Workshop on Algebraic Geometry, Complex Dynamics and their Interaction
(4 - 7 Jan 2011)

~ Abstracts ~


The Noether function arising from the geography
Meng Chen, Fudan University, China

We report some new advances on the geography in dimension 3. This is a joint work with Jungkai A. Chen.

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Kodaira dimension of irregular varieties
Jungkai Alfred Chen, National Taiwan University, Taiwan

Let f: X -> Y be an albegraic fiber space with general fiber F. Iitaka's conjecture predicts that the Kodaira dimensions satisfy the inequality k(X) = k(F)+ k(Y).

In my recent joint work with Hacon, we show that this holds when the base variety Y is of maximal Albanese. In particular, if X has k(X)=0 and f = alb: X -> Alb(X) is the Albanese map, then the general fiber F has k(F)=0. This verifies second part of Ueno's conjecture K.

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Bifurcation currents and equidistribution in parameter space
Romain Dujardin, École Polytechnique, France

The analysis of bifurcations is a central issue in dynamical systems. In any parameter space of, say, holomorphic dynamical systems, there is a stability/bifurcation dichotomy, which parallels the Fatou-Julia decomposition of dynamical space. It turns out that in many situations, one can construct dynamically meaningful closed positive currents in parameter space, describing the asymptotic distribution of natural families of hypersurfaces. In the talk, I will review this construction in various settings. This includes joint work with Charles Favre and Bertrand Deroin.

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Foliations by Riemann surfaces
John Fornaess, University of Michigan, USA

If we have a real vector field in the plane, the integral curves foliate the plane. Similarly, if we have a complex vector field in C2 then the complex integral curves foliate space by Riemann surfaces. An example of a question one might ask, is whether these Riemann surfaces are dense in space. In the case of vector fields in the plane, this is impossible, but if we do it on a torus instead, it is easy. One method to do analysis on a Riemann surface is to use the current of integration. For a foliation one needs to average suitably over the individual Riemann surfaces to get good currents. We will explore these currents in these lectures. The work covered is mostly joint work with Sibony and Wold.

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Kähler cones on non-Kähler surfaces
Akira Fujiki, Osaka University, Japan

In the study of period maps of the families of anti-self-dual bi-hermitian structures on hyperbolic Inoue surfaces constructed in a joint work with Pontecorvo we are led to formulate the notion of Kähler cones for any anti-self-dual metrics. Indeed, the notion actually depends only on the underlying locally conformally Kähler structures and the latter structures have rather recently turned out to exist on almost all the known compact non- Kähler surfaces. Therefore it would be worth developing a general theory of Kähler cones. In our talk we shall explain some basic results and examples related to the Kähler cones.

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Dynamics of Kaehler-Einstein metrics
Vincent Guedj, Université Aix-Marseille, France

Let X be a Kaehler-Einstein Fano manifold. Perelman has shown that the normalized Kaehler-Ricci flow converges towards a Kaehler-Einstein metric. In a joint project with R.Berman, S.Boucksom, P.Eyssidieux and A.Zeriahi, we establish a discretized version of this result. This boils down to study the dynamics of the inverse Ricci operator. Our approach also applies to a singular setting.

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Some results about the fundamental groups of hyperelliptic fibrations and applications
Rajendra V. Gurjar, Tata Institute of Fundamental Research, India

We will discuss the proof of the following result.
Theorem. Let X be a smooth projective surface with a hyperelliptic fibration f:X\to C such that all the fibers of f are irreducible. If f is not isotrivial then the image of the fundamental group of a general fiber of f in the fundamental group of X is finite. As an application we show that any covering space of X is holomorphically convex. This is a strong form of the Shafarevich Conjecture in this situation. We will also prove similar results for open subsets of X and similarly for arbitrary (non-isotrivial) elliptic fibrations.

This is a joint work with Shameek Paul and B.P. Purnaprajna.

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Injectivity radius and gonality of a compact Riemann surface
Jun-Muk Hwang, Korea Institute for Advanced Study, Korea

A compact Riemann surface of genus at least 2 is a hyperbolic manifold and at the same time an algebraic curve. We discuss how these two different geometric aspects interact via the concepts of injectivity radius and gonality. The key ingredient is a certain volume inequality in a tubular neighborhood of the diagonal. This is a joint work with Wing-Keung To.

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Pluripotential theory in a non-archimedean setting
Mattias Jonsson, University of Michigan, USA

Pluripotential theory is the study of plurisubharmonic functions and plays an important role in complex algebraic geometry and dynamics. For various reasons, it is of interest to develop a similar theory over non-archimedean fields. I will discuss some basic results in this direction. This is joint work with Sebastien Boucksom and Charles Favre.

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A remark on the abundance conjecture
Yujiro Kawamata, University of Tokyo, Japan

We prove that the existence conjecture of minimal models in dimension n and the abundance conjecture in dimension less than n as well as the non-vanishing conjecture in dimension n imply the abundance conjecture in dimension n.

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Algebraic Montgomery-Yang problem on homology projective planes
JongHae Keum, Korea Institute for Advanced Study, Korea

Montgomery-Yang problem predicts that every pseudofree differentiable circle action on the 5-dimensional sphere has at most 3 non-free orbits. Using a certain one-to-one correspondence, Janos Kollar formulated the algebraic version of the Montgomery-Yang problem: every projective surface S with quotient singularities having the second Betti number 1 has at most 3 singular points if its smooth locus S^0 is simply connected. I will report recent progress on this problem. This is a joint work with DongSeon Hwang.

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Extremal Gromov-Witten invariants of the Hilbert scheme of points on surfaces
Wei-Ping Li, The Hong Kong University of Science & Technology, Hong Kong

Hilbert schemes of points on a surface are crepant resolutions of the symmetric product of the surface. There is only one extremal ray. The Gromov-Witten invariants of the Hilbert scheme with respect to this ray are computed. The result is equivalent to the computation of the quantum first Chern class operator of the tautological vector bundle on the Hilbert scheme. Thus we obtained the explicit formula for the quantum first Chern class operator in terms of Nakajima operators. The techniques include the reduction method developed by Kiem-Li and a result of Okounkov and Pandharipande. This is a joint work with Jun Li.

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Entire holomorphic curves and the quasi-Albanese map
Steven Lu, Université du Québec à Montréal, Canada

I will present a solution of the strong Lang conjecture on entire holomorphic curves for varieties with maximal quasi-Albanese dimension and in particular for surfaces of log irregularity two or more. It has interesting consequences for the structure of the quasi-Albanese map in general in terms of special varieties. This partly a report of a joint work with Joerg Winkelmann.

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Birational Arakelov geometry
Atsushi Moriwaki, Kyoto University, Japan

Roughly speaking, studies of birational geometry are nothing more than analyses of big divisors. Birational Arakelov geometry is its arithmetic analogue. In this talk, I would like to explain the recent developments of birational Arakelov geometry due to Chen, Yuan and myself. Especially, I will discuss the following topics:

1) arithmetic R-divisor
(2) arithmetic volume function
(3) continuity of arithmetic volume function and it's application
(4) arithmetic Fujita's approximation theorem
(5) Zariski decomposition on arithmetic surfaces
(6) Dirichlet's unit theorem on arithmetic varieties

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Slodowy slices and universal Poisson deformations
Yoshinori Namikawa, Kyoto University, Japan

Brieskorn and Slodowy constructed the semi-universal deformation of an ADE surface singularity by using a slice (Slodowy slice) to a subregular nilpotent element of the corresponding Lie algebra. We will generalise this theorem to an arbitrary Slodowy slice. A Poisson deformation will play a crucial role in this generalisation. This is a joint work with M. Lehn and C. Sorger.

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On Enriques manifolds
Keiji Oguiso, Osaka University, Japan

After introducing the notion of Enriques manifold, I would like to give several concrete examples (and counter examples). If time would be allowed, I also would like to present their explicit complex dynamical and/or birational geometrical aspects in examples. This is a joint work with Stefan Schroeer.

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Stability conditions and Bogomolov-Gieseker type inequalities
Yukinobu Toda, The Unviversity of Tokyo, Japan

In this talk, I construct some interesting t-structures on the derived category of coherent sheaves on projective threefolds, which conjecturally give Bridgeland?s stability conditions near the large volume limit. Our conjecture is reduced to showing a Bogomolov-Gieseker type inequality for certain objects in the derived category, and I show another version of it without the third chern character. This is a joint work with A. Bayer and E. Macri.

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Extensions of multiply twisted pluri-canonical forms
Chin-Lung Wang, National Taiwan University, Taiwan

We study the problem of extending "multiply twisted" pluri-canonical forms from smooth divisors in a projective manifold. Our result is a multiple version of extension theorem of Ohsawa-Takegoshi type for pseudo-effective line bundles. This is a joint work with C.-Y. Chi and S.-S. Wang.

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