Period Matrices of hyperelliptic curves.
Quantization of canonical cones of algebraic curves
We introduce a quantization of the graded algebra of functions on the canonical cone of an algebraic curve , based on the theory of formal pseudodifferential operators. When is a complex curve with Poincaré uniformization, we propose another, equivalent construction, based on the work of Cohen-Manin-Zagier on Rankin-Cohen brackets. We give a presentation of the quantum algebra when is a rational curve, and discuss the problem of constructing algebraically “differential liftings”.
Quantum Singularity Theory for and -Spin Theory
We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity of type our construction of the stack of -curves is canonically isomorphic to the stack of -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the...
Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function
A lattice model with exponential interaction, was proposed and integrated by M. Toda in the 1960s; it was then extensively studied as one of the completely integrable (differential-difference) equations by algebro-geometric methods, which produced both quasi-periodic solutions in terms of theta functions of hyperelliptic curves and periodic solutions defined on suitable Jacobians by the Lax-pair method. In this work, we revisit Toda’s original approach to give solutions of the Toda lattice in terms...
Random matrices and permutations, matrix integrals and integrable systems
Reality conditions of loop solitons genus : hyperelliptic am functions.
Spectral curves of operators with elliptic coefficients.
Spectral curves, opers and integrable systems
The associated map of the nonabelian Gauss-Manin connection
The Gauss-Manin connection for nonabelian cohomology spaces is the isomonodromy flow. We write down explicitly the vector fields of the isomonodromy flow and calculate its induced vector fields on the associated graded space of the nonabelian Hogde filtration. The result turns out to be intimately related to the quadratic part of the Hitchin map.
The characteristic polynomial of a monodromy transformation attached to a family of curves.
The complex geometry of an integrable system
In this paper, a finite dimensional algebraic completely integrable system is considered. We show that the intersection of levels of integrals completes into an abelian surface (a two dimensional complex algebraic torus) of polarization and that the flow of the system can be linearized on it.
Vector bundles on plane cubic curves and the classical Yang–Baxter equation
In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical -matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic -matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin’s classication and prove...
Конечнозонные решения модифицированных уравнений Веселова-Новикова, их спектральные свойства и приложения