Hard Lefschetz theorem for simple polytopes
نویسندگان
چکیده
McMullen’s proof of the Hard Lefschetz Theorem for simple polytopes is studied, and a new proof of this theorem that uses conewise polynomial functions on a simplicial fan is provided.
منابع مشابه
Hard Lefschetz Theorem for Nonrational Polytopes
The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well defined even for nonrational polytopes when there is no variety associated to it. We prove the Hard Lefschetz theorem for the intersection cohomology of a gener...
متن کاملAn analog of the Hard Lefschetz theorem for convex polytopes simple in edges
Proofs of these relations may be found in [1, 2, 3]. If the polytope ∆ is not simple, then the relations above are not true. A polytope ∆ is said to be integral provided all its vertices belong to the integer lattice. With each integral convex polytope ∆ one associates the toric variety X = X(∆) (see [4, 5, 6, 7]). This is a projective complex algebraic variety, singular in general. It turns ou...
متن کاملEhrhart Theory for Lawrence Polytopes and Orbifold Cohomology of Hypertoric Varieties
We establish a connection between the orbifold cohomology of hypertoric varieties and the Ehrhart theory of Lawrence polytopes. More specifically, we show that the dimensions of the orbifold cohomology groups of a hypertoric variety are equal to the coefficients of the Ehrhart δ-polynomial of the associated Lawrence polytope. As a consequence, we deduce a formula for the Ehrhart δ-polynomial of...
متن کاملOn the Generalized Lower Bound Conjecture for Polytopes and Spheres
In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h0, h1, . . . , hd) satisfies h0 ≤ h1 ≤ · · · ≤ h⌊ d2 ⌋. Moreover, if hr−1 = hr for some r ≤ d2 then P can be triangulated without introducing simplices of dimension ≤ d− r. The first part of the conjecture was solved by Stanley ...
متن کاملMixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations
Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose severe restrictions on the cohomology algebra of a smooth compact Kähler manifold or on the intersection cohomology of a projective toric variety; they restrict the local monodromy of a polarized variation of Hodge structure; they impose conditions o...
متن کامل