Coherent state embeddings, polar divisors and Cauchy formulas

For arbitrary quantizable compact Kähler manifolds, relations between the geometry given by the coherent states based on the manifold and the algebraic (projective) geometry realized via the coherent state mapping into projective space, are studied. Polar divisors, formulas relating the scalar products of coherent vectors on the manifold with the corresponding scalar products on projective space (Cauchy formulas), two-point, three-point and more generally cyclic m-point functions are discussed. The three-point function is related to the shape invariant of geodesic triangles in projective space.