Mirror symmetry for Berglund-Hübsch Milnor fibers

Abstract

We explain how to calculate the Fukaya category of the Milnor fiber of a Berglund-Hübsch invertible polynomial, mostly proving a conjecture of Yankı Lekili and Kazushi Ueda on homological mirror symmetry. As usual, we begin by calculating the "very affine" Fukaya category; afterwards, we deform it, generalizing an earlier calculation of David Nadler. The main step of our calculation may be understood as determining a certain canonical extension of a perverse schober.

Figures (4)

• Figure 1: The fan of $\mathbb{P}^2$ superimposed on the tailored amoeba of its mirror hypersurface $H = \{x+y+\frac{1}{xy}=0\}.$ Note how this tailored amoeba is precisely "tropical" away from the vertices.
• Figure 2: The skeleta $\mathbb{L}_{\Sigma_{\mathbb{A}^n}}$ of the Liouville sectors mirror to $\mathbb{A}^n$ for $n=1,2.$
• Figure 3: The degeneration of the tropical hypersurface of $M$ as it approaches the critical value $1$. In the figure on the left, the collapsing region where the disk is to be attached is shaded.
• Figure 4: The Lagrangians $\mathbb{L}_M$ (left) and $\mathring{\mathbb{L}}$ (right) for $n=3.$ Note that the $\mathbb{R}^k\times T^{n-k-2}$ pieces of $\mathbb{L}_M$ become copies of $\mathbb{R}^k\times T^{n-k-1}$ after being swept out by the parallel transport around 0.

Theorems & Definitions (92)

• Conjecture 1.3
• Theorem 1.8
• Example 1.10
• Example 1.11
• Remark 1.12
• Conjecture 1.13: LU1
• Lemma 1.20: Lemma \ref{['lem:colim-undef']} below
• Theorem 1.22
• Conjecture 1.25
• Theorem 1.26