This is an announcement of Sheafhom, a system of programs I have been
developing for studying algebraic topology, especially intersection
homology.
INTRODUCTION
Sheafhom lets the user work with finite-dimensional vector spaces V
over the rational numbers, linear maps V --> W, and chain complexes
V_n --> ... --> V_1 --> V_0. It provides partially ordered sets,
which model simplicial complexes X. One can define sheaves on X with
sections in the category of chain complexes. For example, one can
write programs to construct the intersection homology [IH] sheaves on
X in any perversity, and to find the groups IH_*(X;Q). This includes
the ordinary cohomology H^*(X;Q) and homology H_*(X;Q) as special
cases.
To find these groups, Sheafhom uses spectral sequences. Given the E^1
page, it computes E^2, E^3, E^4,... in turn until it knows the
sequence has stopped. It works directly from the definition of a
spectral sequence, maintaining a finite-dimensional model of the
complex IC_* of all intersection chains on X. We do not need special
conditions like "the sequence collapses at E^2".
At present, Sheafhom does not work over finite fields, or with torsion
modules like Z/nZ. It computes IH_*(X;Q) rather than the full
IH_*(X;Z).
TORIC VARIETIES
Sheafhom currently can find IH_*(X) when X is a toric variety (over
the complex numbers, of any dimension, complete, normal, not
necessarily smooth or projective). This holds in any perversity, and
includes the ordinary H^*(X;Q) and H_*(X;Q).
Methods are already known for computing H^*(X), H_*(X) [see [T]], and
the middle-perversity Betti numbers. My approach is different because
I treat all perversities equally, and because I construct actual
cycles on X that generate the IH groups. In effect, I carry out
Deligne's construction of IH as in [GM], replacing the
infinite-dimensional complexes of injective sheaves with the
finite-dimensional model of IC_*. One can prove the model is "fine
enough" for this purpose.
People have felt that an algorithm like this should exist, but it has
needed care to write it down precisely. Within the next few months, I
will write a paper that translates Sheafhom's toric variety algorithms
into English and proves they are correct.
A goal for the future is to write an algorithm for the intersection
product on IH of toric varieties [including the cup product on H^*(X)
as a special case]. Stanley proved the necessity of McMullen's
conditions on a simplicial convex polytope by using the cup product on
the cohomology of the associated toric variety X, which is essentially
smooth. Understanding the product on IH would help us investigate the
generalization of Stanley's proof to all rational convex polytopes.
"NON-RATIONAL" TORIC VARIETIES AND POLYTOPES
While Sheafhom works over Q, its toric variety algorithms are also
well-defined over the real numbers. Thus the algorithms define "Betti
numbers" for the fictitious toric varieties coming from fans with
irrational slopes. Equivalently, they give Betti numbers for
non-rational convex polytopes (with fixed position relative to the
origin). We do not know if these Betti numbers are non-negative, or
whether they have properties like Poincare Duality and hard Lefschetz.
MORE INFORMATION; OBTAINING SHEAFHOM
The current version of Sheafhom can be downloaded over the Web from
http://www.math.okstate.edu/~mmcconn/shh.html
The Web page also contains tutorial files, an article about Sheafhom,
an encyclopedia of all of Sheafhom's data types and functions, and
previous versions of the code (including a fast implementation for
H_*(X;Q) using [T]).
At present, Sheafhom is written in Common Lisp. To run the code, you
will need a Common Lisp interpreter or compiler that supports CLOS
(the object-oriented extension of Lisp). You can use the toric
variety routines without knowing any Lisp--you can create a text file
that describes the fan data for X according to a simple recipe, and
just load the text file into Lisp. See the toric-variety tutorial for
how to do this. A version of Sheafhom without Lisp is a possibility
for the future.
PLEASE CONTACT ME
at ·······@math.okstate.edu or (405) 744-8220 if you have any
questions or comments. If you use Sheafhom, either briefly or for a
specific project, I'd love to hear about what you're doing and whether
Sheafhom proved to be useful to you.
--Mark McConnell
Assoc. Prof. of Mathematics
Oklahoma State University
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[GM] Mark Goresky and Robert MacPherson, "Intersection Homology II",
Inventiones Math. 72 (1983), pp. 77-130.
[T] Burt Totaro, "Chow groups, Chow cohomology, and linear varieties",
to appear in J. Algebraic Geom.