semicomplete

Semicomplete is a term used in complex analysis and dynamical systems to describe a weaker form of flow completeness for holomorphic vector fields. Specifically, a holomorphic vector field on a complex manifold is called semicomplete if its local flow exists for all nonnegative times, producing a holomorphic action of the time semigroup [0, ∞) on the manifold. This is weaker than complete, which requires a flow defined for all complex times and hence a holomorphic action of the full additive group (C, +).

Formally, let X be a holomorphic vector field on a complex manifold M. For each point p in M, there is a maximal integral curve gamma_p(t) solving gamma_p'(t) = X(gamma_p(t)) with gamma_p(0) = p, defined on some interval I_p containing 0. X is semicomplete if I_p contains [0, ∞) for every p, so the flow exists for all nonnegative times. Equivalently, for each t ≥ 0, there is a holomorphic self-map φ_t: M → M such that φ_0 = id and φ_{t+s} = φ_t ∘ φ_s wherever defined, giving a holomorphic semigroup action.

Examples illustrate the distinction. The vector field X(z) = z on the complex plane C is complete (and thus semicomplete), since its flow z(t) = z(0) e^{t} is defined for all t ∈ R. By contrast, X(z) = z^2 has trajectories that blow up in finite positive time, so it is not semicomplete. On compact complex manifolds, holomorphic vector fields are complete, hence semicomplete.

Semicompleteness plays a role in the study of holomorphic dynamics and foliations, helping to distinguish local behavior near singularities from global behavior and to organize flows into holomorphic semigroups rather than full groups. See also complete vector field, holomorphic flow, and holomorphic foliation.

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