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

TL;DR

This paper proves that certain affine hypersurfaces do not contain exact Lagrangian tori with non-positive curvature, using homological mirror symmetry and symplectic cohomology computations.

Contribution

It establishes a nonexistence result for exact Lagrangian tori in affine conic bundles via homological mirror symmetry techniques.

Findings

No exact Lagrangian tori with non-positive curvature exist in the studied hypersurfaces.

Homological mirror symmetry is used to compute symplectic cohomology.

Finite-dimensionality of symplectic cohomology is demonstrated.

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.