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.