🌴Tropical Geometry Unit 11 – Toric Geometry and Amoebas in Tropical Theory

Key Concepts and Definitions

• Toric varieties algebraic varieties containing algebraic torus as a dense open subset
• Amoebas images of algebraic varieties under the Log map, resembling amoeba shapes
• Tropical geometry studies piecewise linear objects arising as limits of classical algebraic varieties
  • Involves study of tropical curves, tropical hypersurfaces, and tropical varieties
• Newton polytope convex hull of exponent vectors of a polynomial in several variables
  • Plays crucial role in toric geometry and amoeba theory
• Puiseux series formal power series with rational exponents, used in studying amoebas
• Maslov dequantization process of obtaining tropical objects from classical counterparts
• Gröbner bases generating sets of ideals with good computational properties
  • Used in studying toric ideals and tropical varieties

Historical Context and Development

• Toric varieties introduced by Demazure in 1970s as a generalization of projective spaces
  • Studied by Danilov, Fulton, Oda, and others in subsequent years
• Amoeba theory developed by Gelfand, Kapranov, and Zelevinsky in 1990s
  • Investigated complex algebraic varieties under the Log map
• Tropical geometry emerged in early 2000s as a limit of classical algebraic geometry
  • Pioneered by Mikhalkin, Sturmfels, Itenberg, Shustin, and others
• Connections between toric and tropical geometry explored by Speyer, Sturmfels, and others
  • Toric degenerations and tropicalizations of toric varieties
• Applications to mirror symmetry, mathematical physics, and combinatorics discovered
  • Tropical geometry provides new insights and techniques

Toric Varieties: Basics and Construction

• Toric variety associated to a lattice polytope or a fan in a lattice
  • Constructed as a quotient of an open subset of affine space by a torus action
• Affine toric varieties correspond to lattice cones and are spectra of semigroup algebras
• Projective toric varieties obtained by gluing affine toric varieties along torus-invariant subvarieties
  • Correspond to lattice polytopes and normal fans
• Torus action on a toric variety has a dense orbit isomorphic to the algebraic torus
  • Orbit closures give torus-invariant subvarieties
• Toric varieties have a combinatorial description in terms of fans and polytopes
  • Geometric properties encoded in combinatorial data
• Homogeneous coordinate ring of a projective toric variety is a polynomial ring with a grading
  • Determined by the corresponding polytope

Amoebas: Introduction and Properties

• Amoeba of a complex algebraic variety is its image under the Log map
  • Defined as $A_f = \{(\log|z_1|, \ldots, \log|z_n|) : (z_1, \ldots, z_n) \in V(f)\}$
• Amoebas have a tentacle-like structure with complement consisting of convex regions
  • Regions correspond to Laurent series expansions of the defining polynomial
• Spine of an amoeba is a piecewise linear object capturing its combinatorial structure
  • Obtained as a limit under the Log map
• Order map sends an algebraic variety to its corresponding tropical variety
  • Defined using valuations on the field of Puiseux series
• Amoebas have a logarithmic limit set, which is a non-Archimedean amoeba
  • Encodes asymptotic behavior of the amoeba at infinity
• Ronkin function associated to an amoeba is a convex function on its complement
  • Encodes information about the amoeba's shape and structure

Tropical Geometry Fundamentals

• Tropical semiring $(\mathbb{R} \cup \{\infty\}, \oplus, \odot)$ with $a \oplus b = \min(a,b)$ and $a \odot b = a + b$
  • Used to define tropical polynomials and tropical varieties
• Tropical polynomial obtained by replacing addition with $\oplus$ and multiplication with $\odot$
  • Gives a piecewise linear function on $\mathbb{R}^n$
• Tropical hypersurface defined as the corner locus of a tropical polynomial
  • Dual to a subdivision of the Newton polytope
• Tropical varieties defined as intersections of tropical hypersurfaces
  • Have a polyhedral structure and satisfy a balancing condition
• Tropical Grassmannians parametrize tropical linear spaces and are tropicalizations of classical Grassmannians
• Tropical Bézout's theorem relates degrees of tropical varieties to their intersection multiplicities
• Tropical convexity studies convex polyhedra and their polyhedral subdivisions
  • Plays a key role in tropical geometry

Connections Between Toric and Tropical Geometry

• Toric varieties can be tropicalized by taking the Log map of the algebraic torus
  • Gives a tropical variety associated to the fan of the toric variety
• Tropical compactifications of affine space constructed using toric geometry
  • Provide a framework for studying tropical varieties at infinity
• Toric degenerations of algebraic varieties give rise to tropical limits
  • Used to study tropical geometry via classical algebraic geometry techniques
• Tropical Hodge theory relates cohomology of toric varieties to tropical cohomology
  • Provides a tropical analog of classical Hodge theory
• Tropical mirror symmetry relates tropical geometry to mirror symmetry for toric varieties
  • Involves studying Landau-Ginzburg models and tropical disk counts
• Tropical intersection theory developed using toric geometry and intersection theory on toric varieties
  • Allows computation of tropical intersection numbers

Applications and Real-World Examples

• Toric geometry used in geometric modeling and computer-aided design (CAD)
  • Toric patches and toric Bézier surfaces for shape representation
• Amoeba theory applied to study of complex systems and dynamical systems
  • Amoeba-shaped regions in parameter spaces correspond to different behaviors
• Tropical geometry used in optimization and discrete event systems
  • Tropical semiring allows modeling of min-plus systems and scheduling problems
• Applications to phylogenetics and statistical inference
  • Tropical geometry of tree spaces and phylogenetic trees
• Connections to mathematical physics, such as mirror symmetry and string theory
  • Tropical geometry provides new insights and computational tools
• Applications to economics and game theory, such as auction theory and market equilibria
  • Tropical geometry used to study pricing problems and equilibrium conditions

Advanced Topics and Current Research

• Tropical moduli spaces and tropical compactifications of moduli spaces
  • Studied using techniques from toric and tropical geometry
• Tropical analogs of classical theorems in algebraic geometry, such as the Riemann-Roch theorem
  • Developed using tropical intersection theory and tropical cohomology
• Tropical representation theory and tropical cluster algebras
  • Investigate tropical aspects of representation theory and cluster algebras
• Tropical Donaldson-Thomas theory and tropical Gromov-Witten theory
  • Study tropical analogs of enumerative invariants in algebraic geometry
• Tropical Hodge theory and tropical period domains
  • Relate tropical geometry to Hodge theory and period domains
• Connections to non-Archimedean geometry and Berkovich spaces
  • Tropical geometry as a skeleton of non-Archimedean analytic spaces
• Interactions with combinatorics, such as matroid theory and chip-firing games
  • Tropical geometry provides new perspectives and results