环$\mathbb{C}^n$上仿射锥丛中精确Lagrange环面的不存在性
Nonexistence of exact Lagrangian tori in affine conic bundles over $\mathbb{C}^n$
摘要：
设$M\subset\mathbb{C}^{n+1}$是由方程$xy+p(z_1,\cdots,z_{n-1})=1$定义的光滑仿射超曲面，其中$p$是Brieskorn-Pham多项式，$n\geq2$。证明了如果$L\subset M$是可定向的精确拉格朗日子流形，则$L$不允许有非正截面曲率的黎曼度量.证明的关键是建立了$M$的包裹范畴的同调镜像对称性，由此得到了辛上同调群$\mathit{SH}^0(M)$的有限维性，并由此得到了Hochschild上同调计算.
英文摘要：
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.
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