Homological mirror symmetry for projective K3 surfaces

TL;DR

This work establishes Kontsevich's homological mirror symmetry for K3 surfaces in the large-volume limit by relating the Fukaya category of a projective K3 $(X,\omega)$, defined over the Novikov field $\C((q))$, to the derived category of coherent sheaves on a mirror K3 $Y_{\eta}$ with Picard rank $19$. The strategy blends noncompact HMS at the large volume limit with a detailed construction of a compact A-side mirror $M$ and a Gross–Siebert–type upgrade to a Kähler metric, followed by a deformation argument comparing the A- and B-sides through type III degenerations. The authors prove both noncompact HMS $\mathcal{W}(M)\simeq \mathrm{Coh}(Y)$ and compact HMS $\mathcal{F}(M)\simeq \mathrm{Perf}(Y)$, generalising Seidel’s quartic result and connecting to Greene–Plesser mirrors. The deformation-theoretic part shows that the A- and B-sides can be matched along a universal semistable smoothing $\mathcal{Y}/\C[[q]]$, yielding a full HMS statement for the generic fibre $\mathcal{Y}_{\eta}$ over $\C((q))$. This advances HMS for K3 surfaces by providing a robust framework that handles maximal normal crossing degenerations, SYZ-type fibrations, and integral-affine compactifications in a unified way with explicit Lagrangian–line-bundle correspondences.

Abstract

We prove the homological mirror symmetry conjecture of Kontsevich for K3 surfaces in the following form: The Fukaya category of a projective K3 surface is equivalent to the derived category of coherent sheaves on the mirror, which is a K3 surface of Picard rank $19$ over the field $\mathbb{C}((q))$ of formal Laurent series. This builds on prior work of Seidel, who proved the theorem in the case of the quartic surface, Sheridan, Lekili--Ueda, and Ganatra--Pardon--Shende.

Figures (22)

• Figure 2.1: The inclusion of Liouville sectors $T^\ast [0,1] \sqcup T^\ast [0,1] \hookrightarrow (T^\ast S^1)^-$. The fibration $\pi_{[0,1]}$ is given by projecting down vertically.
• Figure 2.2: Stabilised core for $T^\ast[0,1] \times (T^\ast S^1)^-$.
• Figure 2.3: Gluing mirrors to $\mathbb{P}^1$ to get the mirror to $D$: Lagrangian cores.
• Figure 2.4: Two-dimensional mirror to $M_{C_k}$ for $k=3$
• Figure 2.5: The isotopy $\psi$ of $\mathbb{C}_{\operatorname{Re} \geq 0} \times S_i$ used in the proof of Lemma \ref{['lem:fibration-on-M_i']}.

Theorems & Definitions (160)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Lemma 2.4
• proof
• Remark 2.5
• Proposition 2.6
• proof
• Theorem 2.7