Journal of Symplectic Geometry

Volume 18 (2020)

Number 4

Geometric quantization of almost toric manifolds

Pages: 1147 – 1168



Eva Miranda (Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain)

Francisco Presas (Instituto de Ciencias Matemáticas, CSIC–UAM–UC3M–UCM, Madrid, Spain)

Romero Solha (Departamento de Matemática, PUC, Rio de Janeiro, Brazil)


Kostant gave a model for the geometric quantization via the cohomology associated to the sheaf of flat sections of a pre-quantum line bundle. This model is well-adapted for real polarizations given by integrable systems and toric manifolds. In the latter case, the cohomology can be computed by counting integral points inside the associated Delzant polytope. In this article we extend Kostant’s geometric quantization to semitoric integrable systems and almost toric manifolds. In these cases the dimension of the acting torus is smaller than half of the dimension of the manifold. In particular, we compute the cohomology groups associated to the geometric quantization if the real polarization is the one induced by an integrable system with focus-focus type singularities in dimension four. As an application we determine a model for the geometric quantization of K3 surfaces under this scheme.

Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia 2016 Prize and is partially supported by grants with reference MTM2015-69135-P (MINECO-FEDER) and 2017SGR932 (AGAUR). Romero Solha is supported by CAPES and partially supported by MTM2015-69135-P (MINECO/FEDER). Francisco Presas is supported by the grant reference number MTM2016-79400-P (MINECO/FEDER). Eva Miranda and Francisco Presas are supported by an EXPLORA CIENCIA project with reference number MTM2015-72876-EXP and by the excellence project SEV-2015-0554. This article is written within the collaborative agreement of a Doctor Vinculado honorary position at ICMAT-CSIC held by Eva Miranda.

Received 11 May 2018

Accepted 28 May 2019

Published 28 October 2020