Michael Lennox Wong’s research while affiliated with University of Duisburg-Essen and other places

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 bundle on . 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 four‐dimensional cases are expected to be of type ALF.

In this paper we determine the motivic class---in particular, the weight polynomial and conjecturally the Poincar\'e polynomial---of the open de Rham space, defined and studied by Boalch, of certain moduli of irregular meromorphic connections on the trivial 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--Schr\"oer. We finish with constructing natural complete hyperk\"ahler metrics on them, which in the 4-dimensional cases are expected to be of type ALF.

Given a complex manifold X, any K\"ahler class defines an affine bundle over X, and any K\"ahler 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 K\"ahler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Sz\H{o}ke 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.

Let E be a vector bundle over an irreducible projective variety X defined over an algebraically closed field. We give a necessary and sufficient condition for E to be a direct image of a vector bundle on an étale cover, of degree more than one, of X. In fact, we describe all possible ways E can be realized as a direct image. Given a connection D on E, a criterion is given for D to be induced by a connection on a vector bundle whose direct image, by an étale covering map of degree more than one, is E.

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-K\"ahler metric on $T^*(G/P)$, recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.

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-K\"ahler metric on $T^*(G/P)$, recovering Biquard's description of this twistor space, but employing only finite-dimensional, Lie-theoretic means.

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.

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.

We investigate parabolic Higgs bundles and Γ-Higgs bundles on a smooth complex projective variety.