Special Bohr-Sommerfeld Lagrangian submanifolds of algebraic varieties

摘要

In this paper we continue our study of special Bohr-Sommerfeld submanifolds in the case when the ambient symplectic manifold possesses a compatible integrable complex structure (and is thus an algebraic variety). In this case we show how to reduce the special Bohr-Sommerfeld geometry to Morse theory on the complements of ample divisors. This gives rise to a construction of Lagrangian shadows of ample divisors in algebraic varieties, which is an example of 'algebraic v. symplectic' duality. We suggest a condition for the existence of a Lagrangian shadow and give examples of Lagrangian shadows of ample divisors on the projective plane, complex quadrics and flag manifolds.