#
Convex Polytopes and Toric Varieties: Gil Kalai and David Kazhdan

## Weekly seminar: Tuesdays 16-18 Shprinzak 114

Mailing address: Institute of Mathematics, Hebrew University,
Givat-Ram, Jerusalem 91904, Israel
Telephone numbers: Office (972)2-6584729, Home (972)2-6536301,
Fax (972)2-5630702.
Email addresses:
kalai@math.huji.ac.il,
my home page
##
Buzz words:
Convex Polytopes, Face lattices of polytopes,
face numbers, flag numbers, h-numbers, Integral points in polytopes,
toric varieties, homology, Poincare duality, Hard Lefshetz theorem,
Hodge-Riemman-Minkowski relation, intersection homology,
decomposition theorem, Sheaves,
Stanley-Reisner ring, preverse sheaves, polarity, mirror symmetry,
koszul duality, secondary polytopes.
Flag (cliques) complexes, Hopf's conjecture, CAT(0), the Charney-Davis
conjecture, cd-index.

##
Plan of lectures:

###
Lecture I (28/10/2003) Gil Kalai: Polytopes,
face-lattices of polytopes, face numbers,
Euler relation, h-numbers via linear objective functions.

###
Lecture II: (4/11/2003) Gil Kalai: h-numbers for simplicial spheres,
face rings (Stanley-Reisner rings), the Cohen-Macaulay property,
Hard Lefschetz for simplicial polytopes,
intersection homology based h-numbers.

###
On 11/11/2003 there will be no regular meeting of the seminar but
rather a lecture in Mathematics 110 by Enrico Arbarello on "cohomology of
moduli spaces of curves".

###
Lecture III (18/11/2003):
Eran Nevo : The notion of a sheaf. Sheaves defined on fans.
The structure sheaf of a fan.
The geometric construction of the IH-sheaf of modules.
The statement of (HL)

###
Lecture IV (25/11/2003):
Eran Nevo (cont.) : The geometric construction of the IH-sheaf of modules
for complete fans.
The statements of Duality, Hard-Lefschetz, and HRM-relation.

###
Lecture V (2/12/2003):
Eran Nevo (cont.) : The geometric construction of the IH-sheaf of modules
for complete fans.
The statements of Duality, Hard-Lefschetz, and HRM-relation.

###
Lecture VI (9/12/2003): Ilya Tyomkin: Toric Varieties, Algebraic varieties,
affine toric varieties, projective toric varieties

###
Lecture VII (16/12/2003): Ilya Tyomkin: Toric Varieties,
projective toric varieties, equivariant line bundles,

###
Lecture VIII (23/12/2003): Joseph Bernstein: Equivariant Cohomology
General definitions and smooth toric varieties

###
Lecture VIII (30/12/2003): Joseph Bernstein:
Equivariant [intersection] Cohomology - the non simplicial case

###
Lecture IX: (6/1/2004) Gil Kalai:
The cd-index, The Charney-Davis conjecture.

###
Lecture X: (13/1/2004) Gil Kalai:
Incidence algebras and Stanley's general combinatorial settings.
Relations between IH of a polytope and
its dual related to mirror symmetry and koszul duality. The upper
bound theorem. David Kazhdan: I taste of the
paper of Batyrev and Materov on Toric residues and mirror symmetry.

##
Problems and Excercises

Excercises and Problems I
Excercises and Problems II
Excercises and Problems III
##
Papers we want to learn:

###
"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 general polytope."

###
Karu's paper is based on a construction by
Barthel, Brasselet, Fieseler and Kaup
and by Bressler and Lunts and also on a proof of "poincare duality"
by Barthel, Brasselet, Fieseler and Kaup.

A recent paper with a simpler approach than in the original paper
which fits Karu's inductive argument is:

Combinatorial Duality and Intersection Product: A Direct Approach
Authors: Gottfried Barthel, Jean-Paul Brasselet, Karl-Heinz Fieseler,
Ludger Kaup.