11.5 Tropical Hodge theory and toric degenerations

Tropical Hodge theory and toric degenerations offer a combinatorial approach to studying complex algebraic varieties. By associating polyhedral complexes with these varieties, we can explore their topology and geometry using simpler, more tractable methods.

These techniques allow us to define tropical Hodge numbers, which are analogous to classical Hodge numbers. Toric degenerations help us study complex varieties by relating them to toric varieties with rich combinatorial structures, providing valuable insights into their properties.

Tropical Hodge theory fundamentals

• Tropical Hodge theory provides a combinatorial approach to studying complex algebraic varieties by associating polyhedral complexes to them
• Establishes a connection between the topology of complex varieties and the combinatorics of polyhedral complexes
• Introduces tropical Hodge numbers as a tropical analog of classical Hodge numbers, capturing important topological information

Tropical varieties vs complex varieties

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• Tropical varieties are defined over the tropical semiring $(\mathbb{R} \cup \{\infty\}, \min, +)$, while complex varieties are defined over the complex numbers $\mathbb{C}$
• The tropicalization process converts a complex variety into a tropical variety by applying the valuation map coordinatewise (e.g., $(x, y) \mapsto (-\log|x|, -\log|y|)$)
• Tropical varieties inherit certain topological and combinatorial properties from their complex counterparts, allowing for a simpler and more tractable study

Polyhedral complexes in tropical Hodge theory

• Polyhedral complexes, such as tropical fans and tropical polytopes, play a central role in tropical Hodge theory
• These complexes are used to encode the combinatorial structure of tropical varieties
• The intersection theory of tropical cycles can be studied using the intersection theory of polyhedral complexes (e.g., stable intersection of tropical cycles corresponds to the intersection of their associated polyhedral complexes)

Tropical Hodge numbers

• Tropical Hodge numbers $h^{p,q}_{\text{trop}}$ are a tropical analog of the classical Hodge numbers $h^{p,q}$ for complex varieties
• They are defined using the combinatorics of the associated polyhedral complexes, such as the dimensions of certain spaces of piecewise linear functions
• Tropical Hodge numbers satisfy similar symmetries and relations as classical Hodge numbers (e.g., $h^{p,q}_{\text{trop}} = h^{q,p}_{\text{trop}}$ and $\sum_{p+q=k} h^{p,q}_{\text{trop}} = b_k$, where $b_k$ is the $k$-th Betti number)

Toric degenerations overview

• Toric degenerations provide a way to study complex varieties by degenerating them to toric varieties, which have a rich combinatorial structure
• This process allows for the use of combinatorial techniques to study the topology and geometry of the original complex varieties
• Toric degenerations are closely related to tropical geometry, as the tropical limit of a toric degeneration often encodes important information about the original variety

Toric varieties and fans

• Toric varieties are complex algebraic varieties that contain an algebraic torus $(\mathbb{C}^*)^n$ as a dense open subset, with the action of the torus extending to the entire variety
• Toric varieties can be described combinatorially using fans, which are collections of strongly convex rational polyhedral cones in a real vector space
• The geometry of a toric variety is determined by the combinatorial properties of its associated fan (e.g., the orbit-cone correspondence)

Degenerations of complex varieties

• A degeneration of a complex variety $X$ is a family of varieties $\mathcal{X} \to \mathbb{C}$ such that the general fiber is isomorphic to $X$ and the special fiber $\mathcal{X}_0$ (over 0) is a simpler or more tractable variety
• Toric degenerations are degenerations where the special fiber $\mathcal{X}_0$ is a toric variety
• The combinatorial structure of the toric special fiber can provide insights into the topology and geometry of the original variety $X$

Tropical limits of toric degenerations

• The tropical limit of a toric degeneration is a tropical variety that encodes information about the degeneration
• It is obtained by applying the valuation map to the family of varieties in the degeneration and taking the limit as the parameter tends to infinity
• The tropical limit often has a simpler combinatorial structure than the original variety, making it easier to study certain properties (e.g., the limit of a family of Calabi-Yau varieties is often a tropical Calabi-Yau variety)

Connections between tropical Hodge theory and toric degenerations

• Tropical Hodge theory and toric degenerations are closely related, as they both provide combinatorial approaches to studying complex varieties
• The combinatorial data arising from toric degenerations, such as the associated polyhedral complexes, can be used to compute tropical Hodge numbers and other invariants
• The interplay between these two theories allows for a deeper understanding of the topology and geometry of complex varieties

Tropical Hodge numbers vs Hodge numbers

• Tropical Hodge numbers are a tropical analog of classical Hodge numbers, capturing similar topological information
• In some cases, tropical Hodge numbers can be used to compute or estimate classical Hodge numbers (e.g., for certain classes of Calabi-Yau varieties)
• The comparison between tropical and classical Hodge numbers can provide insights into the relationship between tropical and complex geometry

Tropical limits and initial degenerations

• The tropical limit of a toric degeneration is closely related to the initial degeneration of the family of varieties
• The initial degeneration is a flat degeneration obtained by taking the initial ideal of the defining equations with respect to a weight vector
• The combinatorial structure of the tropical limit often encodes information about the initial degeneration, such as the Newton polytope of the defining equations

Combinatorial data in toric degenerations

• Toric degenerations provide a rich source of combinatorial data, such as fans, polytopes, and polyhedral complexes
• This combinatorial data can be used to study various properties of the original complex variety, such as its topology, Hodge theory, and enumerative geometry
• The interplay between the combinatorics of toric degenerations and the geometry of complex varieties is a central theme in tropical Hodge theory

Applications of tropical Hodge theory and toric degenerations

• Tropical Hodge theory and toric degenerations have found numerous applications in various areas of mathematics, including mirror symmetry, enumerative geometry, and moduli theory
• These applications demonstrate the power of combinatorial methods in studying complex geometric objects
• The interplay between tropical geometry, Hodge theory, and toric methods has led to new insights and results in these areas

Mirror symmetry and tropical geometry

• Mirror symmetry predicts a duality between certain pairs of Calabi-Yau varieties, exchanging complex and symplectic geometry
• Tropical geometry has been used to construct mirror pairs and study their properties, using the combinatorial structure of tropical varieties
• The SYZ conjecture, which relates mirror symmetry to special Lagrangian fibrations, has a tropical analog involving tropical Lagrangian fibrations

Enumerative geometry and Gromov-Witten invariants

• Enumerative geometry studies the counting of geometric objects, such as curves or subvarieties, satisfying certain conditions
• Gromov-Witten invariants are enumerative invariants that count holomorphic curves in a variety or symplectic manifold
• Tropical geometry has been used to compute Gromov-Witten invariants by counting tropical curves and relating them to their classical counterparts (e.g., Mikhalkin's correspondence theorem)

Moduli spaces and compactifications

• Moduli spaces are spaces that parameterize geometric objects, such as curves, surfaces, or vector bundles
• Compactifications of moduli spaces are important for studying their global geometry and for defining enumerative invariants
• Tropical geometry and toric degenerations have been used to construct compactifications of moduli spaces (e.g., the moduli space of tropical curves as a compactification of the moduli space of algebraic curves)