辛幾何G-流形上的幾何拓樸相關問題

國立成功大學「發展國際一流大學及頂尖研究中心計畫」

  延攬優秀人才工作報告表

NCKU Project of Promoting Academic Excellence & Developing World Class Research Centers

Work Report Form for Distinguished Scholars

□續聘

■離職

Name of the Employee

酒井洋範

Male Female

聘 期 Period of Employment

from 2010 年(y) 4 月(m) 13 日(d) to 2010 年(y) 12 月(m) 31 日(d) 研究或教學或科技研發與

管理計畫名稱 The project title of research,

teaching, technology development and management

辛幾何 G-流形上的幾何拓樸

相關問題

計畫主持人

（申請單位主管）

Project Investigator (Head of Department/Center)

江孟蓉

補助延聘編號

Grant Number

HUA 99 - 3 - 2 - 013

一、

（由受聘人填寫）

Please summarize the entire research, teaching, or science and technology R&D and management work process (To be completed by the employee)

My work in the project is to reveal a relation between symplectic vortex invariants and Gromov-Witten invariants of closed symplectic orbifolds.

(Orbifold) Gromov-Witten invariants, quantum (orbifold) cohomology, and quantum differential equation are important ingredients in algebraic geometry, symplectic geometry and mirror symmetry [1].

On the other hand, the symplectic vortex invariant is an invariant which is defined by the integration on the solution space of the symplectic vortex equations. The invariants have many applications: Seiberg-Witten theory, Floer theory and Gromov-Witten theory [2]. In particular, for symplectic toric manifolds, the Gromov-Witten invariant coincides with the vortex invariant under a certain topological condition [3]. This fact gives the ring structure of quantum cohomology.

The agreement of Gromov-Witten invariants and vortex invariants is expected to extend to orbifolds.

However, it has been found that the original definition of symplectic vortex invariants fails to apply to symplectic orbifolds. I aim to define and develop symplectic vortex invariants suitably for symplectic orbifolds.

To achieve my aim, I studied the following subjects.

(1) The symplectic geometry of orbifolds.

The symplectic vortex invariants are regarded as invariants of a symplectic quotient. There are two important properties of symplectic quotients.

One is the combinatorial description. If the symplectic quotient is an effective orbifold, its symplectic structure is completely determined by the combinatorial data called a labelled polytope. But orbifold Gromov-Witten theory requires us to deal with non-effective orbifolds. Therefore the notion of labelled polytopes is not enough and the extended notion of labelled polytopes is needed.

Another is that any symplectic smooth quotient can be described as a toric variety. Borisov et al.

introduced the notion of stacky fans to construct effectively a wide class of toric Deligne-Mumford stacks over the category of schemes [4]. This theory is an extension of part of the theory of toric varieties and can be used to study of orbifold Gromov-Witten theory [5]. But there was no symplectic counterpart of the theory of stacky fans.

I established the notion of stacky polytopes as a symplectic counterpart of the theory of stacky fans [D].

This work has following important aspects.

(i) Studying the symplectic geometry of Deligne-Mumford stacks. There are several equivalent definitions of orbifolds. Lerman proposed that the theory of Deligne-Mumford stacks is the best terminology for the geometry of orbifolds [6] and Lerman and Malkin gave a detailed description of symplectic structures and Hamiltonian actions on Deligne-Mumford stacks [7]. Making use of their work, I described symplectic quotients in terms of symplectic Deligne-Mumford stacks in

details.

(ii) Compatibility with labelled polytopes. I gave the relation between stacky polytopes and labelled polytopes and described the classification theorem of symplectic toric effective orbifolds in terms of stacky polytopes.

(iii) The extension of the Delzant construction. The Delzant construction is a procedure for producing a symplectic toric manifold from a Delzant polytope. I gave the procedure for producing a symplectic toric Deligne-Mumford stack from a stacky polytope. This procedure is an extension of the Delzant construction. I also proved that the procedure is compatible with the procedure for constructing a toric Deligne-Mumford stack from a stacky fan.

I gave a talk about these results at NCTS (South) Geometry Seminar [B] and the international workshop held at National Taiwan University [A].

(2) The symplectic vortex equations.

In order to establish the orbifold version of the symplectic vortex invariants, the symplectic vortex equation must extend suitably to orbifolds. Making use of the theory of stacks, I obtained an extension of symplectic vortex equation. The most important aspect is that this extension is natural in terms of differentiable stacks after an extension of the notion of principal bundles over differentiable stacks.

(3) The theory of Frobenius manifolds.

Since it is very difficult to calculate explicitly orbifold Gromov-Witten invariants, we can not compare directly the symplectic vortex invariants with the orbifold Gromov-Witten invariants. For this comparison, a reconstruction theorem of Frobenius manifolds is expected to be applied.

I talked about Gromov-Witten invariants, quantum cohomology and the Dubrovin's theory of Frobenius manifolds at seminars held by Professor Chang-Shou Lin at National Taiwan University [C].

References:

[1] Kontsevich, Maxim. Homological algebra of mirror symmetry. Proceedings of the International Congress of Mathematicians, Vol. 1,2 (Zurich, 1994), 120-139, Birkhauser, Basel, 1995.

[2] Kai Cieliebak, A. Rita Gaio, Ignasi Mundet i Riera, and Dietmar A. Salamon. The symplectic

vortex equations and invariants of Hamiltonian group actions. J. Symplectic Geom., 1(3):543–645, 2002.

[3] Kai Cieliebak and Dietmar A. Salamon. Wall crossing for symplectic vortices and quantum cohomology. Math. Ann., 335(1):133–192, 2006.

[4] Lev A. Borisov, Linda Chen, and Gregory G. Smith. The orbifold Chow ring of toric Deligne-Mumford stacks. J. Amer. Math. Soc., 18(1):193–215 (electronic), 2005.

[5] Hiroshi Iritani. Real and integral structures in quantum cohomology I: toric orbifolds. Preprint, arXiv:math/0712.2204, 2007.

[6] Eugene Lerman. Orbifolds as stacks? Preprint, arXiv:0806.4160, 2008.

[7] Eugene Lerman and Anton Malkin. Hamiltonian group actions on symplectic Deligne-Mumford stacks and toric orbifolds. Preprint, arXiv:0908.0903, 2009.

[A] "Noneffective symplectic toric orbifolds and labelled polytopes", The 2nd TIMS-OCAMI Joint International Workshop on Differential Geometry and Geometric Analysis, National Taiwan University, March 2010.

[B] "Noneffective symplectic toric orbifolds", NCTS (South) Geometry Seminar, National Cheng Kung University, March 2010.

[C] Quantum Cohomology and Integrable Systems I-V, Taida Institute of Mathematical Sciences, April-June 2010.

[D] The symplectic Deligne-Mumford stack associated to a stacky polytope. Preprint, arXiv:1009.3547, 2010.

二、

由計畫主持人或單位主管填寫

Please evaluate the performance of research, teaching or science and technology R&D and management Work: (To be completed by Project Investigator or Head of Department/Center)

(1)是否達到延攬預期目標？

Has the expected goal of recruitment been achieved?

是

(2)研究或教學或科技研發與管理的方法、專業知識及進度如何？

What are the methods, professional knowledge, and progress of the research, teaching, or R&D and management work?

酒井洋範博士於量子拓樸(quantum cohomology)方面學有專長，並有 orbifolds(辛幾何 G-

流形之約化空間)之實際研究經驗。受聘期間致力於代數堆(stacks)與辛幾何之研究，以此

為工具將渦方程(vortex equations)推廣到 orbifolds，作為 orbifold 渦不變量定義之基礎。

(3)受延攬人之研究或教學或科技研發與管理成果對該計畫(或貴單位)助益如何？

How have the research, teaching, or R&D and management results of the employed person given benefit to the project (or your unit)?

受聘人於延攬期間的研究成果，提供了一個了解辛幾何 G-流形拓樸性質的方法，也與代數

幾何關係密切，與成大數學系與南區理論中心幾何組相輔相成，拓展了彼此的研究方向。

(4)受延攬人於補助期間對貴單位或國內相關學術科技領域助益如何？

How has the employed person, during his or her term of employment, benefited your unit or the relevant domestic academic field?

受聘人於延攬期間與幾何組成員積極交流，並受邀在台大數學中心主講系列演講，拓展了辛

幾何和量子拓樸整體的研究方向。

(5)具體工作績效或研究或教學或科技研發與管理成果：

Please describe the specific work performance, or the results of research, teaching, or R&D and management work:

受聘人於延攬期間完成一篇 preprint 並投稿至 SCI 期刊，另在台大數學中心主講系列演講。

(6)

是否續聘受聘人?

□續聘

■不續聘

※ 此報告表篇幅以三～四頁為原則。This report form should be limited to 3-4 pages in principle.

※ 此表格可上頂尖大學網頁/辦法、表格下載/綜合業務組下載。

This report form can be downloaded in http://www.ncku.edu.tw/~top/top_web/C/lawcom.HTM