Hitchin and Calabi–Yau Integrable Systems via Variations of Hodge Structures

Since its discovery by Hitchin in 1987, G-Hitchin systems for a reductive complex Lie group G have extensively been studied. For example, the generic fibers are nowadays well-understood. In this paper, we show that the smooth parts of G-Hitchin systems for a simple adjoint complex Lie group G are isomorphic to non-compact Calabi–Yau integrable systems extending results by Diaconescu–Donagi–Pantev. Moreover, we explain how Langlands duality for Hitchin systems is related to Poincaré–Verdier duality of the corresponding families of quasi-projective Calabi–Yau threefolds. Even though the statement is holomorphic-symplectic, our proof is Hodge-theoretic. It is based on polarizable variations of Hodge structures that admit so-called abstract Seiberg–Witten differentials. These ensure that the associated Jacobian fibration is an algebraic integrable system.

Quantum Langlands duality of representations of -algebras

We prove duality isomorphisms of certain representations of ${\mathcal{W}}$-algebras which play an essential role in the quantum geometric Langlands program and some related results.

Fourier–Mukai and autoduality for compactified Jacobians. I

To every singular reduced projective curve X one can associate, following Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We prove that, for a reduced curve with locally planar singularities, the integral (or Fourier–Mukai) transform with kernel the Poincaré sheaf from the derived category of the generalized Jacobian of X to the derived category of any fine compactified Jacobian of X is fully faithful, generalizing a previous result of Arinkin in the case of integral curves. As a consequence, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence of line bundles coincide on any fine compactified Jacobian, generalizing previous results of Arinkin, Esteves, Gagné, Kleiman, Rocha, and Sawon.The paper contains an Appendix in which we explain how our work can be interpreted in view of the Langlands duality for the Higgs bundles as proposed by Donagi–Pantev.