49 research outputs found
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 denote a reductive algebraic group over and a
nilpotent element of its Lie algebra . The Springer variety
is the closed subvariety of the flag variety of
parameterizing the Borel subalgebras of containing . It
has the remarkable property that the Weyl group of admits a
representation on the cohomology of even though rarely acts
on 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
is what we call parabolic-surjective. The idea is to use localization to
construct an action of on the equivariant cohomology algebra
, where is a certain algebraic subtorus of . This
action descends to via the forgetful map and gives the
desired representation. The parabolic-surjective case includes all nilpotents
of type and, more generally, all nilpotents for which it is known that
acts on for some torus . 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
Convex bodies and multiplicities of ideals
We associate convex regions in R^n to m-primary graded sequences of
subspaces, in particular m-primary graded sequences of ideals, in a large class
of local algebras (including analytically irreducible local domains). These
convex regions encode information about Samuel multiplicities. This is in the
spirit of the theory of Grobner bases and Newton polyhedra on one hand, and the
theory of Newton-Okounkov bodies for linear systems on the other hand. We use
this to give a new proof, as well as a generalization of a Brunn-Minkowski
inequality for multiplicities due to Teissier and Rees-Sharp.Comment: Final version, 20 pages, some typos fixed and minor corrections done.
To appear in Proceedings of Steklov Institute of Mathematic