5,684 research outputs found
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
on the full flag variety 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
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
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 for and the very ample bundle is 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
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
Constructing Mironov cycles in complex Grassmannians
A. Mironov proposed a construction of lagrangian submanifolds in
and ; 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 of complex dimension admits - action and
an anti - holomorphic involution such that the real part has real dimension 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 , 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 .Comment: 5 page
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