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

Abstract

Let $M\subset\mathbb{C}^{n+1}$ be a smooth affine hypersurface defined by the equation $xy+p(z_1,\cdots,z_{n-1})=1$, where $p$ is a Brieskorn-Pham polynomial and $n\geq2$. We prove that if $L\subset M$ is an orientable exact Lagrangian submanifold, then $L$ does not admit a Riemannian metric with non-positive sectional curvature. 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 $\mathit{SH}^0(M)$ follows by a Hochschild cohomology computation.