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 (2021). For a Weinstein manifold M, the existence of an exact Calabi-Yau structure on the wrapped Fukaya category W(M) imposes strong restrictions on its symplectic topology. Under the cyclic open -closed map constructed by Ganatra (2019), an exact Calabi-Yau structure on W(M) induces a class bQ in the degree one equivariant symplectic cohomology SH1S1(M). Any Weinstein manifold admitting a quasi -dilation in the sense of Seidel-Solomon [Geom. Funct. Anal. 22 (2012), 443-477] has an exact Calabi-Yau structure on W(M). We prove that there are many Weinstein manifolds whose wrapped Fukaya categories are exact Calabi-Yau despite the fact that there is no quasi -dilation in SH1(M); a typical example is given by the affine hypersurface {x3 + y3 + z3 + w3 = 1} c C4. 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.