A criterion for multi-polytopes via Duistermaat-Heckman functions

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 multi-fan. It turns out that a multi-fan is a much more general notion than a complete non-singular fan. Shortly after that, a multi-fan 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 multi-fan 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)n-action having one unique dense orbit and other orbits of smaller dimensions. It is well known that there is a one-to-one 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)n-action on a toric variety there corresponds a cone of dimension equal to the codimension of the orbit. This new notion of a multi-fan 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 multi-fan the union of cones in a multi-fan may overlap several times. Moreover, it is an open and intriguing question whether or not there is a one-to-one correspondence between relevant toric varieties and multi-fans. At the moment, we just know that two dierent torus manifolds may correspond to the same multi-fan. Nonetheless, many important topological properties of a torus manifold can be detected by its associated multi-fan. Indeed, in [6] Hattori and Masuda provide several combinatorial invariants of a multi-fan 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 multi-polytope P = (;F) associated to a multi-fan = (;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 multi-polytope is simple, which means that the multi-fan = (;C; !) is complete and simplicial. A multi-polytope 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 Duistermaat-Heckman function, on V minus the ane hyperplanes when P is simple. It can be shown that the Duistermaat-Heckman function is locally con- stant, and Guillemin-Lerman-Sternberg 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 Duistermaat-Heckman function that we are mostly concerned with in this thesis is dened as follow. Denition 1.1. Let (n) denote the n-skeleton 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 Duistermaat-Heckman 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 multi-polytope to be an ordinary polytope in terms of the values of the Duistermaat-Heckman function associated with a multi-polytope. To be more precise, our main result is Theorem 1.2. A multi-polytope P = (;F) is an ordinary polytope if and only if the Duistermaat-Heckman 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 multi-polytope P, called the winding number. It turns out that the values of Duistermaat-Heckman 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 Duistermaat-Heckman 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 Duistermaat-Heckman 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 multi-fan and a multi-polytope, and then introduce certain related notions. The completeness of a multi-fan is one of the most important points in this chapter. The denition of the Duistermaat-Heckman function is given in Chapter 3. As brie y mentioned above, a multi-polytope is a pair P = (;F) of an n-dimensional com- plete multi-fan and a arrangement of hyperplanes F = fFig in H2(BT;R) with the same index set as the set of 1-dimensional cones in . Recall that a multi-polytope is called simple if its as- sociated multi-fan is simplicial. The Duistermaat-Heckmann function DHP associated with a simple multi-polytope P is a locally constant integer-valued 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. 5|하토리와 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 듀이스터매트-핵크만 함수를 정의하고 다중 폴리토프가 보통 폴리토프가 될 필요충분조건을 듀이스터매트-핵크만 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 듀이스터매트-핵크만 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 듀이스터매트-핵크만 함수 는 다중 폴리토프의 내부에서 1의 값을 갖고 외부에서는 0의 값을 가지며 그 역도 성립함을 증명하였다.