27 research outputs found
A tropical characterization of complex analytic varieties to be algebraic
In this paper we study a -dimensional analytic subvariety of the complex
algebraic torus. We show that if its logarithmic limit set is a finite rational
-dimensional spherical polyhedron, then each irreducible component of
the variety is algebraic. This gives a converse of a theorem of Bieri and
Groves and generalizes a result proven in \cite{MN2-11}. More precisely, if the
dimension of the ambient space is at least twice of the dimension of the
generic analytic subvariety, then these properties are equivalent to the volume
of the amoeba of the subvariety being finite.Comment: 7 pages, 3 figure
Higher convexity of coamoeba complements
We show that the complement of the coamoeba of a variety of codimension k+1
is k-convex, in the sense of Gromov and Henriques. This generalizes a result of
Nisse for hypersurface coamoebas. We use this to show that the complement of
the nonarchimedean coamoeba of a variety of codimension k+1 is k-convex.Comment: 14 pages, 5 color figures, minor revision
Analytic varieties with finite volume amoebas are algebraic
In this paper, we study the amoeba volume of a given dimensional generic
analytic variety of the complex algebraic torus (\C^*)^n. When , we show that is algebraic if and only if the volume of its amoeba is
finite. In this precise case, we establish a comparison theorem for the volume
of the amoeba and the coamoeba. Examples and applications to the linear
spaces will be given.Comment: 13 pages, 2 figure
Higher convexity for complements of tropical varieties
We consider Gromov's homological higher convexity for complements of tropical
varieties, establishing it for complements of tropical hypersurfaces and
curves, and for nonarchimedean amoebas of varieties that are complete
intersections over the field of complex Puiseaux series. Based on these
results, we conjecture that the complement of a tropical variety has this
higher convexity, and we prove a weak form of our conjecture for the
nonarchimedean amoeba of a variety over the complex Puiseaux field. One of our
main tools is Jonsson's limit theorem for tropical varieties.Comment: 11 page
Amoebas and coamoebas of linear spaces
We give a complete description of amoebas and coamoebas of -dimensional
very affine linear spaces in . This include an upper bound
of their dimension, and we show that if a -dimensional very affine linear
space in is generic, then the dimension of its (co)amoeba
is equal to . Moreover, we prove that the volume of its
coamoeba is equal to . In addition, if the space is generic and real,
then the volume of its amoeba is equal to .Comment: 17 pages, 4 figures; to appear in the Passare memorial volume; It
will be published by Springer/Birkhuse