We show that the base spaces of the semiuniversal unfoldings of some weighted homogeneous singularities can be identified with moduli spaces of $A_{\infty }$$A_{\infty }$-structures on the trivial extension algebras of the endomorphism algebras of the tilting objects. The same algebras also appear in the Fukaya categories of their mirrors. Based on these identifications, we discuss applications to homological mirror symmetry for Milnor fibers, and give a proof of homological mirror symmetry for an $n$$n$-dimensional affine hypersurface of degree $n+2$$n+2$ and the double cover of the $n$$n$-dimensional affine space branched along a degree $2n+2$$2n+2$ hypersurface. Along the way, we also give a proof of a conjecture of Seidel (Proceedings of the International Congress of Mathematicians, 2002) which may be of independent interest.

中文翻译：

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通过 A∞$A_\infty$-结构模的 Milnor 光纤同调镜像对称

我们证明了一些加权齐次奇点的半泛展开的基空间可以用模空间来识别${\mathrm{一个}}_{\infty }$$A_{\infty }$-倾斜物体的自同态代数的平凡扩展代数上的结构。同样的代数也出现在他们镜子的深谷范畴中。基于这些识别，我们讨论了 Milnor 光纤同调镜像对称的应用，并给出了一个同调镜像对称的证明。$n$$n$度数的一维仿射超曲面$n+2$$n+2$和双盖$n$$n$维仿射空间沿度数分支$2n+2$$2n+2$超曲面。在此过程中，我们还证明了可能具有独立意义的 Seidel 猜想（Proceedings of the International Congress of Mathematicians ，2002）。