Abstract

Paolo Cascini: On the Chern numbers of a smooth threefold

Many birational invariants of a smooth projective three-fold, such as the number and the singularities of its minimal models, are related with the topology of the underlying manifold. Using these methods, I will discuss some recent progress towards a question by Hirzebruch on the Chern numbers of a smooth projective threefold.

Jianxun Hu: Uniruled symplectic orbifold

I will talk about the birational invariance of uniruled symplectic orbifolds. This is a joint work with Bohui Chen.

Joel Fine: The symplectic geometry of Einstein 4-manifolds

This is in parts joint work with Dmitri Panov and Kirill Krasnov. I will describe a new way of writing Einstein's equations in dimension 4 as a gauge theory. The new "independent variable" is an SO(3)-connection over a 4-manifold M, which serves as a potential for the Riemannian metric, analogous to the relationship between the electromagnetic potential and electromagnetic field. The connection also determines a symplectic structure on the associated S2-bundle over M. These symplectic manifolds are very special: they either have c1=0 ("Calabi Yau") or c1=[ω] ("Fano"). There are many examples of connections giving Calabi-Yau 6-manifolds. I will explain how this leads to infinitely many simply-connected non-K√§hler symplectic Calabi-Yau 6-manifolds as well as examples with arbitrary fundamental group. On the other hand, we conjecture that the only Fanos which arise this way are CP3 and the complete flag on C3. I will explain how proving this could be seen as a "gauge theoretic sphere theorem" and discuss its relation to the 4-dimensional classification of symplectic Fanos due to McDuff as well as a conjecture of Donaldson.

Gabriele La Nave: TBA

TBA

Mark McLean: Minimal Discrepancy of Isolated Singularities and Reeb Orbits

Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal discrepancy. We relate the minimal discrepancy with indices of certain Reeb orbits on our link. As a result we show that the standard contact 5 dimensional sphere has a unique Milnor filling up to normalization. This generalizes a Theorem by Mumford.

Jason Starr: Geometry of Spaces of Rational Curves on Fano Manifolds

I will survey several recent results due to Rob Findley, Lloyd Smith, etc., about the geometry of spaces of rational curves on complex Fano manifolds, particularly transversality results proving that Gromov-Witten invariants agree with "classical enumerative invariants". I will also discuss some fixed point theorems for finite group actions on Fano manifolds and "2-Fano manifolds". I will end with speculations for extending these results beyond the regime of complex algebraic geometry.

Zhiyu Tian: Birational invariance of symplectic rational connectedness in special cases

One of the outstanding questions about the symplectic geometry of rationally connected varieties is whether symplectic rational connectedness is a birational invariant. In this talk I will discuss some new observations regarding this question, in particular some new cases of birational invariance in dimension 3. This is joint work with Tom Graber and Y.P. Lee.

Claire Voisin: Curve classes on Calabi-Yau threefolds and rationally connected manifolds

Weiwei Wu: Finite subgroups of symplectic plane Cremona group

In 2006, Dolgachev-Iskovskikh have completely classified the finite subgroups of birational automorphisms on CP2, that is, the plane Cremona group, which is a classical problem in algebraic geometry. In the symplectic category, it is natural to define the symplectic plane Cremona group as the symplectomorphism groups of rational surfaces. Although the symplectomorphism group is vastly different from the algebraic Cremona group, e.g. it is infinite dimensional hence contains much more objects, its finite subgroups maintains similar rigidity as in the algebraic case. I will report on some recent progress on this project with Weimin Chen and Tian-Jun Li.

Weiyi Zhang: Symplectic blowing down in dimension six

Symplectic blow down is a fundamental operation in symplectic birational geometry. Two symplectic manifolds are birationally cobordant equivalent if they are related by a sequence of symplectic blow ups, symplectic blow downs and integral deformations. In this talk, we will explain when a symplectic divisor could be blown down. We will give a cohomological criterion which works particularly well in dimension six. This is based on a joint work with Tian-Jun Li and Yongbin Ruan.


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