DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Compatible complex structure on a symplectic vector space

Definition

Let $(V, ω)$ be a symplectic vector space. A complex structure on $V$ is a real-linear endomorphism $J$ satisfying $J^{2} = - {id}_{V}$ . It is compatible with $ω$ if

$g_{J} (u, v) : = ω (u, J v)$

is a real inner product: it is symmetric and positive definite.

Compatibility implies $ω (J u, J v) = ω (u, v)$ and $g_{J} (J u, J v) = g_{J} (u, v)$ . Thus $J$ preserves both the symplectic form and its associated metric. The definition includes the zero vector space.

Depends on

Symplectic vector space

Used by

A compatible almost-complex structure that is not integrable Counterexample
Compatible almost-Kähler metric Definition
A compatible complex structure on standard symplectic space Example
Compatible complex structures exist on symplectic vector spaces Theorem
Every symplectic manifold admits a compatible almost-complex structure Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)