Morse theory and Euler characteristic of sections of spherical varieties

A theorem due to D. Bernstein states that Euler characteristic of a hypersurface defined by a polynomial f in (C\{0})^n is equal (upto a sign) to n! times volume of the Newton polyhedron of f. This result is related to algebaric torus actions and toric varieties. In this thesis, I prove that one can generalize the above result to actions of reductive groups with spherical orbits. That is, if a reductive group acts linearly on a vector space such that generic orbits are spherical, one can compute the Euler characteristic of generic hyperplane sections of a generic orbit in terms of combinatorial data. Our main tool is Morse theory. We begin with developing a variant of classical Morse theory for algebraic submanifolds of R^n and linear functionals. This will become related to stratification theory of Thom and Whitney as well as Palais-Smale generalized Morse theory.Comment: 69 page

Note on cohomology rings of spherical varieties and volume polynomial

Let G be a complex reductive group and X a projective spherical G-variety. Moreover, assume that the subalgebra A of the cohomology ring H^*(X, R) generated by the Chern classes of line bundles has Poincare duality. We give a description of the subalgebra A in terms of the volume of polytopes. This generalizes the Khovanskii-Pukhlikov description of the cohomology ring of a smooth toric variety. In particular, we obtain a unified description for the cohomology rings of complete flag varieties and smooth toric varieties. As another example we get a description of the cohomology ring of the variety of complete conics. We also address the question of additivity of the moment and string polytopes and prove the additivity of the moment polytope for complete symmetric varieties.Comment: 15 pages. Presentation of paper improved, some minor errors fixed and some references as well as example of complete conics added. Final version, appeared in J. Lie Theor

SAGBI bases and Degeneration of Spherical Varieties to Toric Varieties

Let X \subset Proj(V) be a projective spherical G-variety, where V is a finite dimensional G-module and G = SP(2n, C). In this paper, we show that X can be deformed, by a flat deformation, to the toric variety corresponding to a convex polytope \Delta(X). The polytope \Delta(X) is the polytope fibred over the moment polytope of X with the Gelfand-Cetlin polytopes as fibres. We prove this by showing that if X is a horospherical variety, e.g. flag varieties and Grassmanians, the homogeneous coordinate ring of X can be embedded in a Laurent polynomial algebra and has a SAGBI basis with respect to a natural term order. Moreover, we show that the semi-group of initial terms, after a linear change of variables, is the semi-group of integral points in the cone over the polytope \Delta(X). The results of this paper are true for other classical groups, provided that a result of A. Okounkov on the representation theory of SP(2n,C) is shown to hold for other classical groups.Comment: 17 pages, LaTex, uses the package x

Fixed Points of Torus Action and Cohomology Ring of Toric Varieties

Let X be a smooth simplicial toric variety. Let Z be the set of T-fixed points of X. We construct a filtration for A(Z), the ring of complex-valued functions on Z, such that Gr A(Z) is isomorphic to the cohomology algebra of X. This is the explanation of the general results of Carrell and Liebermann on the cohomology of T-varities, in the case of toric varieties. We give an explicit isomorphism between Gr A(Z) and Brion's description of the polytope algebra.Comment: 15 pages, LaTex file, 2 .eps figures, 1 .eepic figur

On a notion of anticanonical class for families of convex polytopes

The purpose of this note is to give a generalization of the statement that the anticanonical class of a (smooth) projective toric variety is the sum of invariant prime divisors, corresponding to the rays in its fan (or facets in its polytope), to some other classes of varieties with algebraic group actions. To this end, we suggest an analogue of the notion of anticanonical class (of a compact complex manifold) for linear families of convex polytopes. This is inspired by the Serre duality for smooth projective varieties as well as the Ehrhart-Macdonald reciprocity for rational polytopes. The main examples we have in mind are: (1) The family of polytopes normal to a given fan (which corresponds to the case of toric varieties). (2) The family of Gelfand-Zetlin polytopes (which corresponds to the case of the flag variety). (3) The family of Newton-Okounkov polytopes for a (smooth) group compactification.Comment: 16 page

Integrable systems, toric degenerations and Okounkov bodies

Let X be a smooth complex projective variety of dimension n equipped with a very ample Hermitian line bundle L. In the first part of the paper, we show that if there exists a toric degeneration of X satisfying some natural hypotheses (which are satisfied in many settings), then there exists a completely integrable system on X in the sense of symplectic geometry. More precisely, we construct a collection of real-valued functions H_1, ... H_n on X which are continuous on all of X, smooth on an open dense subset U of X, and pairwise Poisson-commute on U. Moreover, we show that in many cases, we can construct the integrable system so that the functions H_1, ..., H_n generate a Hamiltonian torus action on U. In the second part, we show that the toric degenerations arising in the theory of Newton-Okounkov bodies satisfy all the hypotheses of the first part of the paper. In this case the image of the "moment map" \mu = (H_1, ..., H_n): X to R^n is precisely the Okounkov body \Delta = \Delta(R, v) associated to the homogeneous coordinate ring R of X, and an appropriate choice of a valuation v on R. Our main technical tools come from algebraic geometry, differential (Kaehler) geometry, and analysis. Specifically, we use the gradient-Hamiltonian vector field, and a subtle generalization of the famous Lojasiewicz gradient inequality for real-valued analytic functions. Since our construction is valid for a large class of projective varieties X, this manuscript provides a rich source of new examples of integrable systems. We discuss concrete examples, including elliptic curves, flag varieties of arbitrary connected complex reductive groups, spherical varieties, and weight varieties.Comment: 44 pages. Final version. Appeared in Inventiones Mathematica

Springer's Weyl group representation via localization

Let $G$ denote a reductive algebraic group over $\mathbb{C}$ and $x$ a nilpotent element of its Lie algebra $\mathfrak{g}$. The Springer variety $\mathcal{B}_x$ is the closed subvariety of the flag variety $\mathcal{B}$ of $G$ parameterizing the Borel subalgebras of $\mathfrak{g}$ containing $x$. It has the remarkable property that the Weyl group $W$ of $G$ admits a representation on the cohomology of $\mathcal{B}_x$ even though $W$ rarely acts on $\mathcal{B}_x$ itself. Well-known constructions of this action due to Springer et al use technical machinery from algebraic geometry. The purpose of this note is to describe an elementary approach that gives this action when $x$ is what we call parabolic-surjective. The idea is to use localization to construct an action of $W$ on the equivariant cohomology algebra $H_S^*(\mathcal{B}_x)$, where $S$ is a certain algebraic subtorus of $G$. This action descends to $H^*(\mathcal{B}_x)$ via the forgetful map and gives the desired representation. The parabolic-surjective case includes all nilpotents of type $A$ and, more generally, all nilpotents for which it is known that $W$ acts on $H_S^*(\mathcal{B}_x)$ for some torus $S$. Our result is deduced from a general theorem describing when a group action on the cohomology of the fixed point set of a torus action on a space lifts to the full cohomology algebra of the space.Comment: 6 pages, title changed and made shorter, the presentation of the paper totally revised, final version to appear in the Canadian Mathematical Bulleti

Complete intersections in spherical varieties

Let G be a complex reductive algebraic group. We study complete intersections in a spherical homogeneous space G/H defined by a generic collection of sections from G-invariant linear systems. Whenever nonempty, all such complete intersections are smooth varieties. We compute their arithmetic genus as well as some of their h^{p,0} numbers. The answers are given in terms of the moment polytopes and Newton-Okounkov polytopes associated to G-invariant linear systems. We also give a necessary and sufficient condition on a collection of linear systems so that the corresponding generic complete intersection is nonempty. This criterion applies to arbitrary quasi-projective varieties (i.e. not necessarily spherical homogeneous spaces). When the spherical homogeneous space under consideration is a complex torus (C^*)^n, our results specialize to well-known results from the Newton polyhedra theory and toric varieties.Comment: 36 page

Newton polytopes for horospherical spaces

A subgroup H of a reductive group G is horospherical if it contains a maximal unipotent subgroup. We describe the Grothendieck semigroup of invariant subspaces of regular functions on G/H as a semigroup of convex polytopes. From this we obtain a formula for the number of solutions of a generic system of equations on G/H in terms of mixed volume of polytopes. This generalizes Bernstein-Kushnirenko theorem from toric geometry.Comment: 17 page

On the equivariant cohomology of subvarieties of a B-regular variety

By a $B$-regular variety, we mean a smooth projective variety over $C$ admitting an algebraic action of the upper triangular Borel subgroup $B \subset SL_2(C)$ such that the unipotent radical in $B$ has a unique fixed point. A result of M. Brion and the first author describes the equivariant cohomology algebra (over $C$) of a $B$-regular variety $X$ as the coordinate ring of a remarkable affine curve in $X \times P^1$. The main result of this paper uses this fact to classify the $B$-invariant subvarieties $Y$ of a $B$-regular variety $X$ for which the restriction map $i_Y:H^*(X) \to H^*(Y)$ is surjective.Comment: 12 pages, LaTeX. To appear in Transformation Group