Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres

An exact Calabi-Yau structure, originally introduced by Keller, is a special kind of smooth Calabi-Yau structure in the sense of Kontsevich-Vlassopoulos. For a Weinstein manifold $M$, the existence of an exact Calabi-Yau structure on the wrapped Fukaya category $\mathcal{W}(M)$ imposes strong restrictions on its symplectic topology. Under the cyclic open-closed map constructed by Ganatra, an exact Calabi-Yau structure on $\mathcal{W}(M)$ induces a class $\tilde{b}$ in the degree one equivariant symplectic cohomology ${SH}_{S^1}^1(M)$. Any Weinstein manifold admitting a quasi-dilation in the sense of Seidel-Solomon has an exact Calabi-Yau structure on $\mathcal{W}(M)$. We prove that there are many Weinstein manifolds whose wrapped Fukaya categories are exact Calabi-Yau despite the fact the fact there is no quasi-dilation in ${SH}^1(M)$, a typical example is given by the affine hypersurface $\{x^3+y^3+z^3+w^3=1\}\subset {\mathbb{C}}^4$. As an application, we prove the homological essentiality of Lagrangian spheres in many odd-dimensional smooth affine varieties with exact Calabi-Yau wrapped Fukaya categories.