Exact Calabi-Yau categories and odd-dimensional Lagrangian spheres. (English) Zbl 1536.53163
Summary: An exact Calabi-Yau structure, originally introduced by B. Keller [J. Reine Angew. Math. 654, 125–180 (2011; Zbl 1220.18012)], is a special kind of smooth Calabi-Yau structure in the sense of M. Kontsevich et al. [“Pre-Calabi-Yau algebras and topological quantum field theories”, Preprint, arXiv:2112.14667]. 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 S. Ganatra [Geom. Topol. 27, No. 9, 3461–3584 (2023; Zbl 07777465)], an exact Calabi-Yau structure on \(\mathcal{W}(M)\) induces a class \(\tilde{b}\) in the degree one equivariant symplectic cohomology \(\mathrm{SH}^1_{S^1} (M)\). Any Weinstein manifold admitting a quasi-dilation in the sense of P. Seidel and J. P. Solomon [Geom. Funct. Anal. 22, No. 2, 443–477 (2012; Zbl 1250.53078)] 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.
MSC:
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 53D12 | Lagrangian submanifolds; Maslov index |
| 14J33 | Mirror symmetry (algebro-geometric aspects) |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
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