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 $n\ge 2$. 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 ${SH}^0(M)$ follows by a Hochschild cohomology computation.

arXiv comment25 pages, 2 figures. v5: the main result has been further generalized to exclude exact Lagrangian K(π,1) spaces. An appendix discussing central units in group rings has been added. To appear in JSG

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