Research Projects

The study aims to investigate the motives of moduli spaces of connections (principal G-connections, logarithmic connections, parabolic connections) over a complex projective variety and deduce algebro-geometric properties. The study of moduli spaces of vector bundles with connections is a central topic in Algebraic Geometry, and the Betti numbers of these spaces are a difficult problem. Motives have been introduced through various cohomology theories, including de Rham cohomology, Etale cohomology, Weil cohomology, Crystalline cohomology, and Hodge cohomology. Grothendieck proposed the theory of motives to unify different cohomology theories, and several motives have been studied, including Grothendieck motives, Chow motives, Nori motives, and Voevodsky motives. Many algebraic geometer have studied the motive of semistable Higgs bundles of coprime rank and degree on a smooth projective curve over a field in the Voevodsky's trianglulated category under the assumption that the curve has a rational point. The motivic non-abelian Hodge theory is another interesting topic, which sets a correspondence between the integral motives of the Higgs moduli space and de Rham moduli space. The project consists of four main problems: investigating the Chow and Voevodsky motives of moduli spaces of principal G-bundles, principal G-Higgs bundles, and principal G-connections; proving the motivic version of Hodge conjecture for compactified moduli spaces of logarithmic and parabolic connections over a smooth projective curve; considering motivic Torelli type problems for these spaces; computing algebro-geometric invariants such as Chow groups and Brauer group of the moduli space of Lie algebroid connections in positive characteristic; and determining the rationality of the moduli spaces of principal G-connections and parabolic connections over a smooth projective curve.