There is a well-known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns–Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least $3$ and irreducible topological spherical buildings of rank at least $3$.

We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type $\mathsf{M}$ associated with any polar action of cohomogeneity at least $2$ on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type $\mathsf{M}$, its universal Tits cover indeed is a building in all but two exceptional cases. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure, we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least $2$.

Let $X$ be a general cubic $4$-fold. It was observed by Donagi and Markman that the relative intermediate Jacobian fibration $\mathcal{J}_U/U$ (with $U=(\mathbb{P}^5)^\vee\setminus X^\vee$) associated with the family of smooth hyperplane sections of $X$ carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification $\overline{\mathcal{J}}$ of $\mathcal{J}_U$ with the property that the holomorphic symplectic form on $\mathcal{J}_U$ extends to a holomorphic symplectic form on $\overline{\mathcal{J}}$. In particular, $\overline{\mathcal{J}}$ is a $10$-dimensional compact hyper-Kähler manifold, which we show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Kuznetsov and Markushevich.

In this paper, we give a geometric expression for the multiplicities of the equivariant index of a spin$^c$ Dirac operator.