Viterbo's theorem on cotangent bundles $M=T^*N$ tells you in particular that singular cohomology $H^*(M)$ gets embedded in $SH^*(M)$ via the $c^*$ map. Having a Weinstein manifold (or more generally Liouville manifold) $M$, are there any further examples when this occurs?

## 1 Answer

There is a Morse-Bott spectral sequence computing the symplectic cohomology of affine varieties which are complements of normal crossing divisors in smooth projective varieties. For simplicity, let's assume that $M=X\setminus D$ is such an affine variety, and $D$ is smooth, then the first page of the spectral sequence takes the form

$E_1^{p,q}=\left\{\begin{array}{ll}H^{q+n}(M;\mathbb{K}) & p=0; \\ H^{p+q+n-p\mu}(SD;\mathbb{K}) & p<0; \\ 0 & p>0,\end{array}\right.$

where $\mu\in2\mathbb{Z}$ is the Conley-Zehnder index and $SD$ is the circle bundle over $D$, which should be regarded as the ideal contact boundary $\partial M$.

As long as this spectral sequence degenerates at $E_1$, the PSS map $H^\ast(M;\mathbb{K})\rightarrow\mathit{SH}^\ast(M)$ is a ring inclusion. This is the case, for example, for the topological pairs $(X,D)$ considered by Ganatra-Pomerleano: https://arxiv.org/abs/1811.03609. If you are familiar with complex geometry, this amounts to saying that $X\setminus D$ is hyperbolic in some sense. For example, assuming Brody hyperbolicity would be enough. It is proved by Siu-Yeung in the mid 90s that $X\setminus D$ is Brody hyperbolic whenever $X$ is an abelian variety and $D$ is an ample divisor. This would give you plenty of concrete examples.