Symplectic Dolbeault Operators on K\"ahler Manifolds

TL;DR

This paper introduces symplectic Dolbeault operators on K"ahler manifolds, explores their properties on Riemann surfaces and flag manifolds, and demonstrates their utility in distinguishing certain Hermitian structures.

Contribution

It defines symplectic Dolbeault operators using symplectic spinors, analyzes their properties on specific classes of K"ahler manifolds, and connects their behavior to representation theory.

Findings

Symplectic Dolbeault operators are elliptic on Riemann surfaces.

Indices of these operators are computed for Riemann surfaces.

Operators distinguish between flag manifolds of different Lie algebra types.

Abstract

For a K\"ahler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special attention to two special classes of K\"ahler manifolds: Riemann surfaces and flag manifolds ($G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). In the case of flag manifolds, we work with the Hermitian structure induced by the Killing form and a choice of positive roots (this is actually not a K\"ahler structure but is a K\"ahler with torsion (KT) structure). For Riemann surfaces the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Hermitian manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$. We give a thorough analysis of these operators on $\C P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.