Deformation quantization and geometric quantization on K\"ahler manifolds give the mathematical description of the algebra of quantum observables and the Hilbert spaces respectively, where the later forms a representation of quantum observables asymptotically via Toeplitz operators. When there is a Hamiltonian $G$-action on a K\"ahler manifold, there are associated symmetries on both the quantum algebra and representation aspects. We show that in nice cases of coadjoint orbits and K\"ahler-Einstein manifolds, these symmetries are strictly compatible (not only asymptotically).

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For a compact Riemann surface X of positive genus, the space of sections of a certain theta bundle on moduli of bundles of rank r and level k admits a natural map to (the dual of) a similar space of sections of rank k and level r (the strange duality isomorphism). Both sides of the isomorphism carry projective connections as X varies in a family. We prove that this map is (projectively) flat.

Let X be a compact complex, not necessarily K¨ahler, manifold of dimension n. We characterize the volume of any holomorphic line bundle L → X as the supremum R of the Monge-Amp`ere masses X Tn ac over all closed positive currents T in the first Chern class of L, where Tac is the absolutely continuous part of T in its Lebesgue decomposition. This result, new in the non-K¨ahler context, can be seen as holomorphic Morse inequalities for the cohomology of high tensor powers of line bundles endowed with arbitrarily singular Hermitian metrics. It gives, in particular, a new bigness criterion for line bundles in terms of existence of singular Hermitian metrics satisfying positivity conditions. The proof is based on the construction of a new regularization for closed (1, 1)-currents with a control of the Monge-Amp`ere masses of the approximating sequence. To this end, we prove a potential-theoretic result in one complex variable and study the growth of multiplier ideal sheaves associated with increasingly singular metrics.

We show an example of a non-archimedean version of the existence part of the Calabi-Yau theorem in complex geometry. Precisely, we study totally degenerate abelian varieties and certain probability measures on their associated analytic spaces in the sense of Berkovich.