Canonical blow-ups of Lagrangian and Orthogonal Grassmannians

Abstract

Let $\mathbf{LG}(V\oplus V^*)$ and $\mathbf{OG}^+(V\oplus V^*)$ denote the Lagrangian and orthogonal Grassmannians endowed with the natural $\mathbb{G}_m$-actions, respectively. Thaddeus proved that over $\mathbb{C}$, the Hilbert quotients $\mathbf{LG}(V\oplus V^*)\!/\!/\mathbb{G}_m$ and $\mathbf{OG}^+(V\oplus V^*)\!/\!/\mathbb{G}_m$ are isomorphic to the wonderful compactifications of the spaces of symmetric and skew-symmetric matrices of maximal ranks, that is, the spaces of complete quadrics and complete skew-forms, respectively. In this paper, we construct the universal families of these Hilbert quotients by explicitly blowing up the corresponding isotropic Grassmannians, resulting in smooth toroidal compactifications of the spaces of symmetric and skew-symmetric matrices of maximal ranks (before projectivization), which have simple normal crossing boundary divisors and include the spaces of the complete bilinear forms among these divisors. Specifically, we prove that over any algebraically closed field, the universal families of these Hilbert quotients are smooth, and the Hilbert quotients themselves are isomorphic to the spaces of the complete bilinear forms. Over an algebraically closed field of characteristic zero, we prove that the universal families are weak Fano varieties with vanishing higher cohomology groups for their tangent bundles, and are therefore locally rigid. Furthermore, we show that these universal families naturally resolve the Landsberg-Manivel rational maps from projective spaces to isotropic Grassmannians.