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

## TL;DR

This paper constructs and analyzes monotonic non-standard Lagrangian tori in toric and non-toric monotonic symplectic manifolds, extending previous work with new examples and addressing the monotonicity condition.

## Contribution

It introduces methods to construct monotonic non-standard Lagrangian tori, including Chekanov type, in both toric and pseudotoric manifolds, expanding the class of known examples.

## Key findings

- Constructed monotonic Chekanov-type Lagrangian tori.
- Extended constructions to non-toric monotonic manifolds like complex quadrics and flag varieties.
- Provided insights into the topological and monotonicity properties of these tori.

## 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 ${\rm BS}_{can}$ 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 $F^3$.