Copyright © 2000–2016 Matthias Franz
Torhom is a Maple package that provides functions to study the homology with closed support (also called "Borel-Moore homology") of real and complex toric varieties with rational, integer or prime field coefficients. Remember that for complete varieties homology with closed supports coincides with usual (singular) homology.
Toric varieties are in one-to-one correspondence with certain convex-geometric objects called fans. The Torhom package uses the Convex package to deal with cones and fans.
When you start Maple now, the package torhom is defined. It is convenient to load Convex first:
with(convex): |
with(torhom); |
There are plenty of ways to define cones and fans with the Convex package. Here is the absolute minimum one should know:
C1 := poshull([1, 3, -1/2], [0, 5, 1/3]); |
C2 := poshull(Vector([-2, 3/5, 1])); |
F := fan(C1, C2); |
The type MODZ appears in the result of integral homology computations, see below.
E2hom(F::{FAN0, CONE0}, F0::FAN0, rational)::Array(nonnegint) E2hom(F::{FAN0, CONE0}, F0::FAN0, integer)::Array(MODZ) E2hom(F::{FAN0, CONE0}, F0::FAN0, specfunc(prime, integer))::Array(nonnegint)
This functions computes the E^{2} term of a certain spectral sequence converging to the homology with closed support of the complex toric variety given by F. The spectral sequence arises from the filtration of the toric variety by orbit dimension. The second (optional) argument F0 must be a subfan of F; if present, homology relative to F0 is computed. If F0 is left out, then absolute homology is calculated. The third argument determines whether homology is computed with rational, integer or prime field coefficients. (Recall that prime field coefficients require Convex version 1.2.0 or higher.) If the third argument is omitted, rational coefficients are taken.
One knows that for rational coefficients the spectral sequence degenerates on the E^{2} level, hence that the result of E2hom is the homology of the toric variety (after taking the total grading instead of the bigrading — see the function total below). The same holds for integer coefficients provided that the toric variety is regular (smooth) or at most 3-dimensional; in particular, there is no extension problem. Regularity of fans can be tested with the Convex function isregular.
If the support of the fan is known to be a homology manifold (e.g., complete), certain diagonals of the E^{2} term vanish: if the fan is regular in codimension k, the all terms E2[p, q] with p-q > k are zero. (This is essentially due to Brion.) E2hom uses some heuristics to determine whether the support is a homology manifold.
Examples
F := projspace(3); E2 := E2hom(F, integer(5)); |
printarray(E2); |
homreal(F::{FAN0, CONE0}, F0::FAN0, rational)::Array(nonnegint) homreal(F::{FAN0, CONE0}, F0::FAN0, integer)::Array(MODZ) homreal(F::{FAN0, CONE0}, F0::FAN0, specfunc(prime, integer))::Array(nonnegint)
This function computes the homology with closed support of the real toric variety given by F. The meaning of the other arguments is the same as for E2hom.
Example
H := homreal(F, integer): printarray(H); |
H := homreal(F, integer(2)): printarray(H); |
printarray(E::Array)::array
This function shifts the indices of an array so that it is displayed conveniently by Maple's prettyprinter.
total(E::Array(nonnegint))::Array(nonnegint) total(E::Array(MODZ))::Array(MODZ)
This functions computes the total grading of the bigraded (i.e., two-dimensional) array returned by E2hom.
Example
H := total(E2hom(projspace(3))); |
printarray(H); |