Complex Geometry

Job: complex manifolds, holomorphic forms, and Hodge/Kähler structures — the differential-geometric side of complex algebraic geometry.

Core

A complex manifold is locally biholomorphic to open sets in \(\mathbb{C}^n\); the holomorphic tangent bundle and \(\bar\partial\)-complex organize forms and cohomology. On compact Kähler manifolds, Hodge theory decomposes de Rham cohomology into \((p,q)\)-types.

Layer Content
Complex manifolds atlases, holomorphic maps, \(\mathcal{O}_X\), vector bundles
Forms \(\partial,\bar\partial\), Dolbeault cohomology
Hodge \(H^k_{\mathrm{dR}}\cong\bigoplus_{p+q=k}H^{p,q}\) on compact Kähler
Kähler metrics Ricci, Calabi–Yau, YTD pointer

Reading map

Page Focus
Foundations complex manifolds, holomorphic forms, Dolbeault, Hodge, Kodaira embedding pointer
Kähler geometry Kähler metrics, Ricci, Calabi–Yau, YTD one-liner

Dependency order

  1. One-variable complex analysis — complex analysis.
  2. Several complex variables local theory — SCV (Hartogs, \(\bar\partial\), coherence).
  3. Differentiable manifolds / de Rham — diff geom, topology.
  4. Foundations \(\to\) Kähler; then algebraic methods via AG / GAGA.

Interfaces

Area Contact
Several complex variables local holomorphy, \(\bar\partial\), Oka–Cartan
Algebraic geometry projective \(\Rightarrow\) Kähler; Hodge structures on smooth proj. varieties
Complex analysis \(n=1\) case; Riemann surfaces
Mathematical physics Calabi–Yau, mirror symmetry pointers
Differential geometry Hermitian metrics, curvature

Notation (paper-standard)

  • \(T_X^{1,0}\), \(T_X^{0,1}\), \(\Omega_X^p=\bigwedge^p(T^{1,0})^*\), \(\mathcal{A}^{p,q}\)
  • \(\partial,\bar\partial\), \(H^{p,q}_{\bar\partial}(X)\), Hodge filtration \(F^p\)
  • \(\omega\) Kähler form, \(\operatorname{Ric}(\omega)\), \(c_1(X)\)

See it

Hodge diamond of a K3 surface. Compact complex surface with \(K_X\cong\mathcal{O}_X\) and \(H^1(\mathcal{O}_X)=0\): Hodge numbers \(h^{2,0}=1\), \(h^{1,1}=20\), \(h^{1,0}=0\) — a lattice-theoretic and moduli story driven by the Hodge structure on \(H^2\), not just topology. Every K3 is Kähler (Siu); projective ones are algebraic K3s in the AG sense.

Check

Why does every compact complex manifold of dimension \(1\) admit a Kähler metric?

Complex curves are automatically Kähler: any Hermitian metric has Kähler form a positive \((1,1)\)-form, and \(d\omega=0\) is automatic in real dimension \(2\) (or: every Riemann surface is projective/algebraic by... not quite, but compact Riemann surfaces are projective algebraic by Kodaira embedding after choosing a positive line bundle — existence of Hermitian metrics with \(d\omega=0\) is elementary in dim \(1\)).

What fails in the Hodge decomposition without the Kähler condition?

\(\partial\bar\partial\)-lemma and the identification \(H^{p,q}_{\bar\partial}\cong H^q(X,\Omega^p)\) with a direct sum decomposition of \(H^{p+q}_{\mathrm{dR}}\) can fail; Fröhlicher spectral sequence need not degenerate at \(E_1\). Non-Kähler compact complex manifolds (e.g. Hopf surfaces) have torsion in Hodge numbers vs Betti numbers mismatches.

Pitfalls

  • Complex manifold \(\neq\) complex analytic space — singular analytic spaces need coherent sheaves / SCV.
  • Kähler \(\neq\) projective — non-algebraic tori and general K3s exist; Kodaira embedding needs a positive line bundle.
  • Hodge decomposition is for the complexified de Rham cohomology of compact Kähler manifolds — not an arbitrary almost complex manifold.

Refs

  • Huybrechts, Complex Geometry
  • Voisin, Hodge Theory and Complex Algebraic Geometry I
  • Griffiths–Harris, Principles of Algebraic Geometry
  • Wells, Differential Analysis on Complex Manifolds

Next

Foundations · Kähler geometry · SCV · Complex analysis · AG index · Differential geometry