Mathematical Field Notes

Symplectic/Contact Geometry VII at Les Diablerets, Day 1

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I’m currently in Switzerland at the seventh “Symplectic Geometry, Contact Geometry and Interactions” Workshop funded by CAST. This is a yearly conference which started at the same time I started my PhD so I have a great fondness for these workshops. This one is in the mountains, which makes me even fonder…

A mountain.

A mountain.

After three excellent talks today I decided to act as a “maths journalist” and summarise the main ideas from the talks in this blog. I may not be able to keep this up, as there’s six talks tomorrow and too much snow to enjoy. Today’s talks were:

Urs Frauenfelder: A \Gamma-structure on the Lagrangian Grassmannian. (Joint with Peter Albers)

See arXiv:1209.4505.

This was a great way to start the conference – an elementary talk about the topology of one of our favourite spaces, the Lagrangian Grassmannian. Of course, its cohomology has been known for a long time (it’s a homogeneous space U(n)/O(n) and there are spectral sequence techniques for computing such cohomology groups, see for example Toda-Mimura’s books on the topology of Lie groups) – in the case when n is odd the rational cohomology is an exterior algebra on odd -degree generators. But the beauty of this talk was a simple geometric explanation of why we should get an exterior algebra on odd generators!

The key definition of a \Gamma-structure is due to Hopf (1941). It generalises that of a Lie group multiplication m\colon G\times G\to G. The essential topological fact about Lie group multipication maps is that restricting to \{g\}\times G or G\times\{g\} (i.e. left- or right-multiplication by g) gives you degree 1 maps L_g,R_g\colon G\to G. Hopf defined a \Gamma-structure on an oriented, connected manifold M to be a map \Theta\colon M\times M\to M such that the restrictions _m\Theta:=\Theta|_{\{m\}\times M}\colon M\to M and \Theta_m:=\Theta|_{M\times\{m\}}\colon M\to M both have nonzero degree.

This structure gives us (via pullback) a coproduct \Delta\colon H^*(M;\mathbf{Q})\otimes H^*(M;\mathbf{Q})\to H^*(M;\mathbf{Q}) on the cohomology. This is enough to constrain the cohomology ring to being an exterior algebra on odd-dimensional generators (an amazing fact due to Hopf).

The \Gamma-structure on the Lagrangian Grassmannian is most evident when you identify a Lagrangian subspace with the unique orthogonal antisymplectic involution having it as fixed locus. Given two such involutions R,S you can form the “product” RSR and this is the \Gamma-structure (with degrees -1 and 2^{\frac{n-1}{2}}). Urs explained this carefully and clearly and I found it particularly helpful to contemplate the instructive example n=1 (when the Lagrangian Grassmannian is \mathbf{RP}^1).

Yochay Jerby: The symplectic topology of projective varieties with small dual. (Joint with Paul Biran)

See arXiv:1107.0174.

It always pleases me when someone proves a result in algebraic geometry using symplectic techniques. This talk is a beautiful example. The dual of projective space is the space of hyperplanes in projective space (which is isomorphic to projective space). The dual of a subvariety is the variety of hyperplanes containing the tangent spaces of the subvariety. This is generically a hypersurface, but in certain special cases it has higher codimension. For example, the dual of a hyperplane is a point! These are called varieties with small dual and Yochay was talking about smooth varieties X with small dual.

These have been much studied by algebraic geometers and turn out to be very special (for instance there are strong restrictions on their topology, including a 2-periodicity of the cohomology in some range of degrees). From a symplectic viewpoint they turn out to be special too: Biran and Jerby take a hyperplane section \Sigma and prove that the cohomology class of the Fubini-Study form is invertible, considered as an element of quantum cohomology of \Sigma. They do this by showing that it arises as the Seidel element of a loop in the Hamiltonian group. Loops in the Hamiltonian group can be used as clutching functions to describe Hamiltonian fibrations over the sphere, and the Hamiltonian fibration in this case comes very naturally from the projective geometry: you take a line in the dual projective space which is disjoint from the dual variety of X (possible precisely because X has small dual!) and use these hyperplanes to cut out a family of smooth hyperplane sections \Sigma parametrised by this line. The Seidel element associated to this fibration is then an invertible element in quantum cohomology defined using some moduli spaces of sections of the fibration and the main theorem is to compute this and show it is indeed the cohomology class of the symplectic form.

Having an invertible element in degree two means that the quantum cohomology is 2-periodic, but it is also graded by the minimal Chern number and using these two facts allows you to recover a refinement of the classical 2-periodicity theorem of the cohomology of varieties with small dual. Not only is the cohomology 2-periodic in some range but outside that range there is a periodicity of the quantum cohomology which translates into a periodicity of sums of Betti numbers which was not known (I think) via purely algebro-geometric techniques.

Alex Ritter: Floer theory for negative line bundles. (partly joint work with Ivan Smith)

See arXiv:1106.3975 and arXiv:1201.5880.

Faced with the monumental task of introducing (wrapped and unwrapped) Floer homology, symplectic homology and Fukaya categories as well as telling us about his theorem (all in the final hour before dinner), Alex rose to the challenge with a beautiful set of highly illustrated beamer slides.

The spaces Alex was talking about are the total spaces of line bundles over symplectic manifolds (mostly \mathbf{CP}^n) such that the first Chern class is a negative multiple of the cohomology class of the symplectic form. For instance, \mathcal{O}(-1) over \mathbf{CP}^1. These are noncompact symplectic manifolds with symplectic fibres and symplectic base and they are convex in the sense that a sequence of holomorphic curves cannot escape to infinity. The first theorem he proved was that the symplectic homology of such a space is a quotient of the quantum homology (symplectic homology is a Floer theory counting periodic orbits of a Hamiltonian which gets very big quite quickly in the noncompact end of the manifold, quantum homology just counts compact holomorphic spheres!). In particular you quotient by the kernel of a certain map: the quantum cup product with a high power of the first Chern class. When there are no spheres with positive symplectic area in the base (and hence in the total space) the quantum and classical cup products agree and hence a sufficiently large power of the first Chern class vanishes, to the whole quantum cohomology is in its kernel, which recovers an older result of Oancea (that assuming there are no spheres with positive symplectic area the symplectic homology vanishes). The idea of the proof was the following: symplectic homology is defined as a limit of Floer homologies for a sequence of Hamitonians. For a suitable choice of these Hamiltonians you can ensure that each of these Floer homologies is isomorphic to quantum cohomology (roughly speaking you rotate the fibre in such a way as to ensure that all closed orbits lie in the zero section) and the maps in the sequence are precisely multiplications by the first Chern class.

Not satisfied with this, Alex raised the stakes algebraically and introduced the “open-closed string map” (one of the more complicated, though increasingly central, aspects of the Fukaya/Floer story). This is a map from the Hochschild cohomology of the Fukaya category to the quantum homology. I think (hope) I’m right in saying the following. For a single Lagrangian L it takes a collection of cycles (“inputs” – the Hochschild cohomology having as its nth chain group the tensor product of n copies of the Floer chain group of L) to an ambient cycle. The ambient cycle is traced out by a marked point on a holomorphic disc with as many marked points as there are inputs where each point point is required to be mapped to the corresponding cycle. Mad. And then he raised the stakes yet more by introducing the analogous map from the wrapped Fukaya category to the symplectic homology.

Why? Well Abouzaid recently proved a criterion for when a Lagrangian (or collection of Lagrangians) split-generates the Fukaya category (or some part of it) by looking at the image of this open-closed string map. Ritter and Smith have adapted this to the monotone setting they need for these negative line bundles. Using this criterion (namely that the image should contain some invertible element) they prove that you only need a single Lagrangian to split-generate the wrapped Fukaya category of a negative line bundle over \mathbf{CP}^n (for suitably low Chern class of the line bundle). The Lagrangian in question is the circle bundle living over the Clifford torus (which, when taken with various flat connections, generates all the various parts of the Fukaya category). In particular, the wrapped category is generated by a compact Lagrangian, so this proves that all the potential infinite-dimensionality introduced by wrapped Floer cohomology is actually only finite-dimensional (in the same way that the symplectic cohomology reduced to a quotient of quantum cohomology).

At this point, Alex brought the discussion back down to earth by discussing the equator in the sphere (one of the most instructive Lagrangian submanifolds, well worth your contemplation). Then we went for tea.


Written by Jonny Evans

January 11, 2013 at 1:24 am

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