[Paper Review] Kodaira Dimension in Low Dimensional Topology

Tianjun Li|arXiv (Cornell University)|Nov 16, 2015

Algebraic Geometry and Number Theory77 references3 citations

TL;DR

This paper surveys the development of Kodaira dimension in low-dimensional topology, focusing on symplectic 4-manifolds and 3-manifolds since 2006. It introduces and refines the symplectic Kodaira dimension $κ^s$ and topological Kodaira dimension $κ^t$, establishing their behavior under fibrations, surgeries, and geometric decompositions, with key results on additivity, subadditivity, and classification of manifolds by dimension and curvature type.

ABSTRACT

This is a survey on the various notions of Kodaira dimension in low dimensional topology. The focus is on progress after the 2006 survey [78].

Motivation & Objective

• To update and extend the understanding of Kodaira dimension in low-dimensional topology, particularly for symplectic 4-manifolds and 3-manifolds, following the 2006 survey.
• To define and analyze the symplectic Kodaira dimension $\kappa^s$ for 4-manifolds using Taubes' Seiberg-Witten theory and relative invariants.
• To establish a topological Kodaira dimension $\kappa^t$ for 3-manifolds based on Thurston's geometric decomposition and the classification of geometric structures.
• To investigate the behavior of $\kappa^s$ and $\kappa^t$ under surgeries, fibrations, and fiber sum operations, and to explore their invariance and additivity properties.
• To extend the framework to relative and relative-like settings, including symplectic caps and manifolds with concave boundary, and to relate them to log Kodaira dimensions in algebraic geometry.

Proposed method

• Define $\kappa^s(M,\omega)$ for symplectic 4-manifolds using the sign and positivity of $K_\omega \cdot \omega$ and $(K_\omega)^2$, with values in $\{-\infty, 0, 1, 2\}$, based on Taubes' Seiberg-Witten theory.
• Introduce a relative Kodaira dimension $\kappa^s(M,\omega,F)$ for symplectic pairs $(M,\omega)$ with a maximal symplectic surface $F$ without sphere components, using the intersection and square of $K_\omega + [F]$.
• Apply the minimal model program to define $\kappa^s(M,\omega,F)$ via the unique minimal model of the pair, ensuring well-definedness via Lemma 5.1.
• Use the concept of uniruled and Calabi-Yau caps for symplectic 4-manifolds with concave boundary to define relative Kodaira dimensions and relate them to global invariants.
• Analyze the behavior of $\kappa^s$ under Luttinger surgery, genus 0 and positive genus fiber sums, and rational blow-downs, establishing subadditivity and additivity properties.
• Utilize the Yamabe invariant and geometric decomposition of 3-manifolds into Thurston geometries to define $\kappa^t(M^3)$ as $-\infty$, 0, or 1, depending on the presence of geometric types in the decomposition.

Experimental results

Research questions

• RQ1 How does the symplectic Kodaira dimension $\kappa^s$ behave under Luttinger surgery and other 4-manifold surgeries?
• RQ2 What is the relationship between the symplectic Kodaira dimension $\kappa^s$ and the topological Kodaira dimension $\kappa^t$ in 3- and 4-manifolds?
• RQ3 Can the relative Kodaira dimension $\kappa^s(M,\omega,F)$ be consistently defined for symplectic surfaces $F$ with pseudo-holomorphic singularities or weighted structures?
• RQ4 To what extent does the relative Kodaira dimension of a symplectic cap determine the global Kodaira dimension of the ambient closed manifold?
• RQ5 Is the relative Kodaira dimension $\kappa^s(M,\omega,F)$ equivalent to Iitaka's log Kodaira dimension in the context of affine algebraic surfaces?

Key findings

• The symplectic Kodaira dimension $\kappa^s(M,\omega)$ is well-defined and takes values in $\{-\infty, 0, 1, 2\}$, based on the sign and growth of $K_\omega \cdot \omega$ and $(K_\omega)^2$.
• For a maximal symplectic surface $F$ without sphere components, $\kappa^s(M,\omega,F) = -\infty$ if $(K_\omega + [F]) \cdot \omega < 0$ or $(K_\omega + [F])^2 < 0$, and increases to 2 when both intersection and square are positive.
• The relative Kodaira dimension satisfies $\kappa^s(M,\omega,F) \geq \kappa^s(M,\omega)$, and is invariant under homologous maximal surfaces.
• If $\kappa^s(M,\omega) = -\infty$ and $(K_\omega + [F])^2 > 0$, then $(K_\omega + [F]) \cdot \omega > 0$, so no such maximal surface can satisfy $K_\omega + [F] \cdot \omega = 0$.
• For a genus $h \geq 1$ $S^2$-bundle over a surface, $\kappa^s(M,\omega,F) = -\infty$ if $F$ is a section and $\kappa^s(M) = -\infty$.
• Uniruled caps (with $[c_1(P)] \cdot [\omega,\alpha] > 0$) imply $\kappa^s(M,\Omega) = -\infty$ for any closed manifold $M$ containing them, and Calabi-Yau caps (with torsion $c_1(P)$) imply $\kappa^s(M,\Omega) \leq 0$.