What is a stable log map?
Mohammad Farajzadeh-Tehrani, Mohan Swaminathan
TL;DR
The paper builds a bridge between algebraic and symplectic approaches to stable log maps for a smooth variety with a normal crossings divisor. It introduces fine-basic and fs-basic log maps, and proves a precise equivalence: fine algebraic log maps correspond to symplectic log maps, while fs/basic maps correspond to saturated symplectic log maps. A key technical framework is developed around systems of line bundles (slb), a detailed classification of log curves over log points, and a formal presentation theory that encodes log data combinatorially. Saturation and tropicalization notions quantify the finite lifts between the two worlds, yielding a surjective forgetful map with finite fibers and a canonical saturation mechanism akin to normalization. Collectively, these results unify algebraic log Gromov–Witten-type data with symplectic log structures and lay groundwork for a symplectic log Gromov–Witten theory in the almost Kähler setting.
Abstract
Let $X$ be a smooth projective variety over $\mathbb{C}$ with a simple normal crossings divisor $D\subset X$. We compare the notions of stable log maps to $(X,D)$ in algebraic geometry and symplectic topology. In particular, we prove an equivalence between fine (basic) algebraic log maps and symplectic log maps, and we define the symplectic analogue of fine saturated algebraic log maps by refining the notion of log Gromov convergence.