| 研究実績の概要 |
My research achivevements consisted of the construction and characterization of the two-dimensional cohomological Hall algebras of a fixed smooth projective complex curve X.
In the paper arXiv:1801.03482, together with O. Schiffmann, I have defined the cohomological Hall algebra associated with the Dolbeaut moduli stack of X (that is, the moduli stack parameterizing Higgs sheaves on X). We have characterized such an algebra describing, for example, a set of generators of it.
Recall that the de Rham moduli stack is the stack parameterizing vector bundles with flat connections on X, while the Betti moduli stack is the stack parameterizing finite-dimensional representations of the fundamental group of X. In the paper arXiv:1903.07253, together with M. Porta, I constructed cohomological Hall algebras for the de Rham and Betti moduli stacks of X, respectively. This result is a consequence of a more general construction of convolution algebra structures on the bounded derived category of coherent sheaves on the Dolbeaut, de Rham and Betti moduli stacks (this gives rise to categorified Hall algebras). In the Dolbeaut case, the resulting categorified Hall algebra indeed categorifies the algebra constructed with Schiffmann. In addition, I have established some relations between these 3 categorified Hall algebras, which can be interpreted as Hall algebra versions of the Riemann-Hilbert correspondence, in the de Rham & Betti case, and of the non-abelian Hodge correspondence, in the Dolbeaut & de Rham case.
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