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Complex Geometry

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6 domain-specific abstractions whose origin domain is Complex Geometry.

  • Complex polytope — A regular incidence geometry modeled in complex unitary space that generalizes real regular polytopes through complex reflections and phase-valued incidence.
  • Holomorphic tangent bundle — The complex vector bundle of type-(1,0) tangent directions on a complex manifold, with holomorphic transition functions.
  • Positive form — A real differential form of Hodge type (p,p) satisfying a declared positivity cone, with several inequivalent notions in higher bidegree.
  • Quadratic differential — A section of the square of a Riemann surface’s holomorphic cotangent bundle, locally written as a coefficient times the square of a coordinate differential.
  • Siegel upper half-space — The complex manifold of symmetric g-by-g matrices with positive-definite imaginary part, a Hermitian symmetric domain on which the real symplectic group acts transitively and the degree-g setting for Siegel modular forms.
  • Witting polytope — Identify the regular self-dual complex four-dimensional polytope with 240 vertices, 2,160 edges, 2,160 faces, 240 cells, and the associated finite complex-reflection symmetry and incidence configuration.