Mini-workshop on Algebraic geometry and its Applications
"A tropical characterization of an analytic variety to be algebraic"
SPEAKER | Mounir Nisse
INSTITUTE | KIAS
DATE | February 21(Fri), 2014
TIME | 15:10
PLACE | Room 403, Center for Mathematical Challenges (CMC)
ABSTRACT | All tools will be defined in this talk with sample examples. Amoebas are images of subvarieties of the complex algebraic torus under the coordinatewise logarithm map. G. Bergman introduced the logarithmic limit set Lí─(V) of a subvariety V of the torus as the set of limiting directions of points in its amoeba. Bieri and Groves showed that if V is algebraic, then Lí─(V) is a finite rational polyhedral complex of dimension dimC(V)?1 in the sphere Sn?1. Logarithmic limit sets are now called tropical varieties. We show that a generic k-dimensional analytic subvariety of the n-dimensional complex torus is algebraic if and only if its logarithmic limit set is a finite rational complex polyhedral of dimension k?1. In particular, if the dimension of the ambient space is at least 2k, then the last conditions are equivalent to the fact that the volume of the amoeba is finite. I will focus my talk on the main idea behind this result by showing some concrete examples of holomorphic curves.