2022/09/09 by Zupan, Alexander
#32Q55 #57K40 #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2209.04274
We demonstrate that the geometric, topological, and combinatorial complexities of certain surfaces in \mathbbCP2 are closely related: We prove that a positive genus surface K in \mathbbCP2 that minimizes genus in its homology class is isotopic to a complex curve Cd if and only if 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 \mathbbCP2 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 \mathbbCP2. Finally, we show that that the varieties Vd = \[z1:z2:z3] ∈ \mathbbCP2 : z1z2d-1 + z2z3d-1 + z3z1d-1 = 0\ and V'd = \[z1:z2:z3] ∈ \mathbbCP2 : z1d-1z2 + z2d-1z3 + z3d-1z1 = 0\ are in efficient bridge position with respect to the standard Stein trisection of \mathbbCP2, 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 \mathbbCP3 admit efficient Stein trisections, partially answering a question of Lambert-Cole and Meier.