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International conference "Analysis and Singularities" dedicated to 70th anniversary of Vladimir Igorevich Arnold
August 21, 2007 10:05, Moscow

Abelian functions and singularity theory

V. M. Buchstaber

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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V. M. Buchstaber

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Abstract: Theory of Abelian functions was a central topic of the 19th century mathematics. In mid-seventies of the last century a new wave arose of investigation in this field in response to the discovery that Abelian functions provide solutions of a number of challenging problems of modern Theoretical and Mathematical Physics.
In a cycle of our joint papers with V. Enolskii and D. Leykin we have developed a theory of multivariate sigma-function, an analogue of the classic Weierstrass sigma-function.
A sigma-function is defined on a cover of $U$, where $U$ is the space of a bundle $p\colon U\to B$ defined by a family of plane algebraic curves of fixed genus. The base $B$ of the bundle is the space of the family parameters and a fiber $J_b$ over $b\in B$ is the Jacobi variety of the curve with the parameters $b$. A second logarithmic derivative of the sigma-function along the fiber is an Abelian function on $J_b$.
Thus, one can generate a ring $F$ of fiber-wise Abelian functions on $U$. The problem to find derivations of the ring $F$ along the base $B$ is a reformulation of the classic problem of differentiation of Abelian functions over parameters. Its solution is relevant to a number of topical applications.
The talk presents a solution of this problem recently found by the speaker and D. Leykin.
A precise modern formulation of the problem involves the language of Differential Geometry. We obtained explicit expressions for the generators of the module of differentiations of a ring of Abelian functions. The families of curves, which we work with, are special deformations of the singularities $y^n-x^s$, where $\operatorname{gcd}(n,s)=1$. The choice of this type of families allows us to use methods and results of Singularity Theory, especially Arnold's convolution of invariants and the theorem of Zakalyukin on holomorphic vector fields tangent to the discriminant variety.
  • 1. V. M. Buchstaber, V. Z. Enolskii, D. V. Leikin, Kleinian functions, hyperelliptic Jacobians and applications, Reviews in Mathematics and Math. Physics, I. M. Krichever, S. P. Novikov Editors, v. 10, part 2, Gordon and Breach, London, 1997, 3–120.
  • 2. V. M. Buchstaber, D. V. Leykin, Polynomial Lie algebras, Funct. Anal. Appl. 36:4 (2002), 267–280.
  • 3. V. M. Buchstaber, D. V. Leykin, The heat equations in a nonholonomic frame, Funct. Anal. Appl. 38:2 (2004), 88–101.
  • 4. V. M. Buchstaber, D. V. Leykin, Addition laws on Jacobian varieties of plane algebraic curves, Proc. Steklov Math. Inst. 251 (2005), 49–120.
  • 5. V. M. Buchstaber, D. V. Leykin, Differentiation of Abelian functions over its parameters, Russian Math. Surveys 62:4 (2007), 787–789.

    Materials: buch.pdf (171.5 Kb)

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