arXiv subjects

Dmitry Korotkin

Publications and source records attributed to Dmitry Korotkin.

21 records · Page 2Linked to original sources

arXiv2014

Baker-Akhiezer spinor kernel and tau-functions on moduli spaces of meromorphic differentials

In this paper we study Baker-Akhiezer spinor kernel on moduli spaces of meromorphic differentials on Riemann surfaces. We introduce the Baker-Akhiezer tau-function which is related to both Bergman tau-function (which was studied before in the context of Hurwitz spaces and spaces of holomorphic and quadratic differentials) and KP tau-function on such spaces. In particular, we derive variational formulas of Rauch-Ahlfors type on moduli spaces of meromorphic differentials with prescribed singularities: we use the system of homological coordinates, consisting of absolute and relative periods of the meromorphic differential, and show how to vary the fundamental objects associated to a Riemann surface (the matrix of $b$-periods, normalized Abelian differentials, the Bergman bidifferential, the Szegö kernel and the Baker-Akhiezer spinor kernel) with respect to these coordinates. The variational formulas encode dependence both on the moduli of the Riemann surface and on the choice of meromorphic differential (variation of the meromorphic differential while keeping the Riemann surface fixed corresponds to flows of KP type). Analyzing the global properties of the Bergman and Baker-Akhiezer tau-functions we establish relationships between various divisor classes on the moduli spaces.

nlin.SI↗

arXiv2012

On higher genus Weierstrass sigma-function

The goal of this paper is to propose a new way to generalize the Weierstrass sigma-function to higher genus Riemann surfaces. Our definition of the odd higher genus sigma-function is based on a generalization of the classical representation of the elliptic sigma-function via Jacobi theta-function. Namely, the odd higher genus sigma-function $σ_χ(u)$ (for $u\in \C^g$) is defined as a product of the theta-function with odd half-integer characteristic $β^χ$, associated with a spin line bundle $χ$, an exponent of a certain bilinear form, the determinant of a period matrix and a power of the product of all even theta-constants which are non-vanishing on a given Riemann surface. We also define an even sigma-function corresponding to an arbitrary even spin structure. Even sigma-functions are constructed as a straightforward analog of a classical formula relating even and odd sigma-functions. In higher genus the even sigma-functions are well-defined on the moduli space of Riemann surfaces outside of a subspace defined by vanishing of the corresponding even theta-constant.

nlin.SI↗

arXiv2011

Tau function and moduli of differentials

The tau function on the moduli space of generic holomorphic 1-differentials on complex algebraic curves is interpreted as a section of a line bundle on the projectivized Hodge bundle over the moduli space of stable curves. The asymptotics of the tau function near the boundary of the moduli space of 1-differentials is computed, and an explicit expression for the pullback of the Hodge class on the projectivized Hodge bundle in terms of the tautological class and the classes of boundary divisors is derived. This expression is used to clarify the geometric meaning of the Kontsevich-Zorich formula for the sum of the Lyapunov exponents associated with the Teichmüller flow on the Hodge bundle.

math.AG↗