Monotonic Lagrangian Tori of Standard and Nonstandard Types in Toric and Pseudotoric Fano Varieties

Abstract

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 BScan 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 4dimensional quadric and full flag variety F 3 . * The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. N 14.641.31.0001 their symplectic areas which gives the set of periods (p a1 , ..., p an ) defined up to Z. Clearly any lagrangian S is BS k if and only if the periods belong to 1 k Z; any other choice of the basis corresponds to a transformation of the period vector by SL(n, Z), therefore for any lagrangian torus one can define its Bohr -Sommerfeld level as the minimal k such that every kp ai belongs to Z if it exists or if it does not saying that it is BS ∞ .Fixing any almost complex structure J on M , compatible with ω, we get the complex determinant line bundle K −1 M = detT 1,0 M which we call anticanoncial line bundle following the tradition. It depends on the choice of J, but its first Chern class does not being integer valued therefore it is a topological invariant of symplectic manifold. Consider the case of monotonic symplecitc manifold namely when c 1 (K −1 M ) = k[ω] for certain integer k. For this case we say that a lagrangian torus S ⊂ M is Bohr -Sommerfeld with respect to the anticanonical bundle (or BS can for short) if it is Bohr -Sommmerfeld of level k.Remark. This assignment looks a bit artificial in general symplectic setup, but from the point of view of algebraic geometry it looks more natural. Indeed, consider a Fano variety X. By the very definition its anticanonical line bundle K −1 X is ample therefore certain power (K −1 X ) m induces an embedding of X to the projective space. Then choosing a standard Kahler form Ω on the projective space and restricting it to the image of X we get a symplectic form mω on X such that sing...

Mironov Lagrangian cycles in algebraic varieties

“…The Main theorem says that T γ,ci is a smooth lagrangian torus: if γ is non contractible in < D + , D − > \{p ± } then the torus is of the standard type, if γ is contractible then the resulting torus is of exotic type, see [4]. Moreover, in [5] one proves that if X admits a standard monotone lagrangian torus then it exists an exotic lagrangian torus which is monotone as well.…”

“…In this non toric situation the base CP 1 contains three points which underly singular fibers therefore we have more complicated picture since a smooth loop on the complement "base minus three points" can represent different classes being non contractible. In this setup one can reconstruct the famous Gelfand -Zeytlin lagrangian sphere, see [4]; different types of smooth lagrangian tori can be derived from the picture, see [5].…”