Nonexistence of exact Lagrangian tori in affine conic bundles over $\mathbb{C}^n$

Let $M \subset \mathbb{C}^{n+1}$ be a smooth affine hypersurface defined by the equation $xy + p(z_1, \dotsm , z_{n-1}) = 1$, where $p$ is a Brieskorn–Pham polynomial and $n \geq 2$. We prove that if $L \subset M$ is a closed, orientable, exact Lagrangian submanifold, then $L$ cannot be a $K(\pi,1)$ space. The key point of the proof is to establish a version of homological mirror symmetry for the wrapped Fukaya category of $M$, from which the finite-dimensionality of the symplectic cohomology group $SH^0 (M)$ follows by a Hochschild cohomology computation.

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Received 10 May 2021

Accepted 15 December 2021

Published 24 April 2023