Complex Geometry: Foundations¶

Job: complex manifolds, holomorphic differential forms, Dolbeault cohomology, and the Hodge decomposition on compact Kähler manifolds; Kodaira embedding as the bridge to projective algebraic geometry.

Core¶

Complex manifolds¶

A complex manifold $X$ of dimension $n$ is a second-countable Hausdorff space with an atlas of charts to open sets in $\mathbb{C}^n$ whose transition maps are biholomorphic. Morphisms: holomorphic maps (holomorphic in charts).

Object	Notes
Holomorphic functions	$\mathcal{O}_X(U)=$ holo maps $U\to\mathbb{C}$
Holomorphic vector bundle	locally free $\mathcal{O}_X$-module of finite rank; $T_X=T^{1,0}X$
Almost complex structure	$J$ on $T_{\mathbb{R}}X$, $J^2=-\mathrm{id}$; integrable iff Newlander–Nirenberg $\Rightarrow$ complex manifold
Hermitian metric	$h$ on $T^{1,0}$; associated $(1,1)$-form $\omega$

Real underlying manifold has dimension $2n$. Orientation from the complex structure.

Examples. Open sets in $\mathbb{C}^n$; $\mathbb{P}^n_{\mathbb{C}}$; complex tori $\mathbb{C}^n/\Lambda$; smooth complex projective varieties (analytification); Hopf manifolds $(\mathbb{C}^n\setminus\{0\})/q^\mathbb{Z}$.

Holomorphic forms and bigraded forms¶

Decompose $\mathbb{C}\otimes T^*_{\mathbb{R}}X=(T^{1,0})^*\oplus(T^{0,1})^*$ and set $$ \mathcal{A}^{p,q}=\Gamma\Bigl(\bigwedge^p(T^{1,0})^\otimes\bigwedge^q(T^{0,1})^\Bigr),\qquad \mathcal{A}^k_{\mathbb{C}}=\bigoplus_{p+q=k}\mathcal{A}^{p,q}. $$ Exterior derivative splits $d=\partial+\bar\partial$ with $$ \partial:\mathcal{A}^{p,q}\to\mathcal{A}^{p+1,q},\qquad \bar\partial:\mathcal{A}^{p,q}\to\mathcal{A}^{p,q+1}, $$ $\partial^2=\bar\partial^2=0$, $\partial\bar\partial+\bar\partial\partial=0$.

Holomorphic $p$-forms: $\Omega_X^p=\bigwedge^p(T^{1,0})^*$ (locally free sheaf); sections are forms with only $dz_{i_1}\wedge\cdots\wedge dz_{i_p}$ coefficients holomorphic. Canonical bundle $K_X=\Omega_X^n=\det(T^{1,0})^*$.

Dolbeault cohomology¶

Dolbeault complex in fixed $p$: $(\mathcal{A}^{p,\bullet},\bar\partial)$. $$ H^{p,q}_{\bar\partial}(X):=H^q(\mathcal{A}^{p,\bullet},\bar\partial). $$

Theorem (Dolbeault). $H^{p,q}_{\bar\partial}(X)\cong H^q(X,\Omega_X^p)$ (sheaf cohomology), via the fine resolution $$ 0\to\Omega_X^p\to\mathcal{A}^{p,0}\xrightarrow{\bar\partial}\mathcal{A}^{p,1}\xrightarrow{\bar\partial}\cdots. $$ (Compare $\bar\partial$-Poincaré lemma: locally $\bar\partial$-closed $(p,q)$-forms with $q\ge 1$ are $\bar\partial$-exact.)

$n=1$	Meaning
$H^{0,0}$	global holom. functions (constants if compact connected)
$H^{0,1}$	$\cong H^1(X,\mathcal{O}_X)$; related to deformations / Picard for curves
$H^{1,0}$	holomorphic $1$-forms; dim $=g$ on compact Riemann surface

Hodge decomposition (compact Kähler)¶

A Hermitian metric is Kähler if $d\omega=0$ (equiv. $\nabla J=0$ for Chern/Levi-Civita). Compact Kähler $\Rightarrow$ Hodge theory for the Laplacian $\Delta_d=2\Delta_{\bar\partial}=2\Delta_\partial$.

Theorem (Hodge decomposition). $X$ compact Kähler: $$ H^k(X,\mathbb{C})\cong\bigoplus_{p+q=k}H^{p,q}(X),\qquad H^{p,q}\cong H^{p,q}_{\bar\partial}(X)\cong H^q(X,\Omega_X^p), $$ and $H^{p,q}=\overline{H^{q,p}}$. Hodge numbers $h^{p,q}=\dim H^{p,q}$ satisfy $h^{p,q}=h^{q,p}$, $b_k=\sum_{p+q=k}h^{p,q}$.

Hard Lefschetz / primitive decomposition (pointer). Lefschetz operator $L=\,\wedge\omega$ induces isomorphisms $L^{n-k}:H^k\to H^{2n-k}$; primitive cohomology and Hodge–Riemann bilinear relations.

$\partial\bar\partial$-lemma. On compact Kähler, a $d$-closed form that is $\partial$-exact (or $\bar\partial$-exact) is $\partial\bar\partial$-exact — key for Bott–Chern cohomology coinciding with Dolbeault and for moduli of complex structures.

Kodaira embedding (pointer)¶

Theorem (Kodaira). Compact complex $X$ is projective algebraic iff it admits a positive holomorphic line bundle $L$ (curvature form of a Hermitian metric on $L$ is a positive $(1,1)$-form — then $c_1(L)$ is a Kähler class). Equiv.: some power $L^{\otimes m}$ is very ample, embedding $X\hookrightarrow\mathbb{P}^N$.

Thus compact Kähler with a rational Kähler class in $H^{1,1}\cap H^2(X,\mathbb{Q})$ (Kodaira) is projective. Non-algebraic complex tori: Kähler but no positive line bundle.

Details of positivity and vanishing: Kodaira vanishing $H^q(X,K_X\otimes L)=0$ for $q>0$, $L$ positive — proof via Bochner / harmonic forms on Kähler manifolds.

See it¶

Complex tori vs abelian varieties. $X=\mathbb{C}^n/\Lambda$ always admits flat Kähler metrics. It is projective (an abelian variety) iff it admits a positive line bundle iff the lattice satisfies Riemann bilinear relations (a polarization exists). Hodge decomposition on $H^1$ is tautological ($H^{1,0}\cong\mathbb{C}^n$); the $(1,1)$ part of $H^2$ encodes possible polarizations — pure Hodge theory distinguishing algebraic from transcendental tori.

Fröhlicher spectral sequence. $E_1^{p,q}=H^{p,q}_{\bar\partial}\Rightarrow H^{p+q}_{\mathrm{dR}}$. Degenerates at $E_1$ for compact Kähler (Hodge); can fail for non-Kähler compact complex manifolds.

Check¶

Prove $h^{1,0}=h^{0,1}$ for compact Kähler $X$, and give a non-Kähler compact complex surface where Betti vs Hodge numerics warn you.

Hodge symmetry $h^{p,q}=h^{q,p}$ from conjugation on harmonic forms. Hopf surface $S^1\times S^3$ has $b_1=1$ odd, so cannot be Kähler (Kähler $\Rightarrow$ $b_1$ even, since $b_1=2h^{1,0}$).

Why does Dolbeault’s theorem need the $\bar\partial$-Poincaré lemma?

Sheaf resolution of $\Omega^p$ by the fine sheaves $\mathcal{A}^{p,\bullet}$ is exact precisely because every $\bar\partial$-closed form is locally $\bar\partial$-exact in positive antiholomorphic degree; fine/soft sheaves are acyclic for sheaf cohomology, so hypercohomology reduces to global-section cohomology of the complex.

Pitfalls¶

Hermitian $\neq$ Kähler in $\dim_{\mathbb{C}}\ge 2$; closedness of $\omega$ is a PDE constraint.
Analytic spaces with singularities: replace free $\Omega^p$ by Kähler differentials / dualizing sheaves; Hodge theory needs resolutions (e.g. rational singularities, HH filterings).
GAGA: coherent analytic cohomology on projective $X^{\mathrm{an}}$ matches algebraic — do not confuse $C^\infty$ Dolbeault representatives with algebraic Čech cocycles without a comparison theorem.
Almost complex Hodge numbers without integrability are not sheaf cohomology of $\Omega^p$.

Refs¶

Huybrechts, Complex Geometry
Voisin, Hodge Theory and Complex Algebraic Geometry I
Griffiths–Harris, Principles of Algebraic Geometry
Kodaira–Morrow, Complex Manifolds

Next¶

Kähler geometry · CG index · SCV foundations · SCV / Stein & Oka — the Stein manifolds, the Oka principle, the holomorphic convexity · Complex analysis · AG cohomology · Differential geometry

\(n=1\)	Meaning
\(H^{0,0}\)	global holom. functions (constants if compact connected)
\(H^{0,1}\)	\(\cong H^1(X,\mathcal{O}_X)\); related to deformations / Picard for curves
\(H^{1,0}\)	holomorphic \(1\)-forms; dim \(=g\) on compact Riemann surface