A criterion for multipolytopes via DuistermaatHeckman functions
 Author(s)
 조미주
 Issued Date
 2016
 Abstract
 In the paper [8], Masuda rst introduced the notion of a uni
tary toric manifold which properly contained a compact non
singular toric variety, and associated with it a combinatorial
object, called a multifan. It turns out that a multifan is a
much more general notion than a complete nonsingular fan.
Shortly after that, a multifan as a purely combinatorial ob
ject which generalizes an ordinary fan in algebraic geometry has
been greatly developed by Hattori and Masuda in [6]. One typi
cal geometric realization of a multifan is a torus manifold, while
an ordinary fan is associated with a toric variety. Here a toric
variety means a normal complex algebraic variety of dimension
n with a (C)naction having one unique dense orbit and other
orbits of smaller dimensions. It is well known that there is a
onetoone correspondence between toric varieties and fans (see
[2], [10], and [1] for more details). Roughly speaking, the fan associated with a toric variety is a collection of cones in Rn with
apex at the origin, and to each orbit of a (C)naction on a
toric variety there corresponds a cone of dimension equal to the
codimension of the orbit.
This new notion of a multifan shares many important prop
erties with the ordinary fan. On the other hand, there is one
important peculiar feature, compared to the ordinary fan, that
in case of a multifan the union of cones in a multifan may
overlap several times. Moreover, it is an open and intriguing
question whether or not there is a onetoone correspondence
between relevant toric varieties and multifans. At the moment,
we just know that two dierent torus manifolds may correspond
to the same multifan. Nonetheless, many important topological
properties of a torus manifold can be detected by its associated
multifan. Indeed, in [6] Hattori and Masuda provide several
combinatorial invariants of a multifan which correspond to the
ordinary topological invariants of the associated torus manifold.
Associated to an ordinary fan, there is a notion of a convex
polytope. Analogously, there is a notion of a multipolytope
P = (;F) associated to a multifan = (;C; !) (refer to
Chapter 2 for more precise denitions and notations). Indeed,
let N be a lattice of rank n which is isomorphic to Zn, and let
M be the dual lattice Hom(N;Z). Let V := NR = N
Z R.
From now on, we also assume that a multipolytope is simple, which means that the multifan = (;C; !) is complete and
simplicial.
A multipolytope P = (;F) then denes an arrangement of
ane hyperplanes Fi (1 i d) in V , and one can associate
with P a function, called a DuistermaatHeckman function, on
V minus the ane hyperplanes when P is simple. It can be
shown that the DuistermaatHeckman function is locally con
stant, and GuilleminLermanSternberg formula ([4], [5]) tells
us that it agrees with the density function of a Duistermaat
Heckman measure, when P arises from a moment map. More
precisely, the DuistermaatHeckman function that we are mostly
concerned with in this thesis is dened as follow.
Denition 1.1. Let (n) denote the nskeleton of , and let I
denote the characteristic function dened over a suitably dened
convex cone associated with I 2 (n). We then dene a function
DHP on V n
Sd
i=1 Fi by
DHP :=
X
I2(n)
(1)I!(I)I ;
and call it the DuistermaatHeckman function associated with
P. We refer the reader to Chapter 3 for more details.
The primary aim of this paper is to provide a criterion for
a multipolytope to be an ordinary polytope in terms of the
values of the DuistermaatHeckman function associated with a
multipolytope. To be more precise, our main result is Theorem 1.2. A multipolytope P = (;F) is an ordinary
polytope if and only if the DuistermaatHeckman function DHP
dened on V n
Sd
i=1 Fi satises the following identity:
DHP(u) =
(
1; if u lies in the interior P of P;
0; otherwise.
There is another locally constant function dened on the com
plement of the hyperplanes fFig associated to a multipolytope
P, called the winding number. It turns out that the values
of DuistermaatHeckman function is exactly same as those of
the winding number (see [6]). Moreover, the winding num
ber also satises a wall crossing formula entirely similar to the
DuistermaatHeckman function. Hence Theorem 1.2 can be
stated in terms of the winding numbers instead of the Duistermaat
Heckman functions. In the forthcoming thesis [9], Moon will give
a direct proof of this fact without using the equivalence of the
values of DuistermaatHeckman functions and winding numbers.
Now we brie
y explain the contents of each chapter, as fol
lows. In Chapter 2, we give denitions of a multifan and a
multipolytope, and then introduce certain related notions. The
completeness of a multifan is one of the most important points
in this chapter. The denition of the DuistermaatHeckman
function is given in Chapter 3. As brie
y mentioned above, a
multipolytope is a pair P = (;F) of an ndimensional com
plete multifan and a arrangement of hyperplanes F = fFig in H2(BT;R) with the same index set as the set of 1dimensional
cones in . Recall that a multipolytope is called simple if its as
sociated multifan is simplicial. The DuistermaatHeckmann
function DHP associated with a simple multipolytope P is a
locally constant integervalued function with bounded support
dened on the complement of the hyperplanes fFig. The wall
crossing formula which describes the dierence of the values of
the function on adjacent components plays an important role in
the proof of our main Theorem 1.2. In addition, several inter
esting examples will be provided in Chapter 4. Finally, Chapter
5 is devoted to proving our main Theorem 1.2.
5하토리와 마수다에 의해 발견된 다중 팬은 토러스 다양체와 기하학적으로 깊은 관련을 가지고 있다. 다중 팬은 보통의 팬과 여러 가지 다른 성질을 가지고 있는 반면, 또한 비슷한 중요한 성질도 함께 공유하고 있다. 본 논문에서는 하토리와 마수다의 결과를 확장하여, 다중 팬과 다중 폴리토프에 대응하는 듀이스터매트핵크만 함수를 정의하고 다중 폴리토프가 보통 폴리토프가 될 필요충분조건을 듀이스터매트핵크만 함수의 값을 이용하여 찾았다. 좀 더 구체적으로, 를 차원 격자라 하고 를 이라 할 때, 의 쌍대공간 에 있는 아핀 초평면 의 합집합 의 여집합 를 정의역으로 갖는 듀이스터매트핵크만 함수 는
으로 정의된다. 여기서, 는 다중 팬 의 차원 골격을 나타내고 와 는 각각 에 대응하는 가중치와 특성함수를 나타낸다. 본 논문에서 다중 폴리토프 가 보통의 폴리토프이면 듀이스터매트핵크만 함수 는 다중 폴리토프의 내부에서 1의 값을 갖고 외부에서는 0의 값을 가지며 그 역도 성립함을 증명하였다.
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