Symplectic domination

TL;DR

This paper constructs a symplectic manifold of the same dimension as a given manifold, with a map of positive degree, using deep results from Riemannian geometry and symplectic topology.

Contribution

It introduces a novel construction of symplectic manifolds related to negatively curved manifolds and symplectic divisors, expanding methods in symplectic topology.

Findings

Constructs a symplectic manifold of the same dimension as the original manifold.

Establishes a map of strictly positive degree from the symplectic manifold to the original.

Utilizes deep theorems from Ontaneda and Donaldson to achieve the construction.

Abstract

Let M be a compact oriented even-dimensional manifold. This note constructs a compact symplectic manifold S of the same dimension and a map f from S to M of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold N of tightly pinched negative curvature which admits a map to M of degree equal to one; the second is a result of Donaldson on the existence of symplectic divisors. Given Ontaneda's negatively curved manifold N, the twistor space Z is symplectic. The manifold S is then a suitable multisection of the twistor space, found via Donaldson's theorem.

Equations

$$G\cdot (R,e^{i\theta })=(R{G}^{-1},\det (G)e^{i\theta })$$

Full text

This result says, in some sense, that there are “a lot” of symplectic manifolds. This fits with the philosophy behind a folklore conjecture in symplectic topology, stated as Conjecture 6.1 in the article [4] of Eliashberg. The conjecture asserts that if $X$ is a compact manifold of dimension $2n\geqslant 6$, which admits an almost complex structure and a cohomology class $\kappa\in H^{2}(X,\mathbb{R})$ with $\kappa^{n}\neq 0$, then $M$ carries a symplectic structure.

Theorem 1 follows rather quickly from two deep results (stated as Theorems 2 and 4 below). The first is a spectacular construction by Ontaneda of Riemannian manifolds with tightly pinched negative curvatures.

This is the main result of a lengthy preprint [9], which was subsequently broken up into a series of articles for publication [10, 11, 12, 13, 14, 15, 16]. The pinched manifolds constructed by Ontaneda are smoothings of singular negatively curved manifolds constructed by Charney and Davis using a procedure called strict hyperbolisation [2]. This in turn builds on the hyperbolisation of polyhedra by Gromov [7].

For our purposes, the important consequence of the curvature pinching is that the twistor space of $N$ carries a natural symplectic form. We recall that the twistor space $Z\to N$ of an oriented Riemannian manifold $N$ is the bundle of compatible almost complex structures on the tangent spaces. I.e. the fibre of $Z$ over $x\in N$ is the set of all linear orthogonal complex structures on $T_{x}N$ which induce the given orientation. The fibres are homogeneous spaces, identified with $F=\operatorname{SO}(2n)/\operatorname{U}(n)$. The symplectic form on $Z$ is provided by a construction due to Reznikov (which is, in fact, a special case of Weinstein’s “fat bundles” [18]).

Since this is central to the proof of Theorem 1, we explain briefly how the construction goes. The key to the existence of an integral closed 2-form is that the model fibre $F=\operatorname{SO}(2n)/\operatorname{U}(n)$ of twistor space is a homogeneous integral symplectic manifold. In other words, there is a principle $S^{1}$-bundle $P_{F}\to F$ with a connection $A_{F}$ whose curvature is a symplectic form on $F$; moreover $P_{F}$ carries an action of $\operatorname{SO}(2n)$ covering the action on $F$ and leaving $A_{F}$ invariant.

This can be seen via the theory of integral coadjoint orbits (see, for example, [8]), but it is also simple to describe it explicitly. Consider the action of $G\in\operatorname{U}(2n)$ on $\operatorname{SO}(2n)\times S^{1}$ by

[TABLE]

We denote the quotient by $P_{F}=\operatorname{SO}(2n)\times_{\operatorname{U}(n)}S^{1}$. The $S^{1}$-action on $\operatorname{SO}(2n)\times S^{1}$ given by multiplication on the second factor commutes with the diagonal action of $\operatorname{U}(n)$ and so descends to $P_{F}$ making it a principle $S^{1}$-bundle over $F$. Moreover, the $\operatorname{SO}(2n)$-action on $\operatorname{SO}(2n)\times S^{1}$ given by multiplication on the left of the first factor also commutes with the diagonal action of $\operatorname{U}(n)$, and so descends to an $\operatorname{SO}(2n)$-action on $P_{F}$, where it covers the $\operatorname{SO}(2n)$-action on $F$. Finally, to see the connection consider the derivative $\mathfrak{so}(2n)\to T_{p}P_{F}$ of the $\operatorname{SO}(2n)$-action at a point $p\in P_{F}$. It is transverse to the $S^{1}$-orbit through $p$, the image gives the horizontal distribution defining the $\operatorname{SO}(2n)$-invariant connection $A_{F}$.

We now return to the twistor space $Z\to N$ and carry out this construction on every fibre. The result is a principle $S^{1}$-bundle $P\to Z$ fitting together the fibrewise bundles $P_{F}\to F$. Moreover, the connection $A_{F}$ gives a fibrewise connection in $Z$. To promote this to a genuine connection in all of $P\to Z$ we must specify the horizontal distribution transverse to the fibres of $Z\to N$; but this is precisely what the Levi-Civita connection does. This gives a connection $A$ in $P\to Z$ whose curvature determines a closed integral 2-form $\omega$ which is symplectic on each fibre.

One can now ask for $\omega$ to be symplectic, which becomes a curvature inequality for the Riemannian metric on $N$. Reznikov observed that this inequality is satisfied by hyperbolic space and so, by openness, it is also satisfied by all negatively curved metrics which are sufficiently pinched. In the case $\dim N=4$, the article [5] gives the full curvature inequality explicitly.

The next step in the proof is to invoke another deep theorem, namely Donaldson’s result on symplectic hypersurfaces.

Suppose first that $\dim N=4$. In this case, the twistor space $Z\to N$ has fibres $S^{2}$. By Reznikov’s result we know that there is an integral symplectic form on $Z$ for which the twistor fibres are symplectic. Now let $S\subset Z$ be a Donaldson hypersurface, with $[S]=k[\omega]$ for $k>0$. The twistor projection restricts to a smooth map $f\colon S\to N$ and we claim the degree of this map is positive. To prove this write $[F]$ for the homology class of a fibre of $Z\to N$. The intersection number $[S]\cdot[F]=k\int_{F}\omega$ is positive since it is a positive multiple of the symplectic area of $F$. It follows that $f$ is surjective. Now Sard’s theorem implies the existence of a point $x\in N$ which is not a critical value of $f$. This means that $S$ meets the fibre $F_{x}$ over $x$ transversely. The local degree of $f$ at each point of $F_{x}\cap S$ is equal to the local intersection of $F_{x}$ and $S$ at that point, hence the degree of $f$ equals $[S]\cdot[F]$ which we have just seen is positive.

In higher dimensions the argument is similar. When $\dim N=2n$ the twistor space has dimension $n(n+1)$ and the fibre has dimension $n(n-1)$. We start as before with a Donaldson hypersurface $S_{1}\subset Z$, with $[S_{1}]$ Poincaré dual to $k_{1}[\omega]$. We apply Donaldson’s theorem again, this time to $(S_{1},\omega_{S_{1}})$, to obtain a symplectic submanifold $S_{2}\subset S_{1}\subset Z$, where $S_{2}$ has codimension 4 in $Z$ with $[S_{2}]$ Poincar dual to $k_{2}[\omega]^{2}$. We continue in this way, producing a chain $S_{d}\subset S_{d-1}\subset\cdots\subset S_{1}\subset Z$ of symplectic submanifolds where $d=n(n-1)/2$. Each $S_{j}$ is a symplectic submanifold of $Z$ of codimension $2j$ and so $S_{d}$ has complimentary dimension to a fibre of $Z\to N$. Moreover, $[S_{d}]$ is Poincar dual to $k[\omega]^{d}$ for some $k>0$. It follows that $[S_{d}]\cdot[F]=k\int_{F}\omega^{d}$ which is positive since it is a positive multiple of the symplectic volume of the fibre. From here the same argument as before shows that the twistor projection $f\colon S_{d}\to N$ has positive degree. ∎

We close with a remark, that the symplectic manifolds $(S,\omega)$ produced in the proof of Theorem 1 are of “general type” in the sense that $c_{1}(S)=-p[\omega]$ where $p>0$. This follows from adjunction and the fact, proved in [6], that when $dimN=2n$, the symplectic structures on the twistor space satisfy $c_{1}(Z)=(n-2)[\omega]$.

Acknowledgements

We would like to thank Igor Belegradek and Anton Petrunin for discussions. JF was supported by ERC consolidator grant 646649 “SymplecticEinstein”.