[Paper Review] Kodaira dimensions of almost complex manifolds II

Haojie Chen, Weiyi Zhang|arXiv (Cornell University)|Apr 27, 2020

Geometry and complex manifolds54 references18 citations

TL;DR

This paper introduces a second version of Kodaira dimension and Iitaka dimension for compact almost complex manifolds using pseudoholomorphic pluricanonical maps, proving that top Kodaira dimension implies integrability of the almost complex structure. It establishes structural results for 4-manifolds with Kodaira dimension one, generalizes vanishing theorems, and shows the almost Kodaira dimension is bounded above by the symplectic Kodaira dimension for tamed symplectic 4-manifolds via a probabilistic combinatorics argument.

ABSTRACT

This is the second of a series of papers where we study the plurigenera, the Kodaira dimension and the Iitaka dimension on compact almost complex manifolds. By using the pseudoholomorphic pluricanonical map, we define the second version of Kodaira dimension as well as Iitaka dimension on compact almost complex manifolds. We show the almost complex structures with the top Kodaira dimension are integrable. For compact almost complex 4-manifolds with Kodaira dimension one, we obtain elliptic fibration like structural description. Some vanishing theorems in complex geometry are generalized to the almost complex setting. For tamed symplectic $4$-manifolds, we show that the almost Kodaira dimension is bounded above by the symplectic Kodaira dimension, by using a probabilistic combinatorics style argument. The appendix also contains a few results including answering a question of the second author that there is a unique subvariety in exceptional curve classes for irrational symplectic $4$-manifolds, as well as extending Evans-Smith's constraint on symplectic embeddings of certain rational homology balls by removing the assumption on the intersection form.

Motivation & Objective

• To define a second version of Kodaira and Iitaka dimensions on compact almost complex manifolds using the image dimension of pseudoholomorphic pluricanonical maps.
• To investigate the geometric and topological implications of high Kodaira dimension in the almost complex setting, especially in dimension 4.
• To generalize vanishing theorems from complex geometry to the almost complex category.
• To relate the almost Kodaira dimension to symplectic invariants, particularly for tamed symplectic 4-manifolds.
• To explore the behavior of these invariants on specific spaces like $\mathbb{C}P^n$, $S^6$, and homogeneous spaces $G/T$.

Proposed method

• Define the pseudoholomorphic pluricanonical map $\Phi_m: X \setminus B \to \mathbb{C}P^N$ using $(J,\mathcal{J})$-pseudoholomorphic sections of $\mathcal{K}_X^{\otimes m}$.
• Prove that the image of $\Phi_m$ is an open analytic subvariety in $\mathbb{C}P^N$, enabling the definition of the Iitaka dimension as the maximal complex dimension of the image.
• Use the structure of pseudoholomorphic maps and Hodge theory to show that $\kappa_J(X) = \dim_{\mathbb{C}} X$ implies integrability of $J$, via analysis of the pluricanonical system.
• Apply a probabilistic combinatorics-style argument to bound the almost Kodaira dimension $\kappa_J(X)$ by the symplectic Kodaira dimension $\kappa^s(X,\omega)$ for tamed symplectic 4-manifolds.
• Construct bundle almost complex structures on line bundles and define the $m$-th plurigenus $P_m(X,J)$ as the dimension of the space of pseudoholomorphic sections of $\mathcal{K}_X^{\otimes m}$.
• Study the Iitaka dimension for line bundles on homogeneous spaces $G/T$ by decomposing the root system into non-integrable almost complex structures via non-standard root decompositions $R = R_1 \sqcup R_2$ with $R_1 = -R_2$.

Experimental results

Research questions

• RQ1 Does the second version of Kodaira dimension, defined via the image dimension of the pluricanonical map, detect integrability of the almost complex structure when it achieves the maximal value?
• RQ2 Can the almost Kodaira dimension be bounded above by the symplectic Kodaira dimension in tamed symplectic 4-manifolds, and what techniques enable such a bound?
• RQ3 What is the behavior of the Iitaka dimension for the anti-canonical bundle on non-integrable almost complex structures on $G/T$ constructed from non-integrable root decompositions?
• RQ4 Are the Kodaira and Iitaka dimensions birational invariants for compact almost complex 4-manifolds under pseudoholomorphic birational maps?
• RQ5 Do the invariants $\kappa_J$, $\kappa_{J}$, and $h^{p,0}$ on $M \times S^1$ induce meaningful invariants for almost contact structures on $3$-manifolds?

Key findings

• If the Kodaira dimension $\kappa_J(X)$ of a compact almost complex manifold $(X,J)$ equals $\dim_{\mathbb{C}} X$, then $J$ is integrable.
• For compact almost complex 4-manifolds with Kodaira dimension one, the pluricanonical map induces a pseudoholomorphic elliptic fibration structure over a complex manifold.
• The almost Kodaira dimension $\kappa_J(X)$ is bounded above by the symplectic Kodaira dimension $\kappa^s(X,\omega)$ for tamed symplectic 4-manifolds, proven via a probabilistic combinatorics-style argument.
• The plurigenera $P_m(X,J)$ are birational invariants for compact almost complex 4-manifolds, implying $\kappa_J(X)$ is also a birational invariant in dimension four.
• For irrational symplectic 4-manifolds, there exists a unique subvariety in each exceptional curve class, answering a question of the second author.
• The Iitaka dimension of the anti-canonical bundle on $G/T$ with non-integrable almost complex structures (from non-standard root decompositions) can be computed, extending Evans-Smith’s constraint on symplectic embeddings to general intersection forms.