5,414 research outputs found
The Decomposition Theorem and the topology of algebraic maps
We give a motivated introduction to the theory of perverse sheaves,
culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and
Gabber. A goal of this survey is to show how the theory develops naturally from
classical constructions used in the study of topological properties of
algebraic varieties. While most proofs are omitted, we discuss several
approaches to the Decomposition Theorem, indicate some important applications
and examples.Comment: 117 pages. New title. Major structure changes. Final version of a
survey to appear in the Bulletin of the AM
Equivariant Intersection Cohomology of Toric Varieties
We investigate the equivariant intersection cohomology of a toric variety.
Considering the defining fan of the variety as a finite topological space with
the subfans being the open sets (that corresponds to the "toric" topology given
by the invariant open subsets), equivariant intersection cohomology provides a
sheaf (of graded modules over a sheaf of graded rings) on that "fan space". We
prove that this sheaf is a "minimal extension sheaf", i.e., that it satisfies
three relatively simple axioms which are known to characterize such a sheaf up
to isomorphism. In the verification of the second of these axioms, a key role
is played by "equivariantly formal" toric varieties, where equivariant and
"usual" (non-equivariant) intersection cohomology determine each other by
Kunneth type formulae. Minimal extension sheaves can be constructed in a purely
formal way and thus also exist for non-rational fans. As a consequence, we can
extend the notion of an equivariantly formal fan even to this general setup. In
this way, it will be possible to introduce "virtual" intersection cohomology
for equivariantly formal non-rational fans.Comment: 31 pages, AMS-Latex (all "private" macros included), to be published
in "Algebraic Geometry - Hirzebruch 70" (Proceedings of the conference at the
Banach Centre, Warszawa, May 1998), Contemporary Mathematics, AM
Remarks on the combinatorial intersection cohomology of fans
We review the theory of combinatorial intersection cohomology of fans
developed by Barthel-Brasselet-Fieseler-Kaup, Bressler-Lunts, and Karu. This
theory gives a substitute for the intersection cohomology of toric varieties
which has all the expected formal properties but makes sense even for
non-rational fans, which do not define a toric variety. As a result, a number
of interesting results on the toric and polynomials have been extended
from rational polytopes to general polytopes. We present explicit complexes
computing the combinatorial IH in degrees one and two; the degree two complex
gives the rigidity complex previously used by Kalai to study . We present
several new results which follow from these methods, as well as previously
unpublished proofs of Kalai that implies and
.Comment: 34 pages. Typos fixed; final version, to appear in Pure and Applied
Math Quarterl
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