Math 2.0

Several complex variables and analytic spaces Definition

q-convex manifold

Builds on Complex manifoldLevi formCompact setSmooth function

Definition

Let XX be a complex manifold of complex dimension nn, and let qq be an integer with 1≤q≤n1 \le q \le n. Let φ ⁣:X→ℝ\varphi \colon X \to \R be a smooth function. It is strongly qq-convex at a point xx if its Levi form at xx, in local holomorphic coordinates, has at least n−q+1n - q + 1 positive eigenvalues. The manifold XX is qq-convex if there are a smooth function φ ⁣:X→ℝ\varphi \colon X \to \R and a compact set K⊆XK \subseteq X such that:

  1. φ\varphi is an exhaustion: the set {x∈X:φ(x)<c}\set{x \in X : \varphi(x) < c} is relatively compact in XX for every c∈ℝc \in \R.
  2. φ\varphi is strongly qq-convex at every point of X∖KX \setminus K.

The manifold XX is qq-complete if this holds with K=∅K = \emptyset.

Remarks

This is the convention of Andreotti and Grauert; some authors shift the index by one. The number of positive eigenvalues of the Levi form does not depend on the choice of local coordinates. Example: ℂn\C^n is 11-complete with φ(z)=∣z∣2\varphi(z) = \abs{z}^2, whose Levi form ∣w∣2\abs{w}^2 has nn positive eigenvalues. Non-example: a compact complex manifold of positive dimension is qq-convex with K=XK = X, but it is not 11-complete. Indeed a strongly 11-convex function is strictly plurisubharmonic, so it has no maximum, while every continuous function on a compact manifold has one. The 11-complete manifolds are exactly the Stein manifolds, and the 11-convex manifolds are also called strongly pseudoconvex manifolds.

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To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Compact set, concept, no evidence levelCompact setconceptComplex manifold, concept, no evidence levelComplex manifoldconceptLevi form, concept, no evidence levelLevi formconceptSmooth function, concept, no evidence levelSmooth functionconceptq-convex manifold, definition, no evidence levelq-convex manifolddefinition
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  1. Compact set concept
  2. Complex manifold concept
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  4. Smooth function concept
  5. q-convex manifold definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Andreotti%E2%80%93Grauert_theorem
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Kinddefinition
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MSC32F10
Definesq-convex manifold
UsesComplex manifold, Levi form, Compact set, Smooth function

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