User:Tazerenix/∂∂̅-Lemma

In complex geometry, the $$\partial \bar \partial$$-lemma (sometimes called the $$dd^c$$-lemma) is a mathematical lemma about the de Rham cohomology class of a complex differential form. The $$\partial \bar \partial$$-lemma is a result of Hodge theory on a compact Kähler manifold. It asserts that if $$(X,\omega)$$ is a compact Kähler manifold and $$\alpha \in \Omega^{p,q}(X)$$ is a complex differential form of bidegree (p,q) whose class $$[\alpha] \in H_{dR}^{p+q}(X,\mathbb{C})$$ is zero in de Rham cohomology, then there exists a form $$\beta\in \Omega^{p-1,q-1}(X)$$ of bidegree (p-1,q-1) such that $$\alpha = i\partial \bar \partial \beta,$$where $$\partial$$ and $$\bar \partial$$ are the Dolbeault operators of the complex manifold $$X$$. The form $$\beta$$ is called the $$\partial \bar \partial$$-potential of $$\alpha$$. The inclusion of the factor $$i$$ ensures that $$i\partial \bar \partial$$ is a real differential operator, that is if $$\alpha$$ is a differential form with real coefficients, then so is $$\beta$$.

This lemma should be compared to the Poincaré lemma of de Rham cohomology, which applies to any smooth manifold but states the weaker result that if $$\alpha$$ is a closed differential k-form whose class is zero in de Rham cohomology, then $$\alpha = d\gamma$$ for some differential (k-1)-form $$\gamma$$ called the $$d$$-potential (or just potential) of $$\alpha$$, where $$d$$ is the exterior derivative. Indeed since the Dolbeault operators sum to give the exterior derivative $$d = \partial + \bar \partial$$ and square to give zero $$\partial^2 = \bar \partial^2 = 0$$, the $$\partial \bar \partial$$-lemma implies that $$\gamma = \bar \partial \beta $$, therefore refining the Poincaré lemma by relating the $$d$$-potential to the $$\partial \bar \partial$$-potential in the setting of compact Kähler manifolds.

Sometimes this lemma is known as the $$dd^c$$-lemma due to the use of a related operator $$d^c = -\frac{i}{2}(\partial - \bar \partial)$$. The relation between the two operators is $$i\partial \bar \partial = dd^c$$ and so $$\alpha = dd^c \beta$$.

The most significant consequence of the $$\partial \bar \partial$$-lemma occurs when the complex differential form has bidegree (1,1). In this case the lemma states that an exact differential form $$\omega \in \Omega^{1,1}(X)$$ has a $$\partial \bar \partial$$-potential given by a smooth function $$f\in C^{\infty}(X,\mathbb{C})$$:

$$\alpha = i\partial \bar \partial f.$$In particular this occurs in the case where $$\alpha = \omega$$ is a Kähler form restricted to a small open subset $$U \subset X$$ of a Kähler manifold, where the aforementioned Poincaré lemma ensures that it is an exact differential form. This leads to the notion of a Kähler potential. Another important case is when $$\alpha = \omega - \omega'$$ is the difference of two Kähler forms which are in the same de Rham cohomology class $$[\omega] = [\omega']$$. In this case $$[\alpha] = [\omega] - [\omega'] = 0$$ in de Rham cohomology so the $$\partial \bar \partial$$-lemma applies. By allowing Kähler forms to be completely described using a single function (the Kähler potential) which is automatically a plurisubharmonic function, the study of compact Kähler manifolds can be undertaken using techniques of pluripotential theory and convex analysis, for which many analytical tools are available. For example the $$\partial \bar \partial$$-lemma is used to rephrase the Kähler–Einstein equation in terms of potentials, transforming it into a complex Monge–Ampère equation for the Kähler potential. Variational techniques in convex analysis can then be used to prove existence of solutions to the equation under certain conditions.

Complex manifolds which are not necessarily Kähler but still happen to satisfy the $$\partial \bar \partial$$-lemma are known as $$\partial \bar \partial$$-manifolds. For example compact complex manifolds which are Fujiki class C satisfy the $$\partial \bar \partial$$-lemma but are not necessarily Kähler.

Proof
The $$\partial \bar \partial$$-lemma is a consequence of Hodge theory applied to a compact Kähler manifold.

The Hodge theorem for an elliptic complex may be applied to any of the operators $$d, \partial, \bar \partial$$ and respectively to their Laplace operators $$\Delta_d, \Delta_{\partial}, \Delta_{\bar \partial}$$. To these operators one can define spaces of harmonic differential forms given by the kernels:

$$\begin{align} \mathcal{H}_d^k &= \ker \Delta_d : \Omega^k(X) \to \Omega^k(X)\\ \mathcal{H}_{\partial}^{p,q} &= \ker \Delta_{\partial}: \Omega^{p,q}(X) \to \Omega^{p,q}(X)\\ \mathcal{H}_{\bar \partial}^{p,q} &= \ker \Delta_{\bar \partial}: \Omega^{p,q}(X) \to \Omega^{p,q}(X)\\ \end{align} $$The Hodge decomposition theorem asserts that there are three orthogonal decompositions associated to these spaces of harmonic forms, given by

$$\begin{align} \Omega^k(X) &= \mathcal{H}_d^k \oplus \mathrm{im}\, d \oplus \mathrm{im}\, d^*\\ \Omega^{p,q}(X) &= \mathcal{H}_{\partial}^{p,q} \oplus \mathrm{im}\, \partial \oplus \mathrm{im}\, \partial^*\\ \Omega^{p,q}(X) &= \mathcal{H}_{\bar \partial}^{p,q} \oplus \mathrm{im}\, \bar \partial \oplus \mathrm{im}\, \bar \partial^* \end{align} $$where $$d^*, \partial^*, \bar \partial^* $$ are the formal adjoints of $$d,\partial, \bar\partial $$ with respect to the Riemannian metric of the Kähler manifold, respectively. These decompositions hold separately on any compact complex manifold. The importance of the manifold being Kähler is that there is a relationship between the Laplacians of $$d,\partial,\bar \partial $$ and hence of the orthogonal decompositions above. In particular on a compact Kähler manifold $$\Delta_d = 2 \Delta_{\partial} = 2 \Delta_{\bar \partial} $$which implies an orthogonal decomposition $$\mathcal{H}_d^k = \bigoplus_{p+q=k} \mathcal{H}_{\partial}^{p,q} = \bigoplus_{p+q=k} \mathcal{H}_{\bar \partial}^{p,q}   $$where we have the further relations $$\mathcal{H}_{\partial}^{p,q} = \overline{\mathcal{H}_{\bar \partial}^{q,p}}  $$ relating the spaces of $$\partial $$ and $$\bar \partial $$-harmonic forms.

As a result of the above decompositions, one can prove the following lemma.

Indeed let $$\alpha\in \Omega^{p,q}(X) $$ be a closed (p,q)-form on a compact Kähler manifold $$(X,\omega)$$. It follows quickly that (d) implies (a), (b), and (c). Moreover the orthogonal decompositions above imply that any of (a), (b), or (c) imply (e). Therefore the main difficulty is to show that (e) implies (d).

To that end, suppose that $$\alpha$$ is orthogonal to the subspace $$\mathcal{H}_{\bar \partial}^{p,q} \subset \Omega^{p,q}(X)$$. Then we have $$\alpha \in \mathrm{im}\, \bar \partial \oplus \mathrm{im}\, \bar \partial^*$$. Since <math since="" $$\alpha$$ is $$d$$-closed and $$d=\partial + \bar \partial$$, it is also $$\bar \partial$$-closed (that is $$\bar \partial \alpha = 0$$). If $$\alpha = \alpha' + \alpha$$ where $$\alpha' \in \mathrm{im}\, \bar \partial$$ and $$\alpha = \bar \partial^* \gamma$$ is contained in $$\mathrm{im}\, \bar \partial^*$$ then since this sum is from an orthogonal decomposition with respect to the inner product $$\langle -, - \rangle$$ induced by the Riemannian metric, we must have $$\langle \alpha, \alpha\rangle = \langle \alpha, \alpha \rangle = \langle \alpha, \bar \partial^* \gamma \rangle = \langle \bar \partial \alpha, \gamma \rangle = 0$$ or in other words $$\|\alpha\|^2 = 0$$ and $$\alpha'' = 0$$. Thus it is the case that $$\alpha=\alpha'\in \mathrm{im}\, \bar \partial$$. This allows us to write $$\alpha = \bar \partial \eta$$ for some differential form $$\eta \in \Omega^{p,q-1}(X)$$. Applying the Hodge decomposition for $$\partial$$ to $$\eta$$, we have

$$\eta = \eta_0 + \partial \eta' + \partial^* \eta$$where $$\eta_0$$ is $$\Delta_\partial$$-harmonic, $$\eta'\in \Omega^{p-1,q-1}(X)$$ and $$\eta \in \Omega^{p+1,q-1}(X)$$. The equality $$\Delta_\bar \partial = \Delta_\partial$$ implies that $$\eta_0$$ is also $$\Delta_{\bar \partial}$$-harmonic and therefore $$\bar \partial \eta_0 = \bar \partial^* \eta_0 = 0$$. Thus we have $$\alpha = \bar \partial \partial \eta' + \bar \partial \partial^* \eta''$$. However since $$\alpha$$ is $$d$$-closed, it is also $$\partial$$-closed. Then using a similar trick to above, we observe $$\langle \bar \partial \partial^* \eta, \bar \partial \partial^* \eta\rangle = \langle \alpha, \bar \partial \partial^* \eta \rangle = - \langle \alpha, \partial^* \bar \partial \eta \rangle = - \langle \partial \alpha, \bar \partial \eta'' \rangle = 0$$where we have used the Kähler identity that $$\bar \partial \partial^* = -\partial^* \bar \partial $$. Thus we conclude $$\alpha = \bar \partial \partial \eta'$$ and setting $$\beta = i \eta' $$ produces the $$\partial \bar \partial$$-potential.

Bott–Chern cohomology
The Bott–Chern cohomology is a cohomology theory for compact complex manifolds which depends on the operators $$\partial$$ and $$\bar \partial$$, and measures the extent to which the $$\partial \bar \partial$$-lemma fails to hold. In particular when a compact complex manifold is a Kähler manifold, the Bott–Chern cohomology is isomorphic to the Dolbeault cohomology, but in general it contains more information.

The Bott–Chern cohomology groups of a compact complex manifold are defined by$$H_{BC}^{p,q}(X) = \frac{ \ker (\partial: \Omega^{p,q} \to \Omega^{p+1,q}) \cap \ker (\bar \partial: \Omega^{p,q} \to \Omega^{p,q+1})}{\mathrm{im}\, (\partial \bar \partial: \Omega^{p-1,q-1} \to \Omega^{p,q})}.$$Since a differential form which is both $$\partial$$ and $$\bar \partial$$-closed is $$d$$-closed, there is a natural map $$H_{BC}^{p,q}(X) \to H_{dR}^{p+q}(X,\mathbb{C})$$ from Bott–Chern cohomology groups to de Rham cohomology groups. There are also maps to the $$\partial$$ and $$\bar \partial$$ Dolbeault cohomology groups $$H_{BC}^{p,q}(X) \to H_{\partial}^{p,q}(X), H_{\bar \partial}^{p,q}(X)$$. When the manifold $$X$$ satisfies the $$\partial \bar \partial$$-lemma, for example if it is a compact Kähler manifold, then the above maps from Bott–Chern cohomology to Dolbeault cohomology are isomorphisms, and furthermore the map from Bott–Chern cohomology to de Rham cohomology is injective. As a consequence, there is an isomorphism $$H_{dR}^{k}(X,\mathbb{C}) = \bigoplus_{p+q=k} H_{BC}^{p,q}(X)$$ whenever $$X$$ satisfies the $$\partial \bar \partial$$-lemma. In this way, the kernel of the maps above measure the failure of the manifold $$X$$ to satisfy the lemma, and in particular measure the failure of $$X$$ to be a Kähler manifold.