Exact Calabi–Yau categories and odd-dimensional Lagrangian spheres

Abstract

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 \mathcal{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 \mathcal{W}(M) induces a class \tilde{b} in the degree one equivariant symplectic cohomology \mathrm{SH}_{S^1}^{1}(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 \mathcal{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 \mathrm{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.