International Conference in Geometry

December 19 - 22, 2006

Room 502A, Academic Building No. 1, The Chinese University of Hong Kong

 Speakers: Chang, Shu-Cheng (Tsing Hua University, Taiwan) Chen, Jiun Cheng (Tsing Hua University, Taiwan) Hassett, Brendan (Rice University, USA) Li, Peter (University of California, Irvine, USA) Li, Weiping (Hong Kong University of Science and Technology) Sheng, Mao (Johannes Gutenberg Universitat Mainz, Germany) Wang, Chin-Lung (National Central University, Taiwan) Wang, Jiaping (University of Minnesota, USA) Wu, Siye (University of Hong Kong) Zhang, De-Qi (National University of Singapore) Zhang, Qi S. (UC Riverside, USA) Zhou, Jian (Tsing Hua University, China) Zhu, Xi-Ping (Sun Yat-Sen University, China)

Information:

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Program:

 Titles: Characteristic subvarieties and disproof of modularity of moduli space of certain Calabi-Yau manifolds Sheng, Mao Characterizing projective spaces Chen, Jiun Cheng Filtered geometry of formal path spaces Zhou, Jian Geometrical approach to dynamics of automorphisms on algebraic manifolds Zhang, De-Qi Incidence Hilbert schemes and infinite dimensional algebras Li, Weiping Invariance of big quantum ring under simple flops Wang, Chin-Lung Number of minimal graphs with disjoint support Wang, jiaping Projective flatness in the geometric quantisation of bosons and fermions Wu, Siye Sharp estimate on the spectrum of a complete Kahler manifold Li, Peter Some new estimates for the conjugate heat equation under Ricci flow Zhang, Qi S. Sections of rationally connected fibrations through prescribed points Hassett, Brendan Subgradient estimate and Li-Yau Hamilton inequality in pseudohermitian 3-manifold Chang, Shu-Cheng The Existence of Type II Singularity to the Ricci Flow on Compact Manifolds Zhu, Xi Ping

## Characteristic subvarieties and disproof of modularity of moduli space of certain Calabi-Yau manifolds

### Sheng, Mao 16:00 – 17:00, 21st, December, 2006

This talk is a joint work with Ralf Gerkmann and Kang Zuo. In this work, we study the modularity property of the moduli space of Calabi-Yau manifolds as resolutions of double covers of P^n branched along 2n+2 hyperplanes in general positions. Different from n<=2 case, we disproved the modularity of the moduli space for n=3.

We found a new type of invariants defined over the moduli space pointwisely, which we call characteristic subvarieties. In the case of bounded symmetric domain, for certain Calabi-Yau like variation of Hodge structures, the characteristic subvarieties are identified back with the characteristic bundles of N.Mok. The possiblity of the identification was firstly observed by K.Zuo.
We were able to use the identification to disprove the modularity in the claimed case by comparing the characteristic subvarieties with the characteristic bundles at one point.

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## Characterizing projective spaces

### Chen, Jiun Cheng 17:05 – 18:05, 20th, December, 2006

I will present my recent work on characterizing projective spaces for possibly  singular varieties.   We use the space of twisted stable maps (into a Deligne-Mumford stack) and study possible degeneration types. I will also discuss applications in studying Mukai conjecture.

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## Filtered geometry of formal path spaces

### Zhou, Jian 11:15 – 12:15, 21st, December, 2006

We study some natural local coordinates on formal path spaces.

Their transformations indicate that the tangent spaces of formal path spaces have natural structures of filtered vector spaces. Furthermore, the spaces of differential forms and cohomology on formal path spaces have very rich algebraic structures related to vertex operator algebras and elliptic genera. This study is inspired by nonlinear supersymmetric sigma models.

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## Geometrical approach to dynamics of automorphisms on algebraic manifolds

### Zhang, De-Qi 16:00 – 17:00, 20th, December, 2006

We show that dynamics of automorphisms on projective complex manifolds are built up from those on only three kinds of manifolds: Complex Tori, Weak Calabi-Yau Manifolds and Rationally Connected Manifolds.

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## Incidence Hilbert schemes and infinite dimensional algebras

### Li, Weiping 14:30 – 15:30, 20th, December, 2006

Let S be a smooth projective surface. Using correspondences, we construct an infinite dimensional Lie algebra that acts on the direct sum H=\bigoplus_{m=0}^{+\infty}H^*(S^[m,m+1]) of the cohomology groups of the incidence Hilbert schemes S^[m,m+1]. The algebra is related to an extension of an infinite dimensional Heisenberg algebra. The space $H$ is a highest weight representation of this algebra. Our result provides a representation-theoretic interpretation of Cheah's generating function of Betti numbers of the incidence Hilbert schemes. As a consequence, an additive basis of H^*(S^[m,m+1]) is obtained. We also prove similar results for some Donaldson-Thomas moduli spaces of ideal sheaves on a three-fold X admitting a locally trivial fibration to S. This is the joint work with Zhenbo Qin.

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## Invariance of big quantum ring under simple flops

### Wang, Chin-Lung 10:00 – 11:00, 20th, December, 2006

While K-equivalent manifolds are known to share with the same Hodge numbers and complex elliptic genera, it is well-known that the homotopy types are in general different and the ordinary cohomology ring structure is not preserved even under the simplest type of K-equivalent maps, namely flops. It has however been conjectured that the quantum product should be preserved under K-equivalence in a certain sense.

In this talk I will explain some of my recent progress (joint work with Y.-P. Lee and H.-W. Lin) on this problem. For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincare pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kaehler moduli space. The main tools used are localizations and degenerations.

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## Number of minimal graphs with disjoint support

### Wang, Jiaping 10:00 – 11:00, 21st, December, 2006

We explain a joint result with Peter Li on the finiteness of the number of minimal graphs with disjoint Euclidean domains.

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## Projective flatness in the geometric quantisation of bosons and fermions

### Wu, Siye 14:30 – 15:30, 21st, December, 2006

Geometric quantization procedure requires choosing a real or complex polarisation. Quantum physics is independent of the choice if there exists a projectively flat connection on the vector bundle of Hilbert spaces over the space of polarizations. In this talk, I will begin with the example of symplectic vector spaces and study the relation with the Maslov index. Quantisation of fermionic systems will be studied and is related to Clifford algebra and spinors.

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## Sharp estimate on the spectrum of a complete Kahler manifold

### Li, Peter 11:05 – 12:05, 22nd, December, 2006

In this talk, we will discuss a sharp upper estimate for the greatest lower bound of the spectrum of the Laplacian on a complete Kahler manifold. The case when the estimate is realized will also be discussed.

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## Some new estimates for the conjugate heat equation under Ricci flow

### Zhang, Qi S. 16:00 – 17:00, 19th, December, 2006

We present a proof for an upper bound for the fundamental solution of the conjugate heat equation and a gradient estimate for all positive solutions of the same equation. This contrasts to the Perelman gradient estimate which does not work for all positive solutions.

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## Sections of rationally connected fibrations through prescribed points

### Hassett, Brendan 11:05 – 12:05, 20th, December, 2006

Let X-->B be a projective variety fibered over a smooth complex curve. `Weak approximation holds' if, given an
arbitrary collection of horizontal jet data, there exists a section s: B-->X with these jets. We prove this provided
the fibers are smooth and rationally connected. This builds on previous work of Koll'ar-Miyaoka-Mori and
Graber-Harris-Starr. We shall also discuss results at places of bad reduction in special cases like cubic surfaces.
(joint work with Yuri Tschinkel)

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## Subgradient estimate and Li-Yau Hamilton inequality in pseudohermitian 3-manifold

### Chang, Shu-Cheng 10:00 – 11:00, 22nd, December, 2006

In this talk I will report on the joint work with H.-L. Chiu and C.-T. Wu. First we get a subgradient estimate on a pseudohermitian 3-manifold which is served as the CR analogue of Yau.s gradient estimate. With its application, it is shown that the natural analogue of Liouville-type theorems holds for the sub-
Laplacian on a complete noncompact pseudohermitian 3-manifold. Second, we deform the contact form by the amount of the Tanaka-Webster curvature minus its mean value on a closed spherical CR 3-manifold. We show that if a contact form evolves with free torsion from initial data with positive Tanaka-Webster curvature, then we obtain a time-reversed Li-Yau-Hamilton inequality for the Tanaka-Webster
curvature. As a consequence, we get the global existence and asymptotic convergence of the Yamabe .ow on a closed spherical CR 3-manifold of zero torsion.

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## The Existence of Type II Singularity to the Ricci Flow on Compact Manifolds

### Zhu, Xi Ping 14:30 – 15:30, 19th, December, 2006

According to Hamilton, the singularities of the Ricci flow on compact manifolds can be divided into two types. All known existence results are forming Type I singularities. In this talk we will prove the existence of Type II singularities to the Ricci flow on S^n for each n > 2. This is a joint work with Huiling Gu.

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Organizers:

• Conan Leung (The Institute of Mathematical Science, CUHK, Hong Kong)
• S.T. Yau (Director, The Institute of Mathematical Science, CUHK, Hong Kong)

Enquiry:

For enquiry, please contact Prof. Conan Leung at ncleung@math.cuhk.edu.hk