# The singular cohomology embeds into the symplectic cohomology

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?

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.