11.2 Amoebas of algebraic varieties

Amoebas are a fascinating way to visualize complex algebraic varieties in real space. They're created by applying a logarithmic map to algebraic varieties, transforming complex equations into geometric shapes with tentacles and holes.

Amoebas have a rich structure that reflects properties of the original varieties. They're used to study asymptotic behavior, topology, and convergence domains of complex functions, bridging algebraic and tropical geometry in surprising ways.

Definition of amoebas

• Amoebas are images of algebraic varieties under the logarithmic map, providing a way to visualize complex algebraic geometry in real space
• The logarithmic map is defined as $Log: (ℂ^*)^n → ℝ^n$, where $Log(z_1, ..., z_n) = (\log|z_1|, ..., \log|z_n|)$
• For an algebraic variety $V ⊂ (ℂ^*)^n$, its amoeba is $\mathcal{A}(V) = Log(V) ⊂ ℝ^n$

Amoebas vs coamoebas

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• Coamoebas are another way to visualize algebraic varieties, defined using the argument map instead of the logarithmic map
• The argument map is $Arg: (ℂ^*)^n → (S^1)^n$, where $Arg(z_1, ..., z_n) = (\arg(z_1), ..., \arg(z_n))$
• For an algebraic variety $V ⊂ (ℂ^*)^n$, its coamoeba is $\mathcal{C}(V) = Arg(V) ⊂ (S^1)^n$

Amoebas of hypersurfaces

• Hypersurfaces are algebraic varieties defined by a single polynomial equation $f(z_1, ..., z_n) = 0$
• The amoeba of a hypersurface $V_f = \{z ∈ (ℂ^*)^n : f(z) = 0\}$ is $\mathcal{A}(V_f) = \{x ∈ ℝ^n : ∃z ∈ (ℂ^*)^n, Log(z) = x, f(z) = 0\}$
• Amoebas of hypersurfaces have a rich geometric structure and are well-studied in tropical geometry

Amoebas of varieties

• Algebraic varieties are defined by a set of polynomial equations $f_1(z) = ... = f_k(z) = 0$
• The amoeba of a variety $V = \{z ∈ (ℂ^*)^n : f_1(z) = ... = f_k(z) = 0\}$ is $\mathcal{A}(V) = Log(V)$
• Amoebas of varieties can be more complicated than those of hypersurfaces, but still exhibit interesting properties

Structure of amoebas

• Amoebas have a rich geometric and topological structure that reflects properties of the original algebraic varieties
• Key structural elements of amoebas include Harnack curves, the order map, contours, and complement components

Harnack curves

• Harnack curves are real algebraic curves that are maximally nested, meaning they have the maximum number of compact connected components in their complement
• The amoeba of a Harnack curve has a specific shape, with tentacles extending to infinity in prescribed directions
• Harnack curves play a crucial role in understanding the topology of amoebas and their complements

Order map

• The order map is a piecewise linear map from the Newton polygon of a polynomial to the amoeba of the corresponding hypersurface
• It relates the combinatorial structure of the Newton polygon to the geometry of the amoeba
• The order map is used to study the asymptotic behavior of amoebas and their complements

Contour of amoebas

• The contour of an amoeba is the boundary of its complement components
• It consists of critical points of the logarithmic map restricted to the algebraic variety
• The contour encodes information about the shape and topology of the amoeba

Complement components

• The complement of an amoeba $ℝ^n \setminus \mathcal{A}(V)$ consists of connected components, each corresponding to a lattice point in the Newton polygon of the defining polynomial
• The number and arrangement of complement components are related to the combinatorics of the Newton polygon
• Complement components play a key role in the amoeba theorem and its applications

Amoeba theorem

• The amoeba theorem, also known as the Gelfand-Kapranov-Zelevinsky theorem, is a fundamental result in the theory of amoebas
• It establishes a correspondence between the Newton polygon of a polynomial and the complement components of its amoeba

Maslov dequantization

• Maslov dequantization is a procedure that relates the algebra of polynomials to the geometry of their amoebas
• It involves a deformation of the multiplication operation on polynomials, controlled by a parameter $h$
• As $h → 0$, the deformed multiplication approaches the tropical multiplication, and the amoeba of the polynomial approaches its tropical variety

Litvinov-Maslov correspondence

• The Litvinov-Maslov correspondence is a generalization of the amoeba theorem to the case of non-Archimedean amoebas
• It relates the Newton polygon of a polynomial over a non-Archimedean field to the complement components of its amoeba
• The correspondence is established using the Maslov dequantization procedure and tropical geometry

Topology of amoebas

• The topology of amoebas is a rich area of study, with connections to complex analysis, algebraic geometry, and combinatorics
• Key topological invariants of amoebas include their genus, tentacles, and the structure of amoebas of lines

Genus of amoebas

• The genus of an amoeba is the minimal genus of a surface in which the amoeba can be embedded
• It is related to the number of holes or handles in the amoeba
• The genus of an amoeba can be computed using the Newton polygon of the defining polynomial and the order map

Tentacles of amoebas

• Tentacles are the unbounded components of an amoeba that extend to infinity
• The number and direction of tentacles are determined by the Newton polygon of the defining polynomial
• Tentacles play a crucial role in the asymptotic behavior of amoebas and their complements

Amoebas of lines

• Lines are the simplest examples of algebraic varieties, defined by linear equations
• The amoeba of a line is a convex subset of $ℝ^n$, bounded by hyperplanes orthogonal to the line
• Amoebas of lines serve as building blocks for understanding the structure of more complex amoebas

Applications of amoebas

• Amoebas have found applications in various areas of mathematics, including complex analysis, real algebraic geometry, and mirror symmetry
• They provide a bridge between algebraic geometry and tropical geometry, allowing for the transfer of ideas and techniques between the two fields

Amoebas in complex analysis

• In complex analysis, amoebas are used to study the asymptotic behavior of holomorphic functions
• The complement components of an amoeba correspond to the domains of convergence of Laurent series expansions of the function
• Amoebas also appear in the study of the Ronkin function, a convex function associated with a polynomial that encodes information about its amoeba

Amoebas in real algebraic geometry

• In real algebraic geometry, amoebas are used to study the topology of real algebraic varieties
• The amoeba of a variety contains information about the number and arrangement of its real connected components
• Amoebas also play a role in the study of Hilbert's 16th problem, which concerns the topology of real algebraic curves

Amoebas in mirror symmetry

• Mirror symmetry is a duality between complex geometry and symplectic geometry, with applications in string theory and enumerative geometry
• Amoebas provide a way to relate the complex structure of a variety to the symplectic structure of its mirror partner
• The study of amoebas has led to new insights and conjectures in mirror symmetry, such as the Strominger-Yau-Zaslow conjecture