Geometric Structures Modeled on Smooth Projective Horospherical Varieties of Picard Number One

Abstract

Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric structures modeled on horospherical varieties. Using Cartan geometry, we prove that a geometric structure modeled on a smooth projective horospherical variety of Picard number one is locally equivalent to the standard geometric structure when the geometric structure is defined on a Fano manifold of Picard number one.

Characterizations of smooth projective horospherical varieties of Picard number one

“…We will take o as the base point of the quasi-projective variety X. Proposition 4.4 (Section 2, Proposition 48 and Propositon 49 of [14]). Let X be a smooth nonhomogeneous projective horospherical variety (L, α, β) of Picard number one.…”

“…Most of the arguments in [5] can also be applied to the case when X is of type (B m , α m−1 , α m ) or of type (F 4 , α 2 , α 3 ), which is done in Section 5. For the geometric structures associated to them, the second author suggested a way to construct a Cartan connection which solves the local equivalence problem by showing that the condition (C) in [17] is satisfied (Proposition 53 of [14]). Due to a gap in the proof of Theorem 17 of [14], instead, we use the prolongation methods developed in [4].…”

Standard Embeddings of Smooth Schubert Varieties in Rational Homogeneous Manifolds of Picard Number 1

“…Recently, Hong [5] showed that a smooth horospherical variety X of Picard number 1 can be embedded as a linear section into a rational homogenous manifold of Picard number 1 except when X is (B n , α n−1 , α n ) for n ≥ 7. For a description of their tangent space based on weights and roots, see Proposition 2.6 of Kim [18]. Example 2.3 (Odd symplectic Grassmannian (C n , α k , α k−1 )).…”

“…Let Z be a smooth Schubert variety of type (C 2 , α 2 , α 1 ). By Lemma 4.2 (2) and Proposition 25 of Kim [18], after proper shifting of the gradation on the tangent space of Z, we let g ′ −1 ⊕ g ′ −2 be the induced gradation on the tangent space of Z from X. Then…”

“…Let Z be a smooth Schubert variety of type (B 3 , α 2 , α 3 ). The gradation on the tangent space of Z described in Proposition 25 of Kim [18] could be embedded in the gradation on the tangent space…”