Kähler Geometry¶

Job: Kähler metrics, Ricci curvature, the Calabi–Yau theorem, and a one-line Yau–Tian–Donaldson pointer — analytic PDE methods on complex manifolds.

Core¶

Kähler metrics¶

A Hermitian metric $g$ on a complex manifold $X$ has associated real $(1,1)$-form $$ \omega(X,Y)=g(JX,Y). $$ Kähler: $d\omega=0$. Locally (Kähler potential) $\omega=\frac{i}{2}\partial\bar\partial\varphi$ on contractible charts (more precisely $\omega=\omega_0+i\partial\bar\partial\varphi$ in a fixed class).

Equivalent conditions (local)	Notes
$d\omega=0$	closed fundamental form
$\nabla^{\mathrm{LC}} J=0$	Levi-Civita preserves complex structure
Holonomy $\subset\mathrm{U}(n)$	Riemannian holonomy reduction
Christoffel $\Gamma_{ij}^k$ only (no mixed)	holomorphic normal coords

Kähler class $[\omega]\in H^{1,1}(X,\mathbb{R})\cap H^2(X,\mathbb{R})$. Volume $\int_X\omega^n/n!$.

Examples. Fubini–Study on $\mathbb{P}^n$; flat metrics on tori; induced metrics on complex submanifolds of Kähler manifolds (restriction of $\omega$ remains closed); hyperkähler metrics (holonomy in $\mathrm{Sp}(n)$).

Curvature and Ricci¶

Chern connection on $T^{1,0}$ coincides with Levi-Civita for Kähler metrics. Riemann curvature tensor with symmetries; Ricci form $$ \operatorname{Ric}(\omega)=-i\partial\bar\partial\log\det(g_{i\bar j}) $$ is a closed real $(1,1)$-form representing $2\pi c_1(X)$ (first Chern class of $T_X$ / of $K_X^{-1}$).

Notion	Definition
Ricci-flat	$\operatorname{Ric}(\omega)=0$
Kähler–Einstein	$\operatorname{Ric}(\omega)=\lambda\omega$
Scalar curvature $S$	trace of Ricci; Calabi functional $\int S^2$

Global $\partial\bar\partial$-lemma $\Rightarrow$ any representative of $c_1(X)$ is $\operatorname{Ric}(\omega_0)+i\partial\bar\partial f$ for a potential $f$, reducing KE / Ricci-flat equations to Monge–Ampère PDEs on potentials.

Calabi–Yau theorem¶

Theorem (Yau; Calabi conjecture). Let $X$ be compact Kähler, $[\omega]$ a Kähler class, and $\rho$ any closed real $(1,1)$-form with $[\rho]=2\pi c_1(X)$. Then there exists a unique Kähler metric $\omega'\in[\omega]$ with $\operatorname{Ric}(\omega')=\rho$.

Corollary (Ricci-flat metrics). If $c_1(X)=0$ in $H^2(X,\mathbb{R})$, every Kähler class contains a unique Ricci-flat Kähler metric. Such $X$ with holonomy in $\mathrm{SU}(n)$ (simply connected, torsion-free $K_X$) are Calabi–Yau manifolds in the differential-geometric sense.

Proof outline: reformulate as complex Monge–Ampère $$ (\omega+i\partial\bar\partial\varphi)^n=e^{F+\mathrm{const}}\omega^n, $$ a priori $C^0$ (Yau), $C^2$, then Evans–Krylov / bootstrapping; continuity method in the space of potentials.

Kähler–Einstein and YTD (one-liner)¶

Kähler–Einstein: $\operatorname{Ric}(\omega)=\lambda\omega$. Sign of $\lambda$ ties to the sign of $c_1$: - $\lambda<0$ ($c_1<0$, general type): Aubin–Yau existence. - $\lambda=0$: Calabi–Yau as above. - $\lambda>0$ (Fano): existence is obstructed; Yau–Tian–Donaldson: on smooth Fano manifolds, existence of KE metrics $\Leftrightarrow$ K-polystability (algebro-geometric GIT/stability of $\mathbb{Q}$-Fano varieties) — proved by Chen–Donaldson–Sun (2012–2015).

Bundle geometry (brief)¶

Hermitian–Yang–Mills metrics on holomorphic vector bundles: Donaldson–Uhlenbeck–Yau theorem relates HYM to slope polystability (again a stability $\Leftrightarrow$ Hermitian metric PDE dictionary).

See it¶

Quintic threefold. Smooth hypersurface $X\subset\mathbb{P}^4$ of degree $5$ has $K_X\cong\mathcal{O}_X$ by adjunction, hence $c_1(X)=0$. Yau’s theorem supplies a Ricci-flat Kähler metric in each Kähler class (here $H^{1,1}$ is $1$-dimensional, spanned by $\mathcal{O}(1)|_X$). Mirror symmetry predicts Hodge number swaps $h^{2,1}\leftrightarrow h^{1,1}$ with a mirror family — the Ricci-flat metric is the geometric input for $N=2$ string compactifications, though almost never explicit in coordinates.

Fubini–Study is Kähler–Einstein. $\operatorname{Ric}(\omega_{\mathrm{FS}})=(n+1)\omega_{\mathrm{FS}}$ on $\mathbb{P}^n$; positive KE, and $\mathbb{P}^n$ is K-stable (as a Fano).

Check¶

Why does $c_1(X)=0$ not alone produce a Ricci-flat metric without a Kähler class?

The Calabi–Yau theorem deforms within a fixed Kähler class. Non-Kähler complex manifolds with $c_1=0$ (some non-Kähler Calabi–Yau-type constructions) need different analytic setups; the Monge–Ampère method as stated uses a background Kähler form $\omega$.

Sketch why $\operatorname{Ric}(\omega)$ represents $2\pi c_1(X)$.

Transition functions of $K_X^{-1}=\det T^{1,0}$ are $\det(\partial z'/\partial z)$; the Chern curvature of the induced metric $\det g$ is $-\partial\bar\partial\log\det g$, and Chern–Weil gives $c_1=(i/2\pi)$ times the curvature form of $\det T^{1,0}$.

Pitfalls¶

Calabi–Yau is overloaded: trivial canonical bundle / $c_1=0$ / holonomy $\mathrm{SU}(n)$ / existence of Ricci-flat metric — related but not identical without compactness and Kähler assumptions.
Uniqueness of Ricci-flat metrics is within a fixed Kähler class; different classes give different metrics.
Singular KE / Kähler–Einstein currents on $\mathbb{Q}$-Fano varieties need pluripotential theory beyond smooth YTD.
Hyperkähler vs Calabi–Yau threefolds: holonomy $\mathrm{Sp}(n)$ vs $\mathrm{SU}(n)$; $h^{2,0}$ distinguishes.

Refs¶

Besse, Einstein Manifolds
Joyce, Compact Manifolds with Special Holonomy
Huybrechts, Complex Geometry
Yau, “On the Ricci curvature of a compact Kähler manifold…”
Chen–Donaldson–Sun, Kähler–Einstein metrics on Fano manifolds (J. AMS)

Next¶

Foundations · CG index · SCV · SCV / analytic methods — the Bergman kernel / Bergman metric, the Kodaira embedding, the Tian–Yau / Zelditch · AG foundations · PDE foundations · Mathematical physics · Mirror symmetry / GW — Calabi–Yau / Kähler geometry as the B-model input to mirror symmetry · Higher Teichmüller theory — Higgs bundles / non-abelian Hodge theory live on Kähler moduli · Geometric Langlands — the Hitchin system / opers as a section of the Hitchin fibration · Elliptic PDE — the Calabi–Yau / Monge–Ampère PDE as fully nonlinear elliptic (Caffarelli)