David Farris (Indian Institute of Science/UC Berkeley) The embedded contact homology of circle bundles over Riemann surfaces

The abstract:

Embedded contact homology is a topological invariant of three-manifolds defined by choosing a contact structure on the manifold and studying pseudoholomorphic curves in the four-dimensional symplectization. We compute this invariant for circle bundles over Riemann surfaces (prequantization spaces).


David gave a brief introduction to ECH and then sketched the proof of his thesis result. He computed the ECH invariant for non-trivial circle bundles over surfaces by taking the prequantization contact form. Note that since ECH is a smooth invariant, one is free to take any contact structure on this.

The main result is that, with $Y$ denoting the circle bundle and $\Sigma$ the base,
\[
ECH(Y) = \oplus_{d \ge 1} Sym^d( H_*(\Sigma) )
\]
Furthermore, recall that ECH has a filtration by $H_1(Y)$, representing the homology class of the generators in question. We note that $H_1(Y) = \mathbb{Z}/|e| \oplus \mathbb{Z}^{2g+2}$, where $g$ is the genus of $\Sigma$. The torsion factor is the only relevant one. Then,
\[
ECH(Y, l) = ???see notes??
\]

The key difficulty is that for an $S^1$-invariant complex structure, one can show that the index $1$ moduli spaces are empty... however, these $S^1$-invariant acs are not generic. David solves this problem by first showing the moduli spaces are empty for a domain dependent perturbation. He is able to carry out the Cieliebak-Mohnke/Fabert methods, even though he has genus, essentially because the formation of a ghost bubble with genus is codimension at least 2. (This uses the fact we are in dimension 4 quite crucially). Then, the next tricky point is to show that the count of embedded curves remains $0$ as he deforms the $S^1$-invariant domain-dependent almost complex structure to a generic domain-independent complex structure. This relies on a theorem of Hutchings and Taubes about bifurcations, since David can rule out connectors in the compactness argument.

In thinking about this, I wonder if there is a way of making sense of this as some statement about moduli spaces with an $S^1$ action. This is an idea Jian He explained to me, and is the heart of the geometric picture behind his work on descendants. The general principle (stolen from localization in algebraic geometry) is that if you have a non-generic $J$ and a moduli space of which a connected component has a free $S^1$ action, then after perturbation, this connected component still can be arranged to have an $S^1$ action, in particular then, not contribute to the count of isolated solutions. If one could understand this principle in dimension 4 for embedded curves, it might then be possible to extend David's result to the case of a trivial circle bundle. (Ironically, that case seems harder.)

Welcome

This is a blog I am starting to keep track of the notes I take during talks. Usually, I won't include a long discussion, but will instead mention an interesting idea or two.

The latex rendering is done by mathjax. I followed the instructions I found at irreducible representations.

Masahiro Futaki (Universiteit Antwerpen) On the Sebastiani-Thom theorem for directed Fukaya categories

The abstract:

The directed Fukaya category is an A-model counterpart for the derived category of toric Fano B-model under the Fano/Landau-Ginzburg mirror correspondence. Auroux-Katzarkov-Orlov conjectured that the directed Fukaya category of the direct sum of two potentials splits as tensor product (up to derived equivalence) and pointed out that this is the case for the mirror of $\mathbb{P}^1 \times \mathbb{P}^1$. We show that the conjecture holds if one of the potentials is of complex dimension 1, partially generalizing Seidel's suspension theorem for directed Fukaya categories.


If I understood the talk correctly, part of the challenge is to make sense of the tensor product for $A_\infty$ categories. Alas, most of the aspects of this I understand superficially at best.

This conjecture of AKO is related to some things I do understand, however. Oancea, in his thesis, proved a Künneth formula for the symplectic homology of the product of two Liouville manifolds. By the appendix to Bourgeois-Ekholm-Eliashberg's surgery exact sequence written by Ganatra and Maydanski (and conjectured by Seidel), symplectic homology of the total space of an exact Lefschetz fibration is the Hochschild [Co?]homology of the derived Fukaya category. In this sense, the AKO conjecture is saying that Oancea's theorem actually lives at the level of the Fukaya category.

I wonder if it would be possible to prove this theorem by carefully using the BEE surgery construction.

Clément Hyvrier (Uppsala) Weinstein conjecture in Hamiltonian fibrations

\[ \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Symp}{\operatorname{Symp}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\e}{\operatorname{e}} \newcommand{\pt}{\mathrm{pt}} \newcommand{\tensor}{\otimes} \] Defined Weinstein conjecture. Interested in contact-type hypersurfaces in $(M,\omega)$.
Examples in which known: Viterbo for $(\R^{2n}, \omega_0)$. Hofer-Viterbo 1992 for symplectically rationally connected manifolds under semi-positivity assumption. (Through any 2 points of $M$, there is a holomorphic sphere.)
The actual definition of symplectically rationally connected: exists a non-vanishing GW invariant $GW_0(\pt, \pt, A, B, \dots)$.
Liu-Tian result with separating hypersurface and classes $A_0$, $A_1$ with support on either side.
Lu: If $<\pt, \beta_1, \dots, \beta_n>^{M, A}_{0, n+1} \ne 0$ then result holds. (symplectically uniruled)
proof sketch: This non-vanishing implies $\star = <\pt, PD(\omega), \beta_1, \dots, \beta_n>^{M, A}_{0, n+2} \ne 0$. Now the trick is to find $\hat \omega = \omega - d(\beta \lambda)$ where $\lambda$ is a local primitive of $\omega$ near $\Sigma$ and $\beta$ is a cut-off function. Then, this has support away from $\Sigma$. Now obtain \[ \star = < \pt, \gamma_+, \beta> + <\pt, \gamma_-, \beta> \] since $\star \ne 0$, one of these terms $\ne 0$. The result now follows by Liu-Tian.
Corollary: shWC true for $M \times N$ with product form and $M$ symp uniruled.
Question: can we generalize to Hamiltonian fibrations? \[ (F, \omega) \hookrightarrow P \to (B, \omega_B) \] s.t.

• $P|_{B_1}$ is sympl trivial, where $B_1$ is the 1-skeleton
• exists a connection $P$ such that the holonomy is hamiltonian i.e.~exists $\tau \in \Omega^2(P)$ such that $\tau|_{F} = \omega$ and $d\tau = 0$ with a normalization condition.

Then, $(P, \epsilon \tau + \pi^* \omega_B)$ is symplectic for $\epsilon > 0$ small enough.
THEOREM: Suppose

1. $(P, \pi)$ is cohomologically split over $\Q$, i.e.~ \\[ H^*(P) \sim H^*(B) \otimes H^*(F) \\] as vector spaces. And $i_* : H^*(P) \to H^*(F)$ is surjective.
2. $(B, \omega_B)$ is symplectically uniruled for some primitive class $\sigma_B \in H_2(B)$ for spherical class $B$.
3. $(F, \omega)$ satisfies semi-positivity relative to $P$.

then, $(P, \omega_P)$ is symplectically uniruled and thus shWC applies there.
THEOREM: We can drop condition (1) by asking that $(B, \omega_B)$ is symp rationally connected.
PROOF:
Case 1: the fibre is symplectically uniruled. Result done by Ruan, Tian and Liu. \\ Basic idea: \[ 0 \ne < \pt, \beta>^{F, \sigma}_{0, n} = <\pt, i_* \beta>_{0,n}^{P, i_* \sigma} \] where $\sigma \in H_2(F)$.
Case 2: Suppose $(F, \omega)$ not symp uni. consider: $C$ the image of a curve counted in $<\pt, \beta_B>_{0,n}^{\beta, \sigma_B}$ Now restrict $P|_C$. Gives a fibration over $S^2$. this is then described by a loop in $Ham(F, \omega)$. Since $(F, \omega)$ NOT symp uni, for any symplectic loop, exists $\sigma' \in H_2^{sph}(P|_C)$ such that $<\pt_{0,1}^{P|_C, \sigma'} \ne 0$, where $\sigma'$ is a section class. (i.e.~projects to the fundamental class of the base) Then, $=<\pt, [F], \dots, [F]>_{0, n+1}^{P|_C, \sigma'}$.
THEOREM: \[ < \pt, \pi^{-1} \beta_B >_{0,n}^{P, i_* \sigma'} = < \pt, \beta_B>_{0, n}^{B, \sigma_B} < \pt, [F], \dots, [F]>_{0,n}^{P|_C, \sigma'} \]
Remarkable fact: the Seidel element, when we are in a NOT uniruled case, looks like $ S(\gamma) = [F] \tensor \lambda + x $.