Toric Cycles in the Complement of a Complex Curve in $(\mathbb{C}^{\times})^2$

TL;DR

This paper proves the homological independence of toric cycles in the complement of a complex algebraic curve with maximal amoeba area, enhancing understanding of the topology of amoebas in complex tori.

Contribution

It establishes the homological independence of toric cycles in the complement of a complex curve with maximal amoeba area, a novel result in the topology of amoebas.

Findings

Toric cycles are homologically independent in the complement of certain complex curves.

The result applies specifically to curves with amoebas of maximal area.

Provides new insights into the topology of amoebas in complex algebraic geometry.

Abstract

The amoeba of a complex curve in the 2-dimensional complex torus is its image under the projection onto the real subspace in the logarithmic scale. The complement to an amoeba is a disjoint union of connected components that are open and convex. A toric cycle is a 2-cycle in the complement to a curve associated with a component of the complement to an amoeba. We prove homological independence of toric cycles in the complement to a complex algebraic curve with amoeba of maximal area.