Maslov-Zero Lagrangian Tori in Quartic K3 Surfaces

Definitions and mathematical statement

Question. If $L_0$ is an oriented Lagrangian torus in a quartic K3 surface and its Maslov class $\mu_{L_0}$ vanishes, must its homology class $[L_0]$ be nonzero? In the usual projective interpretation, let $X\subset{\mathbb{P}}^3({\mathbb{C}})$ be a smooth quartic surface, equip it with the restriction $\omega$ of the Fubini--Study form, and let $\iota:L_0\cong{\mathbb{R}}^2/{\mathbb{Z}}^2\hookrightarrow X$ be an oriented smooth embedding with $\iota^*\omega=0$. The conclusion is\[\mu_{L_0}=0\quad\Longrightarrow\quad\iota_*[L_0]\ne0\ \text{in }H_2(X;{\mathbb{Z}}).\] This specifies a conventional symplectic and integral-homology interpretation; the short source statement does not explicitly fix either convention or say whether other symplectic forms are intended.

A K3 surface has a nowhere vanishing holomorphic two-form $\Omega$. For an oriented Lagrangian tangent plane, the complex argument of $\Omega$ evaluated on a positively oriented real basis defines a phase map $\alpha:L_0\to S^1$, independent of the basis after normalization. With the squared-phase convention, its Maslov class is $\mu_{L_0}=(\alpha^2)^*\eta\in H^1(L_0;{\mathbb{Z}})$, where $\eta$ generates $H^1(S^1;{\mathbb{Z}})$. A phase-winding convention changes the factor of two but not the vanishing condition here. The homology class is the pushforward of the oriented fundamental class, not the abstract fundamental class of the torus in its own homology.

The supplied literature triage checked 2026-08-21 labels the record open, yet describes its extraction as supporting geometric evidence rather than a standalone problem and reports later positive results for the underlying Maslov-zero conjecture. The actual original and curated fields state the implication above. The relation between the contextual warning, the cited later results, and this specific quartic formulation is not supplied; no precise theorem or present-day solved/open conclusion is inferred from the triage.

Short English statement

Must an embedded Lagrangian torus with zero Maslov class represent a nonzero class in the homology of its quartic K3 surface? The natural phase and homology conventions are explicit, while the intended symplectic form and the supplied literature-status warning remain to be checked.

Sources

• Local UnsolvedMath registry, permanent ID 30000106, code catalogue entry; Maslov-Zero Lagrangian Tori in Quartic K3 Surfaces, Oberwolfach Reports Open Problems.
• String-Theorie und Geometrie (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/38, pp. 2015--2070. https://doi.org/10.4171/owr/2004/38. Proposed year and year tag: 2004; no proposer named.
• Supplied literature label OWR report: https://doi.org/10.4171/owr/2004/38.
• Supplied literature label Smirnov: https://arxiv.org/abs/2211.00620; no precise theorem scope is supplied.
• Literature triage checked 2026-08-21, registry status open; record updated 2026-09-16. The corrected_verified field reports checking the official report, not independently checked here.