Lagrangian shadows of ample algebraic divisors

TL;DR

This paper explores the concept of Lagrangian shadows associated with ample divisors in algebraic varieties, demonstrating their properties and providing examples, including Gelfand-Zeytlin spheres in flag varieties.

Contribution

It introduces the notion of Lagrangian shadows in the context of Special Bohr-Sommerfeld geometry and illustrates their structure with explicit examples.

Findings

Lagrangian shadows are almost canonical real submanifolds associated with ample divisors.

Examples include Gelfand-Zeytlin spheres as Lagrangian shadows in flag varieties.

The paper connects algebraic divisors with symplectic geometry via Lagrangian submanifolds.

Abstract

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.