Hexagonal lattice diagrams for complex curves in $\mathbb{CP}^2$

Abstract

We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in $\mathbb{CP}^2$ are closely related: We prove that a positive genus surface $\mathcal{K}$ in $\mathbb{CP}^2$ that minimizes genus in its homology class is isotopic to a complex curve $\mathcal{C}_d$ if and only if $\mathcal{K}$ admits a hexagonal lattice diagram, a special type of shadow diagram in which arcs meet only at bridge points and tile the central surface of the standard trisection of $\mathbb{CP}^2$ by hexagons. There are eight families of these diagrams, two of which represent surfaces in efficient bridge position. Combined with a result of Lambert-Cole relating symplectic surfaces and bridge trisections, this allows us to provide a purely combinatorial reformulation of the symplectic isotopy problem in $\mathbb{CP}^2$. Finally, we show that that the varieties $\mathcal{V}_d = \{[z_1:z_2:z_3] \in \mathbb{CP}^2 : z_1z_2^{d-1} + z_2z_3^{d-1} + z_3z_1^{d-1} = 0\}$ and $\mathcal{V}'_d = \{[z_1:z_2:z_3] \in \mathbb{CP}^2 : z_1^{d-1}z_2 + z_2^{d-1}z_3 + z_3^{d-1}z_1 = 0\}$ are in efficient bridge position with respect to the standard Stein trisection of $\mathbb{CP}^2$, and their shadow diagrams agree with the two families of efficient hexagonal lattice diagrams. As a corollary, we prove that two infinite families of complex hypersurfaces in $\mathbb{CP}^3$ admit efficient Stein trisections, partially answering a question of Lambert-Cole and Meier.