12-27【张洋】Homogeneous Einstein Metrics on Flag Supermanifolds

A generalised flag manifold, also known as a Kählerian C-space, is a homogeneous space G/K, where G is a compact Lie group and K is the centraliser of a torus in G. These spaces have rich geometric structures, making them significant in both differential geometry and algebraic geometry.

In this talk, we will introduce flag supermanifolds via Dynkin diagrams of compact Lie supergroups, resembling the construction of generalised flag manifolds in the classical (non-super) setting. We will give explicit Ricci curvature formulas for those supermanifolds, and classify invariant Einstein metrics on specific flag supermanifolds of types A and C. Our results provide examples of compact homogeneous supermanifolds on which the Einstein equation admits no solutions, discrete families of solutions, and continuous families of Ricci-flat solutions among invariant metrics. These findings demonstrate that the classical finiteness conjecture does not extend to supermanifolds and challenge the intuition furnished by Bochner's vanishing theorem. This is joint work with Mark Gould, Artem Pulemotov, and Jorgen Rasmussen.