In this short course, we shall review some techniques and recent results regarding the equivariant asymptotics of the Szegö kernels associated to a compact quantizable manifold with a Hamiltonian action; while our focus will actually be on algebro-geometric Szegö kernels, much of what will be discussed admits a generalization to the symplectic almost complex setting.

The thread of the discussion will be offered by the following basic question: if we are given a Hamiltonian action on a quantizable symplectic manifold, which is ‘quantized’ by a unitary representation, how does the equivariant decomposition into isotypical components of the latter reflect the geometry of the action? We shall consider the asymptotic and local aspects of this relation, and our basic tool will be the description of Szegö kernels as Fourier integral operators, which is due to Boutet de Monvel and Sjöstraend. Thus our development will build on the approach to geometric quantization originally introduced by Boutet de Monvel and Guillemin, and more recently extended, and further unfolded and elaborated, principally, by Shiffman and Zelditch.

References

We’ll touch mainly on the content of the following papers, to which we refer for an adequate bibliography on this theme:

1. S. Zelditch, Szegő kernels and a theorem of Tian. Internat. Math. Res. Notices 1998, no. 6, 317–331
2. B. Shiffman, S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds. J. Reine Angew. Math. 544 (2002), 181–222
3. R. Paoletti, Asymptotics of Szegö kernels under Hamiltonian torus actions. Israel J. Math. 191 (2012), no. 1, 363–403.
4. R. Paoletti, Lower-order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions. Recent advances in algebraic geometry, 321–369
5. A. Galasso, R. Paoletti. Equivariant asymptotics of Szegö kernels under Hamiltonian $U(2)$-actions. Annali di Matematica (2018).
6. A. Galasso, R. Paoletti. Equivariant Asymptotics of Szegö kernels under Hamiltonian $SU(2)$-actions.