Special Bohr–Sommerfeld Lagrangian submanifolds of algebraic varieties

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.