Calabi-Yau structures on the complexifications of rank two symmeric spaces

TL;DR

This work proves the existence of a $G$-invariant $C^{\infty}$-Calabi–Yau structure on the complexification $G^{\mathbb C}/K^{\mathbb C}$ of a rank-two compact-type symmetric space $G/K$, by embedding the problem in the anti-K"ahler Hermann-type action framework. The authors provide a new orbit-geometry-based proof linking the complex Hessian of a $G$-invariant strictly plurisubharmonic potential to the Hessian of a $W$-invariant strictly convex function, using explicit shape operator data of orbits. They construct a $G^{\mathbb C}$-invariant holomorphic volume form $\Omega$ and derive a Monge–Amp\u00e8re equation on the Weyl chamber; applying Caffarelli’s regularity results and Bielawski’s existence/regularity theory yields a smooth invariant solution, which in turn produces a Ricci-flat K"ahler form $\omega_{\rho^h}$ satisfying $\omega_{\rho^h}^n = C\, \Omega\wedge\overline{\Omega}$, with a calibrated special Lagrangian structure. The approach highlights the utility of orbit geometry and Monge–Amp̀re methods in constructing Calabi–Yau structures on anti-K"ahler manifolds arising from complexifications of symmetric spaces, and paves the way for broader applications to complex hyperpolar actions.

Abstract

For a (Reimannian) symmetric space $G/K$ of compact type, the natural action of $G$ on its complexification $G^{\mathbb C}/K^{\mathbb C}$ (which is an anti-Kaehler symmetric space) is one of the isometric actions called ``Hermann type action''. Let $ψ$ be the $G$-invariant strictly plurisubharmonic $C^{\infty}$-function on an open set of $G^{\mathbb C}/K^{\mathbb C}$ arising from a $W$-invariant strictly convex $C^{\infty}$-function $ρ$ on an open set of a maximal abelian subspace $\mathfrak a^d$ of $\mathfrak p^d$, where $\mathfrak p^d$ is the subspace of the Lie algebra $\mathfrak g^d$ of $G^d$ such that $\mathfrak g^d=\mathfrak k\oplus\mathfrak p^d$ gives the Cartan decomposition associated to the dual symmetric space $G^d/K$ of $G/K$ and $W$ is the Weyl group assocaited to $\mathfrak a^d$. In this paper, we first give a new proof of a known relation between the complex Hessian of $ψ$ and the Hessian of $ρ$. This new proof is performed from the viewpoint of the orbit geometry of the Hermann type action $G\curvearrowright G^{\mathbb C}/K^{\mathbb C}$. In more detail, it is performed by using the explicit descriptions of the shape operators of the orbits of the isotropy action $K\curvearrowright G^d/K$ and the Hermann type action $G\curvearrowright G^{\mathbb C}/K^{\mathbb C}$. Next we prove that there exists a $C^{\infty}$-Calabi-Yau structure on the whole of the complexification $G^{\mathbb C}/K^{\mathbb C}$ in the case where $G/K$ is of rank two on the basis of this relation. In the future, the above new proof will be useful to investigate the existence of invariant Calabi-Yau structure on an anti-Kaehler manifold equipped with a certain kind of complex hyperpolar action in more general, where we note that Hermann type actions are complex hyperpolar.