Geometry of Regular Hessenberg Varieties

Abstract

Let g be a complex semisimple Lie algebra. For a regular element x in g and a Hessenberg space H ⊆ g, we consider a regular Hessenberg variety X(x, H) in the flag variety associated with g. We take a Hessenberg space so that X(x, H) is irreducible, and show that the higher cohomology groups of the structure sheaf of X(x, H) vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.

Fano and weak Fano Hessenberg varieties

“…Since L ξ h is ample by the assumption, Lemma 3.5 (1) now implies that all coefficients of ξ h with respect to the fundamental weights must be positive. Hence it follows that k…”

“…Since Hess(S, h) is a smooth projective variety which admits a torus action with finite fixed points [9], the higher cohomology groups of the structure sheaf vanish [8]. This means that there is a natural isomorphism Pic(Hess(S, h)) ∼ = H 2 (Hess(S, h); Z) so that algebraic line bundles L and L ′ over Hess(S, h) are isomorphic if and only if their first Chern classes coincide (see for instance [1,Corollary 5.3]).…”

“…Proof. Since the anti-canonical bundle of Hess(S, h) is ample by the assumption, Proposition 3.2 and Lemma 3.5 (1) imply that the coefficients of ξ h with respect to the fundamental weights ̟ i must be positive, that is,…”

“…In case (1), if s i (ξ h ) = ξ h in addition, then h(i + 1) = h(i). In case (2), if s i+1 (ξ h ) = ξ h in addition, then h(i + 2) = h(i + 1).…”

A formula for the cohomology and K-class of a regular Hessenberg variety

“…The third author was partially supported by Simons Collaboration Grant 359792. theory [IT16]. Recently, Abe, Fujita, and Zeng gave a formula in K-theory [AFZ15,Cor. 4.2].…”