Seminar Geometrija i primene, 18. april 2024.

Naredni sastanak Seminara biće održan u četvrtak, 18. aprila 2024, u sali 301f Matematičkog instituta SANU sa početkom u 17.15.

Predavač: Nikolay Tyurin, JINR Dubna, MI RAN Moscow

Apstrakt: Every algebraic variety by the very definition can be equipped with a symplectic structure, given by the Kahler form of any positive polarization.

Therefore a natural problem is to study lagrangian submanifolds - real submanifolds which posses the lagrangian property. This problem seems to be important itself; moreover certain modern approaches to Mirror Symmetry conjecture are based on a duality between complex and symplectic geometries of Kahler manifolds and therefore one needs to know which lagrangian submanifolds a given algebraic variety admits. At the same time not too much is known even for the simplest and basic cases: f.e. for the projective space CPn one has two natural possibilities  - real part of the complex projective space RPn and Liouville tori Tn subset CPn coming from the fact that one has here the complete set of first integrals.

In 2004 Andrey Mironov presented new construction which gives a number of new examples of lagrangian submanifolds in Cn and CPn  and moreover he showed that these lagrangian submanifolds are Hamiltonian minimal. Leaving aside the minimality story we can present a natural generalization of the Mironov construction which can be applied for much broad case of algebraic variety. Namely suppose that a given compact algebraic variety X  equipped with a symplectic form, given by the Kahler stucture, admits an TK - action by Kahler isometries (here k  can be less that the complex dimension of X) on X which is compatible with a fixed anti holomorphic structure sigma on X. Suppose that the real part X_R subset X with respect to sigma has the maximal possible dimension. Then for each l from 0 to k we can construct a lagrangian submanifold, smooth or with self intersections.

As an example of this generalized construction we present how it works for the case of complex grassmanian Gr(k, n).
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