Abstracts for A.N. Tyurin's Memorial Conference

Moduli spaces of holomorphic vector bundles on a Riemann surface were extensively studied by A. Tyurin [Tyu08]. Such a moduli space carries, beyond its projective variety structure, a certain real Poisson structure that is symplectic when there are no (real) singularities, the real singularity structure actually being finer than the standard complex analytical one. A special case is the degree zero rank two moduli space with trivial determinant over a Riemann surface of genus two. I had discussed that particular case with A. Tyurin.

One problem we have inherited after Andery Tyurin is the interest in lagrangian geometry of algebraic varieties. The lagrangian geometry comes when we consider an additional data - a fixed Kähler form of the Hodge type. E.g. for Fano varieties one has an embedding by certain power of the anticanonical system to the projective space and so the standard Kähler form on the latter induces an (anti) canonical symplectic structure of the variety.

In 1977 Andrei Tyurin proved that every finite rank bundle on the projective ind-space is isomorphic to a direct sum of line bundles. In this talk I will explain that Tyurin's Theorem is valid for a much broader class of ind-varieties.