Noncommutative Geometry and Topology in Prague

A spectral gap for twisted Dolbeault–Dirac operators over irreducible quantum flag manifolds

In this talk we present a geometric approach to understanding the analytic properties of (twisted) Dolbeault–Dirac operators D_{\overline{\partial}} over the irreducible quantum flag manifolds \mathcal{O}_q(G/L_S). We deduce from a noncommutative Akizuki–Nakano identity that twisting a Dolbeault–Dirac operator by a positive line bundle results in a spectral gap for these operators. We then conclude that each positively twisted D_{\overline{\partial}} is an (unbounded) Fredholm operator, and calculate its index using the recently established noncommutative Borel–Weil theorem. (Joint work with Biswarup Das and Petr Somberg