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.

25 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

References in corpus (6)

• Symplectic cohomology and duality for the wrapped Fukaya category
• A counterexample to the unit conjecture for group rings
• Homological mirror symmetry for Milnor fibers of simple singularities
• Geometric and algebraic presentations of Weinstein domains
• Mirror symmetry for Berglund-Hübsch Milnor fibers
• On the embedding complexity of Liouville manifolds