The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some traces of amoebas were appearing from time to time, even before the formal introduction, as auxiliary tools in several problems. After 1994 amoebas have been seen and studied in several areas of mathematics, from algebraic geometry and topology to complex analysis and combinatorics. In particular, amoebas provided a very powerful tool for studying topology of algebraic varieties.
This survey aims to summarize the current state of knowledge about amoebas and to outline the applications to real algebraic geometry and adjacent areas. Most proofs are omitted here. An expanded version of this survey is currently under preparation jointly with Oleg Viro.
This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical varieties whose behavior is governed by algebraic geometry over the so-called tropical semifield. Geometric aspects of tropical algebraic geometry are the content of Part 2. We pay special attention to tropical curves. Both parts also include relevant applications of the theories. Pa…
In this paper, we study the amoeba volume of a given $k-$dimensional generic analytic variety $V$ of the complex algebraic torus $(\C^*)^n$. When $n\geq 2k$, we show that $V$ 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 $k-$linear spaces will be given.
An $n$-dimensional algebraic variety in $({\mathbb C}^\times)^{2n}$ covers its amoeba as well as its coamoeba generically finite-to-one. We provide an upper bound for the volume of these amoebas as well as for the number of points in the inverse images under the amoeba and coamoeba maps.
The paper deals with amoebas of $k$-dimensional algebraic varieties in the algebraic complex torus of dimension $n\geq 2k$. First, we show that the area of complex algebraic curve amoebas is finite. Moreover, we give an estimate of this area in the rational curve case in terms of the degree of the rational parametrization coordinates. We also show that the volume of the amoeba of $k$-dimensional algebraic variety in $(\mathbb{C}^*)^{n}$, with $n\geq 2k$, is finite.
In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via tropical varieties. When we apply this construction to the Teichmuller spaces we see that they can be mapped in a real algebraic hypersurface in such a way that the cone over the boundary is a subpolyhedron of a tropical hypersurface. We want to show how some properties of the boundary becomes straightforward if looked at from this point of view. For example…
We study the non-archimedean counterpart to the complex amoeba of an algebraic variety, and show that it coincides with a polyhedral set defined by Bieri and Groves using valuations. For hypersurfaces this set is also the tropical variety of the defining polynomial. Using non-archimedean analysis and a recent result of Conrad we prove that the amoeba of an irreducible variety is connected. We introduce the notion of an adelic amoeba for varieties over global fields, and establish a form of the local-global principle for them. This principle is used to explain the calculation of the nonexpansiv…
Amoebas and coamoebas are the logarithmic images of algebraic varieties and the images of algebraic varieties under the arg-map, respectively. We present new techniques for computational problems on amoebas and coamoebas, thus establishing new connections between (co-)amoebas, semialgebraic and convex algebraic geometry and semidefinite programming. Our approach is based on formulating the membership problem in amoebas (respectively coamoebas) as a suitable real algebraic feasibility problem. Using the real Nullstellensatz, this allows to tackle the problem by sums of squares techniques and se…
Amoebas are projections of complex algebraic varieties in the algebraic torus under a Log-absolute value map, which have connections to various mathematical subjects. While amoebas of hypersurfaces have been intensively studied in recent years, the non-hypersurface case is barely understood so far. We investigate intersections of amoebas of $n$ hypersurfaces in $(\mathbb{C}^*)^n$, which are canonical supersets of amoebas given by non-hypersurface varieties. Our main results are amoeba analogs of Bernstein's Theorem and Bézout's Theorem providing an upper bound for the number of connected compo…
An amoeba is the image of a subvariety of an algebraic torus under the logarithmic moment map. We consider some qualitative aspects of amoebas, establishing some results and posing problems for further study. These problems include determining the dimension of an amoeba, describing an amoeba as a semi-algebraic set, and identifying varieties whose amoebas are a finite intersection of amoebas of hypersurfaces. We show that an amoeba that is not of full dimension is not such a finite intersection if its variety is nondegenerate and we describe amoebas of lines as explicit semi-algebraic sets.
A generalization of the amoeba and the Ronkin function of a plane algebraic curve for a pair of harmonic functions on an algebraic curve with punctures is proposed. Extremal properties of $M$-curves are proved and connected with the spectral theory of difference operators with positive coefficients.
We define the generalized logarithmic Gauss map for algebraic varieties of the complex algebraic torus of any codimension. Moreover, we describe the set of critical points of the logarithmic mapping restricted to our variety, and we show an analogous of Mikhalkin's result on the critical points of the logarithmic map restricted to a hypersurfaces.
Given a complex algebraic hypersurface~$H$, we introduce a polyhedral complex which is a subset of the Newton polytope of the defining polynomial for~$H$ and enjoys the key topological and combinatorial properties of the amoeba of~$H.$ We provide an explicit formula for this polyhedral complex in the case when the spine of the amoeba is dual to a triangulation of the Newton polytope of the defining polynomial. In particular, this yields a description of the polyhedral complex when the hypersurface is optimal.
The amoeba of an affine algebraic variety V in (C^*)^r is the image of V under the map (z_1, ..., z_r) -> (log|z_1|, ..., log|z_r|). We give a characterisation of the amoeba based on the triangle inequality, which we call testing for lopsidedness. We show that if a point is outside the amoeba of V, there is an element of the defining ideal which witnesses this fact by being lopsided. This condition is necessary and sufficient for amoebas of arbitrary codimension, as well as for compactifications of amoebas inside any toric variety. Our approach naturally leads to methods for approximating hype…
In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexifications.
To any algebraic curve A in a complex 2-torus $(\C^*)^2$ one may associate a closed infinite region in a real plane called the amoeba of A. The amoebas of different curves of the same degree come in different shapes and sizes. All amoebas in have finite area and, furthermore, there is an upper bound on the area in terms of the degree of the curve. The subject of this paper is the curves in a complex 2-torus whose amoebas are of the maximal area. We show that up to multiplication by a constant such curves are defined over real numbers and, furthermore, that their real loci are isotopic to so-ca…
We show that the amoeba of a generic complex algebraic variety of codimension $1<r<n$ do not have a finite basis. In other words, it is not the intersection of finitely many hypersurface amoebas. Moreover we give a geometric characterization of the topological boundary of hypersurface amoebas refining an earlier result of F. Schroeter and T. de Wolff \cite{SW-13}.
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.
We prove a new patchworking theorem for singular algebraic curves, which states the following. Given a complex toric threefold $Y$ which fibers over ${\mathbb C}$ with a reduced reducible zero fiber $Y_0$ and other fibers $Y_t$ smooth, and given a reduced curve $C_0\subset Y_0$, the theorem provides a sufficient condition for the existence of a one-parametric family of curves $C_t\subset Y_t$, which induces an equisingular deformation for some singular points of $C_0$ and certain prescribed deformations for the other singularities. As application we give a comment on a recent theorem by G. Mik…
Given an algebraic variety we get Puiseux type parametrizations on suitable Reinhardt domains. These domains are defined using the amoeba of hypersurfaces containing the discriminant locus of a finite projection of the variety.
We completely characterize sections of the cones of nonnegative polynomials, convex polynomials and sums of squares with polynomials supported on circuits, a genuine class of sparse polynomials. In particular, nonnegativity is characterized by an invariant, which can be immediately derived from the initial polynomial. Furthermore, nonnegativity of such polynomials $f$ coincides with solidness of the amoeba of $f$, i.e., the Log-absolute-value image of the algebraic variety $\mathcal{V}(f) \subset (\mathbb{C}^*)^n$ of $f$. These results generalize earlier works both in amoeba theory and real al…
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.
We derive efficient algorithms for coarse approximation of algebraic hypersurfaces, useful for estimating the distance between an input polynomial zero set and a given query point. Our methods work best on sparse polynomials of high degree (in any number of variables) but are nevertheless completely general. The underlying ideas, which we take the time to describe in an elementary way, come from tropical geometry. We thus reduce a hard algebraic problem to high-precision linear optimization, proving new upper and lower complexity estimates along the way.
In this paper we study a $k$-dimensional analytic subvariety of the complex algebraic torus. We show that if its logarithmic limit set is a finite rational $(k-1)$-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.
For every fibration $f : X \to B$ with $X$ a compact Kähler manifold, $B$ a smooth projective curve, and a general fiber of $f$ an abelian variety, we prove that $f$ has an algebraic approximation.
This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas obtained relate global invariants to singularities of general complex algebraic (or analytic) maps. These results, new even for complex manifolds, are applied to obtain a version of Grothendieck-Riemann-Roch, a calculation of Todd classes of toric varieties, and an explicit formu…
Let $G$ be a connected reductive algebraic group. We develop a Gröbner theory for multiplicity-free $G$-algebras, as well as a tropical geometry for subschemes in a spherical homogeneous space $G/H$. We define the notion of a spherical tropical variety and prove a fundamental theorem of tropical geometry in this context. We also propose a definition for a spherical amoeba in $G/H$. Our work partly builds on the previous work of Vogiannou on spherical tropicalization and in some ways is complementary.
We study 3 basic questions about fundamental groups of algebraic varieties. For a morphism, is being surjective on $π_1$ preserved by base change? What is the connection between openness in the Zariski and in the Euclidean topologies? Which morphisms have the path lifting property?
We survey some recent developments in the study of canonical Kähler metrics on algebraic varieties and their relation with stability in algebraic geometry.
Every algebraic variety can be regarded as a symplectic manifold being equipped with a Kahler form. Therefore it is natural to study lagrangian geometry of any algebraic variety. We present two basic constructions which can be applied to a sufficiently wide set of algebraic varieties.
We present a new notion of speciality which is valid for Bohr - Sommerfeld lagrangian submanifolds. For algebraic varieties it leads to the construction of finite dimensional moduli spaces which are algebraic starting with any ample line bundle. The construction can be exploited both in Geometric Quantization and Mirror Symmetry.
We extend the decomposition theorem for numerically $K$-trivial varieties with log terminal singularities to the Kähler setting. Along the way we prove that all such varieties admit a strong locally trivial algebraic approximation, thus completing the numerically $K$-trivial case of a conjecture of Campana and Peternell.
Applying the Second Main Theorem we deal with the algebraic degeneracy of entire holomorphic curves from the complex plane into a complex algebraic normal variety of positive log Kodaira dimension that admits a finite proper morphism to a semi-abelian variety. We will also discuss applications to the Kobayashi hyperbolicity problem.
We extend the definition of algebraic entropy to endomorphisms of affine varieties. We calculate algebraic entropy of the action of elements of mapping class groups on various character varieties, and show that it is equal to a quantity we call the spectral radius, a generalization of the dilatation of a Pseudo-Anosov mapping class. Our calculations are compatible with all known calculations of the topological entropy of this action.
In this paper we study certain homogeneous spaces, which we call generalised affine Stiefel algebraic varieties. The main aim is to characterise the canonical divisor of generalised affine Stiefel algebraic varieties in terms of group representations. Affine Stiefel algebraic varieties and in particular $S^{n}$ are two special cases of the generalised affine Stiefel algebraic varieties.
We study Minkowski sums and Hadamard products of algebraic varieties. Specifically we explore when these are varieties and examine their properties in terms of those of the original varieties.
We give sufficient cohomological criteria for the classes of given varieties over a field $k$ to be algebraically independent in the Grothendieck ring of varieties over $k$ and construct some examples.
The construction of double point cobordism groups of vector bundles on varieties in the work [Lee-P] (arXiv:1002.1500 [math.AG]) of Yuan-Pin Lee and Rahul Pandharipande gives immediately double point cobordism groups of filtered vector bundles on varieties. We note also that among the four basic operations -- direct sum, tensor product, dual, and Hom -- on vector bundles on varieties, only taking dual is compatible with double point cobordisms of vector bundles on varieties in general, by a demonstration on an example of vector bundles on Calabi-Yau 3-folds. A question on refined and/or higher…
We show that every affine or projective algebraic variety defined over the field of real or complex numbers is homeomorphic to a variety defined over the field of algebraic numbers. We construct such a homeomorphism by choosing a small deformation of the coefficients of the original equations. This method is based on the properties of Zariski equisingular families of varieties. Moreover we construct an algorithm, that, given a system of equations defining a variety $V$, produces a system of equations with algebraic coefficients of a variety homeomorphic to $V$
We develop a moduli theory of algebraic varieties and pairs of non-negative Kodaira dimension. We define stable minimal models and construct their projective coarse moduli spaces under certain natural conditions. This can be applied to a wide range of moduli problems in algebraic geometry.
In this paper, we prove effective upper bounds for effective sections of line bundles on projective varieties and hermitian line bundles on arithmetic varieties in terms of the volumes. They are effective versions of the Hilbert--Samuel formula and the arithmetic Hilbert--Samuel formula.
We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology and for a strong topology that we introduce. Our answers involve a study of seminormalization and saturation for morphisms between algebraic varieties, together with an interpretation in terms of continuous rational functions on the closed points of an algebraic variety. The continuity refers to the strong topology which is the usual Euclidean topology in th…
In this paper, we establish the following version at infinity of Whitney's theorem [6, 7]: Geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of geometric tangent cones at infinity and using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. We also show that tangent cones at infinity of complex algebraic varieties can be computed using Gröbner bases.
