In this paper, we obtain a complete classification of compact hyperbolic Coxeter five-dimensional polytopes with nine facets.
In this paper, we obtain a complete classification of compact hyperbolic Coxeter five-dimensional polytopes with nine facets.
Let (Gamma ) be a discrete subgroup of (text {PU}(1,n)). In this work, we look at the induced action of (Gamma ) on the projective space (mathbb {P}(wedge ^{k+1}mathbb {C}^{n+1})) by the Plücker embedding, where (wedge ^{k+1}) denotes the exterior power. We define a limit set for this action called the k-Chen-Greenberg limit set, which extends the classical definition of the Chen-Greenberg limit set (L(Gamma )), and we show several of its properties. We prove that its Kulkarni limit set is the union taken over all (pin L(Gamma )) of the projective subspace generated by all k-planes that contain p or are contained in (p^{perp }) via the Plücker embedding. We also prove a duality between both limit sets.
After proving that every Schubert variety in the full flag variety of a complex reductive group G is a general Hessenberg variety, we show that not all such Schubert varieties are adjoint Hessenberg varieties. In fact, in types A and C, we provide pattern avoidance criteria implying that the proportion of Schubert varieties that are adjoint Hessenberg varieties approaches zero as the rank of G increases. We show also that in type A, some Schubert varieties are not isomorphic to any adjoint Hessenberg variety.
A hypercomplex structure (I, J, K) on a manifold M is said to be (C^infty )-pure-and-full if the Dolbeault cohomology (H^{2,0}_{partial }(M,I)) is the direct sum of two natural subgroups called the (overline{J})-invariant and the (overline{J})-anti-invariant subgroups. We prove that a compact hypercomplex manifold that satisfies the quaternionic version of the (dd^c)-Lemma is (C^infty )-pure-and-full. Moreover, we study the dimensions of the (overline{J})-invariant and the (overline{J})-anti-invariant subgroups, together with their analogue in the Bott-Chern cohomology. For instance, in real dimension 8, we characterize the existence of hyperkähler with torsion metrics in terms of the dimension of the (overline{J})-invariant subgroup. We also study the existence of special hypercomplex structures on almost abelian solvmanifolds.
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