$-dimensional cases are expected to be of type ALF.

math.AG↗
arXiv2023

E-polynomials of character varieties for real curves

We calculate the E-polynomial for a class of the (complex) character varieties $\mathcal{M}_n^τ$ associated to a genus $g$ Riemann surface $Σ$ equipped with an orientation reversing involution $τ$. Our formula expresses the generating function $\sum_{n=1}^{\infty} E(\mathcal{M}_n^τ) T^n$ as the plethystic logarithm of a product of sums indexed by Young diagrams. The proof uses point counting over finite fields, emulating Hausel and Rodriguez-Villegas.

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arXiv2019

Canonical complex extensions of Kähler manifolds

Given a complex manifold $X$, any Kähler class defines an affine bundle over $X$, and any Kähler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact Kähler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Szőke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having non-negative holomorphic bisectional curvature and big tangent bundle.

math.CV↗
arXiv2018

The universal connection for principal bundles over homogeneous spaces and twistor space of coadjoint orbits

Given a holomorphic principal bundle $Q\, \longrightarrow\, X$, the universal space of holomorphic connections is a torsor $C_1(Q)$ for $\text{ad} Q \otimes T^*X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic connection. When $X\,=\, G/P$, where $P$ is a parabolic subgroup of a complex simple group $G$, and $Q$ is the frame bundle of an ample line bundle, we show that $C_1(Q)$ may be identified with $G/L$, where $L\, \subset\, P$ is a Levi factor. We use this identification to construct the twistor space associated to a natural hyper-Kähler metric on $T^*(G/P)$, recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.

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arXiv2016

Arithmetic and representation theory of wild character varieties

We count points over a finite field on wild character varieties of Riemann surfaces for singularities with regular semisimple leading term. The new feature in our counting formulas is the appearance of characters of Yokonuma-Hecke algebras. Our result leads to the conjecture that the mixed Hodge polynomials of these character varieties agree with previously conjectured perverse Hodge polynomials of certain twisted parabolic Higgs moduli spaces, indicating the possibility of a P=W conjecture for a suitable wild Hitchin system.

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arXiv2012

Root stacks, principal bundles and connections

We investigate principal bundles over a root stack. In case of dimension one, we generalize the criterion of Weil and Atiyah for a principal bundle to have an algebraic connection.

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arXiv2012

An Interpretation of Some Hitchin Hamiltonians In Terms of Isomonodromic Deformation

This paper deals with moduli spaces of framed principal bundles with connections with irregular singularities over a compact Riemann surface. These spaces have been constructed by Boalch by means of an infinite-dimensional symplectic reduction. It is proved that the symplectic structure induced from the Atiyah--Bott form agrees with the one given in terms of hypercohomology. The main results of this paper adapt work of Krichever and of Hurtubise to give an interpretation of some Hitchin Hamiltonians as yielding Hamiltonian vector fields on moduli spaces of irregular connections that arise from differences of isomonodromic flows defined in two different ways. This relies on a realization of open sets in the moduli space of bundles as arising via Hecke modification of a fixed bundle.

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arXiv2011

Hecke Modifications, Wonderful Compactifications and Moduli of Principal Bundles

In this paper, we obtain parametrizations of the moduli space of principal bundles over a compact Riemann surface using spaces of Hecke modifications in several cases. We begin with a discussion of Hecke modifications for principal bundles and give constructions of "universal" Hecke modifications of a fixed bundle of fixed type. This is followed by an overview of the construction of the "wonderful," or De Concini--Procesi, compactification of a semi-simple algebraic group of adjoint type. The compactification plays an important role in the deformation theory used in constructing the parametrizations. A general outline to construct parametrizations is given and verifications for specific structure groups are carried out.

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