Sheaf Cohomology 10: Fine Sheaves, de Rham, and Dolbeault

Sheaf cohomology / 10

Fine sheaves, de Rham, and Dolbeault

Partitions of unity make smooth sheaves acyclic; the Poincare lemma then converts local exactness into global cohomology.

Chapter 12 of the PDF explains why differential forms compute topological and holomorphic cohomology. The mechanism is always the same: resolve a difficult sheaf by fine sheaves, then take global sections.

First picture: closed forms modulo exact forms

Before sheaves enter, de Rham cohomology is a quotient of differential forms:

\[H^k_{\mathrm{dR}}(X)= {\ker(d:\mathcal A^k(X)\to\mathcal A^{k+1}(X)) \over \operatorname{im}(d:\mathcal A^{k-1}(X)\to\mathcal A^k(X))}.\]

The numerator consists of closed forms: forms with no local differential defect, so \(d\omega=0\). The denominator consists of exact forms: forms that are globally derivatives, so \(\omega=d\eta\). Since \(d^2=0\), every exact form is closed. De Rham cohomology asks whether the converse holds globally.

On a small coordinate ball, the Poincare lemma says that every closed positive-degree form is exact; in degree zero, closed functions are locally constant. Therefore de Rham cohomology is not measuring local calculus. It is measuring global information: in positive degree, the failure of local primitives to glue; in degree zero, the locally constant choices on connected components.

Example 1: degree zero cohomology

A 0-form is a smooth function. It is closed exactly when

\[df=0.\]

On each connected component this means \(f\) is constant. Since there are no \((-1)\)-forms, there are no exact 0-forms to quotient by. Thus

\[H^0_{\mathrm{dR}}(X)\cong \{\text{locally constant real functions on }X\}.\]

For a connected manifold, \(H^0_{\mathrm{dR}}(X)\cong\mathbb R\).

Example 2: punctured plane and winding

On \(\mathbb C^*\), write \(z=x+iy\). The 1-form

\[\alpha={-y\,dx+x\,dy\over x^2+y^2}\]

is closed. Locally it is \(d\theta\), the derivative of the polar angle. It is not globally exact, because

\[\int_{|z|=1}\alpha=2\pi.\]

If \(\alpha=df\) globally, Stokes or the fundamental theorem for line integrals would force every integral over a closed loop to vanish. The nonzero period records the hole in \(\mathbb C^*\).

Example 3: area on a closed surface

Let \(X\) be a compact oriented surface with area form \(\omega\). Since \(\omega\) is top degree, \(d\omega=0\) automatically. But \(\omega\) is not exact if

\[\int_X\omega\ne0,\]

because an exact top-degree form \(d\eta\) integrates to \(\int_{\partial X}\eta=0\) on a closed surface. Thus \(H^2_{\mathrm{dR}}(X)\) detects the fundamental class.

This quotient viewpoint is the concrete side of the theorem below. The sheaf-theoretic side explains why this quotient computes \(H^k(X,\underline{\mathbb R})\), the cohomology of the constant sheaf.

Fine sheaves

A sheaf \(\mathcal F\) on a paracompact space is fine if partitions of unity act on it strongly enough to localize sections. For smooth manifolds, the sheaves

\[\mathcal A^k_X\]

of smooth \(k\)-forms are fine because smooth bump functions multiply forms. Fine sheaves are acyclic:

\[H^q(X,\mathcal A^k_X)=0\qquad(q>0).\]

Example 4: smooth functions

The sheaf \(\mathcal C^\infty_X\) is fine. Given a locally finite open cover and a partition of unity \(\{\rho_i\}\) subordinate to it, a local function can be cut off by \(\rho_i\) and extended by zero. This is exactly the operation that fails holomorphically: holomorphic bump functions do not exist on connected complex manifolds.

Example 5: smooth forms

If \(\omega_i\in\mathcal A^k(U_i)\) are local forms, then \(\rho_i\omega_i\) extends smoothly by zero to \(X\). This support control is why smooth form sheaves are acyclic and why de Rham theory can be globalized by local computations.

Poincare lemma and de Rham resolution

The Poincare lemma says that on a sufficiently small ball,

\[d\omega=0,\quad k>0 \qquad\Longrightarrow\qquad \omega=d\eta.\]

Equivalently, the sequence of sheaves

\[0\to\underline{\mathbb R}\to \mathcal A^0\xrightarrow{d}\mathcal A^1\xrightarrow{d} \mathcal A^2\xrightarrow{d}\cdots\]

is exact. Since the \(\mathcal A^k\) are fine, this is an acyclic resolution of \(\underline{\mathbb R}\). Therefore

\[H^q(X,\underline{\mathbb R}) \cong H^q_{\mathrm{dR}}(X).\]

This is the de Rham theorem in sheaf language.

Example 6: the circle

On \(S^1=\mathbb R/2\pi\mathbb Z\), the invariant 1-form usually written \(d\theta\) is locally exact but not globally exact on the circle. Its integral around \(S^1\) is \(2\pi\). Sheaf-theoretically, local primitives fail to glue by a locally constant Cech 1-cocycle, producing \(H^1(S^1,\mathbb R)\cong\mathbb R\).

Example 7: the two-sphere

The area form \(\omega_{S^2}\) is closed and represents a generator of \(H^2(S^2,\mathbb R)\) after normalization. It cannot be exact because

\[\int_{S^2}\omega_{S^2}\ne0,\]

while the integral of an exact 2-form over a closed surface would be zero by Stokes.

Mayer-Vietoris from forms

For \(X=U\cup V\), the short exact sequence of form complexes

\[0\to\mathcal A^\bullet(X)\to \mathcal A^\bullet(U)\oplus\mathcal A^\bullet(V)\to \mathcal A^\bullet(U\cap V)\to0\]

leads to the Mayer-Vietoris long exact sequence in de Rham cohomology. The connecting map is built by taking a form on the overlap, splitting it with a partition of unity, and differentiating the pieces.

Dolbeault resolution

On a complex manifold, smooth complex forms decompose into type \((p,q)\). The operator

\[\bar\partial:\mathcal A^{p,q}\to\mathcal A^{p,q+1}\]

satisfies \(\bar\partial^2=0\). The \(\bar\partial\)-Poincare lemma gives an exact sequence

\[0\to\Omega^p_X\to \mathcal A^{p,0}\xrightarrow{\bar\partial} \mathcal A^{p,1}\xrightarrow{\bar\partial} \mathcal A^{p,2}\to\cdots,\]

where \(\Omega^p_X\) is the sheaf of holomorphic \(p\)-forms. Since the smooth form sheaves are fine,

\[H^q(X,\Omega^p_X)\cong H^{p,q}_{\bar\partial}(X).\]

Example 8: holomorphic functions on compact Riemann surfaces

For \(p=0\) on a compact Riemann surface,

\[H^0(X,\mathcal O_X)\]

is the kernel of \(\bar\partial:\mathcal A^{0,0}(X)\to\mathcal A^{0,1}(X)\). These are global holomorphic functions, hence constants when \(X\) is connected.

Example 9: \(H^1(X,\mathcal O_X)\)

The group \(H^1(X,\mathcal O_X)\) is represented by global \((0,1)\)-forms modulo \(\bar\partial\) of smooth functions:

\[H^1(X,\mathcal O_X)\cong \mathcal A^{0,1}(X)/ \bar\partial\mathcal A^{0,0}(X).\]

For a genus \(g\) compact Riemann surface, this vector space has dimension \(g\).

The lesson

Fine resolutions are what make analysis and sheaf theory meet. Local differential equations provide exactness; partitions of unity provide acyclicity; global cohomology is the finite-dimensional remainder that cannot be solved away.