Action Of Maximal Torus Research Articles

Basic Cohomology of Canonical Holomorphic Foliations on Complex Moment-Angle Manifolds

Abstract We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold, or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg–Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.

Relative algebro-geometric stabilities of toric manifolds

In this paper we study the relative Chow and $K$-stability of toric manifolds. First, we give a criterion for relative $K$-stability and instability of toric Fano manifolds in the toric sense. The reduction of relative Chow stability on toric manifolds will be investigated using the Hibert-Mumford criterion in two ways. One is to consider the maximal torus action and its weight polytope. We obtain a reduction by the strategy of Ono [34], which fits into the relative GIT stability detected by Székelyhidi. The other way relies on $\mathbb{C}^*$-actions and Chow weights associated to toric degenerations following Donaldson and Ross-Thomas [13, 36]. In the end, we determine the relative $K$-stability of all toric Fano threefolds and present counter-examples which are relatively $K$-stable in the toric sense but which are asymptotically relatively Chow unstable.

Torus quotients of Richardson varieties in the Grassmannian

We study the GIT quotient of the minimal Schubert variety in the Grassmannian admitting semistable points for the action of maximal torus T, with respect to the T-linearized line bundle and show that this is smooth when When n = 7 and r = 3 we study the GIT quotients of all Richardson varieties in the minimal Schubert variety. This builds on work by Kumar [21], Kannan and Sardar [18], Kannan and Pattanayak [17], and Kannan et al. [16]. It is known that the GIT quotient of is projectively normal. We give a different combinatorial proof.

Geometry of compact complex manifolds with maximal torus action

We study the geometry of compact complex manifolds M equipped with a maximal action of a torus T = (S1)k. We present two equivalent constructions that allow one to build any such manifold on the basis of special combinatorial data given by a simplicial fan Σ and a complex subgroup H ⊂ Tℂ = (ℂ*)k. On every manifold M we define a canonical holomorphic foliation F and, under additional restrictions on the combinatorial data, construct a transverse Kahler form ωF. As an application of these constructions, we extend some results on the geometry of moment-angle manifolds to the case of manifolds M.

Integral homology of loop groups via Langlands dual groups

Let K K be a connected compact Lie group, and G G its complexification. The homology of the based loop group Ω K \Omega K with integer coefficients is naturally a Z \mathbb {Z} -Hopf algebra. After possibly inverting 2 2 or 3 3 , we identify H ∗ ( Ω K , Z ) H_*(\Omega K,\mathbb {Z}) with the Hopf algebra of algebraic functions on B e ∨ B^\vee _e , where B ∨ B^\vee is a Borel subgroup of the Langlands dual group scheme G ∨ G^\vee of G G and B e ∨ B^\vee _e is the centralizer in B ∨ B^\vee of a regular nilpotent element e ∈ Lie ⁡ B ∨ e\in \operatorname {Lie} B^\vee . We also give a similar interpretation for the equivariant homology of Ω K \Omega K under the maximal torus action.

Del Pezzo surfaces and representation theory

The connection between del Pezzo surfaces and root systems goes back to Coxeter and Du Val, and was given modern treatment by Manin in his seminal book Cubic forms. Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded into a certain projective homogeneous space of the semisimple group with the same root system, equivariantly with respect to the maximal torus action. Computational proofs of this conjecture based on the structure of the Cox ring have been given recently by Popov and Derenthal. We give a new proof of Batyrev’s conjecture using an inductive process, interpreting the blowing-up of a point on a del Pezzo surface in terms of representations of Lie algebras corresponding to Hermitian symmetric pairs.

MAXIMAL TORUS ACTIONS ON SOLVMANIFOLDS AND DOUBLE COSET SPACES

Any compact, connected Lie group which acts effectively on a closed aspherical manifold is a torus Tk with k ≤ rank of [Formula: see text], the center of π1 (M). When [Formula: see text], the torus action is called a maximal torus action. The authors have previously shown that many closed aspherical manifolds admit maximal torus actions. In this paper, a smooth maximal torus action is constructed on each solvmanifold. They also construct smooth maximal torus actions on some double coset spaces of general Lie groups as applications.

Vector fields on some class of complete symmetric varieties

In the previous papers [6], [7], we show that the set of an algebraic homogeneous space G/P fixed under the action of a maximal torus T can be canonically identified with the coset W1 = W/W1 of Weyl group W. We find a T invariant Zariski open set near each element w ∊ W1 and introduce a very nice local coordinate system such that we can express the maximal torus action explicitly. As a result, we become able to apply the study of J. B. Carrell and D. Lieberman [2], [3] to the space G/P and investigate the numerical properties of its characteristic classes and cycles.

Algebro-geometric semistability of polarized toric manifolds

Let $\Delta \subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_{\Delta}$ and a very ample $(\mathbb{C}×)^n$-equivariant line bundle $L_{\Delta}$ on $X_{\Delta}$ associated with $\Delta$. In the present paper, we give a necessary and sufficient condition for Chow semistability of $( X_{\Delta}, {L^i}_{\Delta})$ for a maximal torus action. We then see that asymptotic (relative) Chow semistability implies (relative) K-semistability for toric degenerations, which is proved by Ross and Thomas.