Oka–Grauert principle

Target entity: Oka–Grauert principle
Context triple: [Cartan theorems A and B, influenced, Oka–Grauert principle]

• A. Oka–Weil theorem
  The Oka–Weil theorem is a fundamental result in several complex variables that extends Runge’s approximation theorem by characterizing when holomorphic functions on certain compact sets in complex manifolds can be uniformly approximated by global holomorphic functions.
• B. Kodaira vanishing theorem
  The Kodaira vanishing theorem is a fundamental result in algebraic geometry that gives conditions under which certain cohomology groups of ample line bundles on smooth projective varieties vanish, with deep implications for the classification of complex manifolds.
• C. Cartan theorems A and B
  Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
• D. Bochner–Kodaira–Nakano identity
  The Bochner–Kodaira–Nakano identity is a fundamental formula in complex differential geometry relating the Laplacian on differential forms to curvature terms, with key applications to vanishing theorems and Hodge theory.
• E. Hirzebruch–Riemann–Roch theorem
  The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Oka–Grauert principle
Target entity description: The Oka–Grauert principle is a fundamental result in complex geometry asserting that, for certain complex manifolds, topological and holomorphic classification problems coincide, allowing topological data to determine holomorphic structures.

• A. Oka–Weil theorem
  The Oka–Weil theorem is a fundamental result in several complex variables that extends Runge’s approximation theorem by characterizing when holomorphic functions on certain compact sets in complex manifolds can be uniformly approximated by global holomorphic functions.
• B. Kodaira vanishing theorem
  The Kodaira vanishing theorem is a fundamental result in algebraic geometry that gives conditions under which certain cohomology groups of ample line bundles on smooth projective varieties vanish, with deep implications for the classification of complex manifolds.
• C. Cartan theorems A and B
  Cartan theorems A and B are fundamental results in complex analytic geometry that characterize coherent analytic sheaves on Stein spaces by guaranteeing the existence of enough global sections and the vanishing of higher cohomology.
• D. Bochner–Kodaira–Nakano identity
  The Bochner–Kodaira–Nakano identity is a fundamental formula in complex differential geometry relating the Laplacian on differential forms to curvature terms, with key applications to vanishing theorems and Hodge theory.
• E. Hirzebruch–Riemann–Roch theorem
  The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
• F. None of above. chosen