By the very definition a compact algebriac variety can be embedded to projective space of appropriate dimension and therefore can be equipped with the corresponding Kahler form given by the lifting along this embedding of the standard form of the Fubini - Study metric. This form can be regarded as a real symplectic form and then it is reasonable to speak about the classification of possible lagrangian submanifolds.

The classification naturally depends on the choice of the embedding to projective space so on the choice of polarization and does not depend on the particular choice of the form in the same class since every two forms from the same class are equivalent.

The classification problem splits into steps of different levels:
1) Which classes from $H_n(X,\mathbb{Z})$ can be realized by lagrangian submanifolds?
2) Which topological types of submanifolds can be realized as lagrangian?
3) How many equivalence classes (up to lagrangian deformations) exist for fixed class and topological type?
4) The same question for hamiltonian deformations.

Despite of the fact that algebraic varieties of small dimensions are studied quite well, the classification problem of their lagrangian submanifolds is very far to be solved even in the basic cases. For the projective plane $\mathbb{CP}^2$ itself we know that:
1) evidently any lagrangian surface must be homologically trivial;
2) it can be realized either by $\mathbb{RP}^2$ or $T^2$, and the resting question still have not complete final answers.

Our course is aimed to present certain methods constructing lagrangian submanifolds if some algebraic varieties which admit $T^k$ torus action by Kahler isometries. Observations lied in the base of these methods come from the classical method of characteristics.

First examples which we would like to carefully discuss — lagrangian submanifolds in $\mathbb{CP}^n$, constructed by A.E. Mironov. Then we go to natural generalizations of his construction and present lagrangian tori and spheres in the flag variety (full flag in $\mathbb{C}^3$), and after that we will show how the generalized Mironov construction works for the case of Grassmanian $\mathrm{Gr}(1,n)$.

Writing this abstract the author supposes that in the process of our work we shall find something new (see the program below, especially no. 10).

The course is addressed to the students familiar with the basic notions of symplectic geometry (f.e. one can see the first lectures of my previous course in SEC). Due to the brief format we will focus on geometric interpretations, not on sharp proofs.

Preliminary program

1. Algebraic variety from the point of view of differential geometry. Polarizations, Classification problem of lagrangian submanifold.
2. Geometric formulation of Quantum mechanics. Couple of words on quantization.
3. Torus action of algebraic variety. Pseudotoric structures.
4. Flag variety $F^3$, pseudotoric structure and lagrangian fibrations.
5. Complex projective space: how to fill the gap between $\mathbb{RP}^n$ and $T^n$.
6. Digression: (Hamiltonian) minimality of lagrangian submanifolds.
7. Symplectic reduction and lagrangian submanifolds. First examples in $\mathrm{Gr}(1,n)$.
8. Examples of lagrangian submanifolds in $\mathrm{Gr}(1,n)$ (continuation).
9. Examples of lagrangian submanifolds in $\mathrm{Gr}(1,n)$ (final part). Gelfand - McPherson correspondence.
10. Construction of D. Bykov: flag variety as a lagrangian submanifold in the direct product of projective spaces. Possible generalizations.

Seminar organizer
Tyurin Nikolai Andreevich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center