Special Bohr - Sommerfeld geometry

We present a new approach to special lagrangian geometry which works for Bohr - Sommerfeld lagrangian submanifolds of symplectic manifolds with integer symplectic forms. This leads to construction of finite dimensional moduli spaces of SBS lagrangian cycles over algebraic varieties.Comment: 19 pages, preliminary version, comments are welcom

Lagrangian shadows of ample algebraic divisors

In the framework of Special Bohr - Sommerfeld geometry it was established that an ample divisor in compact algebraic variety can define almost canonically certain real submanifold which is lagrangian with respect to the corresponding Kahler form. It is natural to call it "lagrangian shadow"; below we emphasize this correspondence and present some simple examples, old and new. In particular we show that for irreducible divisors from the linear system $\vert - \frac{1}{2} K_{F^3} \vert$ on the full flag variety $F^3$ their lagrangian shadows are Gelfand - Zeytlin type lagrangian 3 - spheres.Comment: 4 pages, comments are welcom

Three conjectures on lagrangian tori in the projective plane

In this paper we extend the discussion on Homological Mirror Symmetry for Fano toric varieties presented by Hori and Vafa to more general case of monotone symplectic manifolds with real polarizations. We claim that the Hori -- Vafa prediction, proven by Cho and Oh for toric Fano varieties, can be checked in much more wider context. Then the notion of Bohr - Sommerfeld with respect to the canonical class lagrangian submanifold appears and plays an important role. The discussion presents a bridge between Geometric Quantization and Homological Mirror Symmetry programmes both applied to the projective plane in terms of its lagrangian geometry. Due to this relation one could exploit some standard facts known in GQ to produce results in HMS.Comment: 17 pages, no figa

Special Bohr - Sommerfeld geometry on Riemann surfaces: toy problems

Special Bohr - Sommerfeld geometry, first formulated for simply connected symplectic manifolds (or for simple connected algebraic varieties), gives rise to some natural problems for the simplest example in non simply connected case. Namely for any algebraic curve one can define a correspondence between holomorphic differentials and certain finite graphs. Here we ask some natural questions appear with this correspondence. It is a partial answer to the question of A. Varchenko about possibility of applications of Special Bohr -Sommerfeld geometry in non simply connected case. The russian version has been translated.Comment: 4 page

Homological orthogonality of "symplectic" and "lagrangian"- corrected version

In this remark we discuss a relationship between (co)homology classes of a symplectic manifold realized by symplectic and lagrangian objects. We establish some transversality condition for the classes, realized by symplectic divisors and smooth lagrangian tori with some special condition on their intersections.Comment: 4 pages, no figures, LaTe

Maslov class of lagrangian embedding to Kahler manifold

One generalizes the notion of Maslov class of lagrangian embeddings to symplectic vector spaces for the compact case. Topological and geometrical properties of the generalized class is discussed. Certain relationship with the minimality problem is established. Applications are presented.Comment: 23 pages, no figures, submitted to Izvestiya Mat

New example of modified moduli space of special Bohr - Sommerfeld lagrangian submanifolds

We present an example of modified moduli space of special Bohr - Sommerfeld lagrangian submanifolds for the case when the given algebraic variety is the full flag $F^3$ for $\mathbb{C}^3$ and the very ample bundle is $K^{- \frac{1}{2}}_{F^3}$Comment: 6 page

Monotonic lagrangian tori of standard and non standard types in toric and pseudotoric manifolds

In recent papers, summarized in survey [1], we construct a number of examples of non standard lagrangian tori on compact toric varieties and as well on certain non toric varieties which admit pseudotoric structures. Using this pseudotoric technique we explain how non standard lagrangian tori of Chekanov type can be constructed and what is the topological difference between standard Liouville tori and the non standard ones. However we have not discussed the natural question about the periods of the constructed twist tori; in particular the monotonicity problem for the monotonic case was not studied there. In the paper we present several remarks on these questions, in particular we show for the monotonic case how to construct non standard lagrangian tori which satisify the monotonicity condition. First of all we study non standard tori which are Bohr - Sommerfeld with respect to the anticanonical class. This notion was introduced in [2], where one defines certain universal Maslov class for the ${\rm BS}_{can}$ lagrangian submanifolds in compact simply connected monotonic symplectic manifolds. Then we show how monotonic non standard lagrangian tori of Chekanov type can be constructed. Furthemore we extend the consideration to pseudotoric setup and construct examples of monotonic lagrangian tori in non toric monotonic manifolds: complex 4 - dimensional quadric and full flag variety $F^3$

Constructing Mironov cycles in complex Grassmannians

A. Mironov proposed a construction of lagrangian submanifolds in $\mathbb{C}^n$ and $\mathbb{C} \mathbb{P}^n$; there he was mostly motivated by the fact that these lagrangian submanifolds (which can have in general self intersections, therefore below we call them lagrangian cycles) present new example of minimal or Hamiltonian minimal lagrangian submanifolds. However the Mironov construction of lagrangian cycles itself can be directly extended to much wider class of compact algrebraic varieties: namely it works in the case when algebraic variety $X$ of complex dimension $n$ admits $T^k$ - action and an anti - holomorphic involution such that the real part $X_{\mathbb{R}} \subset X$ has real dimension $n$ and is transversal to the torus action. For this case one has families of lagrangian submanifolds and cycles. In the present small text we show how the construction of Mironov cycles works for the complex Grassmannians, resulting in simple examples of smooth lagrangian submanifolds in ${\rm Gr}(k, n+1)$, equipped with a standard Kahler form under the Pl\"{u}cker embedding. For sure the text is not complete but in the new reality we would like to fix it, hoping to continue the investigations and to present in a future complete list of Mironov cycles in ${\rm Gr}(k, n+1)$.Comment: 5 page

Pseudo symplectic geometry as an extension of the symplectic geometry

In this paper we define a new category of almost complex riemannian 4- manifolds and discuss some basic properties of such pseudo symplectic manifolds. Some motivation based on the Seiberg - Witten theory is imposed.Comment: 29 page