Quasitoric manifold explained
-dimensional
manifold is a quasitoric manifold if it admits a smooth, locally standard action of an
-dimensional torus, with
orbit space an
-dimensional
simple convex polytope.
Quasitoric manifolds were introduced in 1991 by M. Davis and T. Januszkiewicz,[1] who called them "toric manifolds". However, the term "quasitoric manifold" was eventually adopted to avoid confusion with the class of compact smooth toric varieties, which are known to algebraic geometers as toric manifolds.[2]
Quasitoric manifolds are studied in a variety of contexts in algebraic topology, such as complex cobordism theory, and the other oriented cohomology theories.[3]
Definitions
Denote the
-th subcircle of the
-torus
by
so that
. Then coordinate-wise multiplication of
on
is called the
standard representation.
Given open sets
in
and
in
, that are closed under the
action of
, a
-action on
is defined to be
locally isomorphic to the standard representation if
, for all
in
,
in
, where
is a
homeomorphism
, and
is an
automorphism of
.
Given a simple convex polytope
with
facets, a
-manifold
is a
quasitoric manifold over
if,
- the
-action is locally isomorphic to the standard representation,
that maps each
-dimensional orbit to a point in the interior of an
-dimensional
face of
, for
.
The definition implies that the fixed points of
under the
-action are mapped to the vertices of
by
, while points where the action is free project to the interior of the polytope.
The dicharacteristic function
A quasitoric manifold can be described in terms of a dicharacteristic function and an associated dicharacteristic matrix. In this setting it is useful to assume that the facets
of
are ordered so that the intersection
is a vertex
of
, called the
initial vertex.
, such that if
is a
codimension-
face of
, then
is a
monomorphism on restriction to the subtorus
in
.
The restriction of λ to the subtorus
corresponding to the initial vertex
is an isomorphism, and so
can be taken to be a basis for the
Lie algebra of
. The
epimorphism of Lie algebras associated to λ may be described as a linear transformation
, represented by the
dicharacteristic matrix
given by
\begin{bmatrix}
1&0&...&0&λ1,&...&λ1,\\
0&1&...&0&λ2,&...&λ2,\\
\vdots&\vdots&\ddots&\vdots&\vdots&\ddots&\vdots\\
0&0&...&1&λn,&...&λn,\end{bmatrix}.
The
th column of
is a primitive vector
in
, called the
facet vector. As each facet vector is primitive, whenever the facets
meet in a vertex, the corresponding columns
form a basis of
, with determinant equal to
. The isotropy subgroup associated to each facet
is described by
for some
in
.
In their original treatment of quasitoric manifolds, Davis and Januskiewicz introduced the notion of a characteristic function that mapped each facet of the polytope to a vector determining the isotropy subgroup of the facet, but this is only defined up to sign. In more recent studies of quasitoric manifolds, this ambiguity has been removed by the introduction of the dicharacteristic function and its insistence that each circle
be oriented, forcing a choice of sign for each vector
. The notion of the dicharacteristic function was originally introduced V. Buchstaber and N. Ray
[4] to enable the study of quasitoric manifolds in complex cobordism theory. This was further refined by introducing the ordering of the facets of the polytope to define the initial vertex, which eventually leads to the above neat representation of the dicharacteristic matrix
as
, where
is the
identity matrix and
is an
submatrix.
[5] Relation to the moment-angle complex
The kernel
of the dicharacteristic function acts freely on the moment angle complex
, and so defines a
principal
-bundle
over the resulting
quotient space
. This quotient space can be viewed as
where pairs
,
of
are identified if and only if
and
is in the image of
on restriction to the subtorus
that corresponds to the unique face
of
containing the point
, for some
.
It can be shown that any quasitoric manifold
over
is
equivariently diffeomorphic to a quasitoric manifold of the form of the quotient space above.
[6] Examples
-dimensional
complex projective space
is a quasitoric manifold over the
-
simplex
. If
is embedded in
so that the origin is the initial vertex, a dicharacteristic function can be chosen so that the associated dicharacteristic matrix is
\begin{bmatrix}
1&0&...&0&-1\\
0&1&...&0&-1\\
\vdots&\vdots&\ddots&\vdots&\vdots\\
0&0&...&1&-1
\end{bmatrix}.
The moment angle complex
is the
-sphere
, the kernel
is the diagonal subgroup
, so the quotient of
under the action of
is
.
[7] - The Bott manifolds that form the stages in a Bott tower are quasitoric manifolds over
-cubes. The
-cube
is embedded in
so that the origin is the initial vertex, and a dicharacteristic function is chosen so that the associated dicharacteristic matrix
has
given by
\begin{bmatrix}
1&0& … &0&0& … &0&0\\
-a(1,2)&1& … &0&0& … &0&0\\
\vdots&\vdots&&\vdots&\vdots&&\vdots&\vdots\\
-a(1,i)&-a(2,i)& … &-a(i-1,i)&1& … &0&0\\
\vdots&\vdots&&\vdots&\vdots&&\vdots&\vdots\\
-a(1,n)&-a(2,n)& … &-a(i-1,n)&-a(i,n)& … &-a(n-1,n)&1\end{bmatrix},
for integers
.
The moment angle complex
is a product of
copies of 3-sphere embedded in
, the kernel
is given by
\{(t1,t
t2,...,t
...
ti,...,
...
tn,
,...,
):ti\inT,1\leqi\leqn\}<T2n
,
so that the quotient of
under the action of
is the
-th stage of a Bott tower.
[8] The integer values
are the tensor powers of the line bundles whose product is used in the iterated sphere-bundle construction of the Bott tower.
[9] The cohomology ring of a quasitoric manifold
Canonical complex line bundles
over
given by
x K(l)Ci\longrightarrowM2n
,
can be associated with each facet
of
, for
, where
acts on
, by the restriction of
to the
-th subcircle of
embedded in
. These bundles are known as the
facial bundles associated to the quasitoric manifold. By the definition of
, the preimage of a facet
is a
-dimensional quasitoric
facial submanifold
over
, whose isotropy subgroup is the restriction of
on the subcircle
of
. Restriction of
to
gives the
normal 2-plane bundle of the embedding of
in
.
Let
in
denote the first
Chern class of
. The integral
cohomology ring
is generated by
, for
, subject to two sets of relations. The first are the relations generated by the
Stanley–Reisner ideal of
; linear relations determined by the dicharacterstic function comprise the second set:
xi=-λi,xn+1- … -λi,xm,for1\leqi\leqn
.
Therefore only
are required to generate
multiplicatively.
Comparison with toric manifolds
- Any projective toric manifold is a quasitoric manifold, and in some cases non-projective toric manifolds are also quasitoric manifolds.
can be constructed as a quasitoric manifold, but it is not a toric manifold.
[10] Notes and References
- M. Davis and T. Januskiewicz, 1991.
- V. Buchstaber and T. Panov, 2002.
- V. Buchstaber and N. Ray, 2008.
- V. Buchstaber and N. Ray, 2001.
- V. Buchstaber, T. Panov and N. Ray, 2007.
- M. Davis and T. Januskiewicz, 1991, Proposition 1.8.
- V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.11.
- V. Buchstaber, T. Panov and N. Ray, 2007, Example 3.12.
- Y. Civan and N. Ray, 2005.
- M. Masuda and D. Y. Suh 2007.