Integrable almost complex structure

An almost complex structure is integrable when it comes from holomorphic coordinate charts. Equivalently, its Nijenhuis tensor vanishes, or the $(0,1)$ vector fields are closed under Lie bracket.

For an almost complex structure, closure of $T^{1,0}$ under the Lie bracket is equivalent to the absence of a $(0,2)$ component in $d\alpha$ for every $(1,0)$-form $\alpha$. Conjugation exchanges the bracket condition with closure of $T^{0,1}$. For sections $U,V$ of $T^{0,1}$, the exterior derivative identity gives $d\alpha (U,V)=-\alpha ([U,V])$. Its vanishing for all $(1,0)$-forms is exactly $[U,V]\in T^{0,1}$.

The Newlander-Nirenberg theorem says that a smooth almost complex structure is integrable if and only if its Nijenhuis tensor vanishes.

A complex $n$-manifold has charts to ${\mathbb{C}}^n$ with holomorphic transition maps. It is a real $2n$-manifold with an integrable almost complex structure.

A map between complex manifolds is holomorphic when its local coordinate expression has holomorphic scalar components. For a smooth map, this is equivalent to its differential being complex-linear at every point. These maps compose by the chain rule. For one-dimensional manifolds this reduces to a holomorphic map between Riemann surfaces; the general definition also applies to multiplication and inversion on a complex Lie group.

A proper holomorphic submersion has compact smooth fibres, and the Ehresmann fibration theorem gives differentiable local trivializations. Their integral cohomology, modulo torsion, forms a local system. If the fibres are Kähler manifolds, their Hodge filtrations form holomorphic subbundles and give a variation of Hodge structure. Properness is needed for this usual conclusion.

An analytic subvariety is a closed subset locally defined by finitely many holomorphic functions. It is irreducible if it is not the union of two proper closed analytic subsets. An irreducible analytic subvariety of complex codimension $k$ in an $n$-dimensional complex manifold has pure complex dimension $n-k$. Global irreducibility does not forbid multiple local branches at a singular point.

A complex surface is a complex manifold of complex dimension two, hence real dimension four. Compact complex surfaces need not admit a Kähler metric; a Hopf surface is an example. Every holomorphic one-form on a compact complex surface is closed, and the Betti and Hodge bounds for compact complex surfaces still apply without Kähler assumptions.

A complex K3 surface is a compact, simply connected complex surface with trivial canonical bundle. For projective algebraic surfaces, one can equivalently require a trivial canonical bundle and $H^1(X,{\mathcal{O}}_X)=0$. Nonprojective complex K3 surfaces are also included, which is essential for the full moduli space of Ricci-flat K3 metrics.

Over $\mathbb{C}$, the four partial derivatives show that this projective surface is smooth. Adjunction makes its canonical bundle trivial, and the hypersurface exact sequence gives $H^1({\mathcal{O}}_X)=0$, so it is a K3 surface. Lines given by $x_0=\zeta x_1$, $x_2=\eta x_3$, with ${\zeta }^4={\eta }^4=-1$, lie on it. Choosing both roots differently gives skew lines, one of which is a section of the elliptic pencil defined by planes through the other.

An integral Hodge isometry between complex K3 surfaces which carries some Kähler class to a Kähler class is the pullback of a unique isomorphism of the surfaces. A Hodge isometry without that cone condition need not itself be geometric: reflection in an effective $(-2)$-curve reverses its class and cannot be induced by an isomorphism.

For nef big systems on a complex K3 surface, the successive alternatives distinguish a monogonal linear system on a K3 surface from a basepoint-free system; then a degree-two map from a birational map; then a projective ideal generated by quadrics from the trigonal and plane-quintic exceptions. The last alternatives must include the low-genus quartic and quadric-cubic models. The birational morphism can contract exactly the orthogonal $(-2)$-curves; it need not embed the original smooth surface.

For a basepoint-free nonhyperelliptic nef big divisor $D$, the trigonal condition means that its general smooth curve is a trigonal curve. An elliptic pencil on a K3 surface with $D\cdot E=3$ restricts to its degree-three pencil. On a smooth quartic containing a line $\Gamma$, the hyperplane class $H$ has this property for $E=H-\Gamma$.

For a basepoint-free nef big divisor $D$, this means that the morphism given by its complete system is generically two-to-one onto its image. Equivalently, its general smooth curve is hyperelliptic. A double cover of ${\mathbb{P}}^2$ branched over a smooth sextic, with the pullback of $\mathcal{O}(1)$, gives the degree-two example.

A primitive effective nef class of square zero on a complex K3 surface defines a basepoint-free pencil with connected genus-one fibres. Its general fibre is a smooth elliptic curve. A section is a smooth rational curve $\Gamma$ with $E\cdot \Gamma =1$ and ${\Gamma }^2=-2$.

For a nef big divisor, the monogonal case is the fixed-section system $|D|=\Gamma +|mE|$, where $E$ is an elliptic pencil on a K3 surface and $\Gamma$ is a section. Thus $D\cdot E=1$. The fixed part persists even though $D$ is nef; the general divisor is reducible. Riemann–Roch theorem for algebraic surfaces gives $h^0(D)=m+1=h^0(mE)$, proving the equality of systems.

Locally, a unit-volume Ricci flat metric on a K3 surface determines a positive three-plane in its real second cohomology, with the K3 intersection lattice fixed. The corresponding Grassmannian has dimension $3\cdot 19=57$. Allowing the volume gives one further positive real parameter; large diffeomorphisms identify lattice-equivalent descriptions. This is the same continuous moduli count as the three-torus Narain moduli space plus its coupling.

// Target: supersymmetry.bigb

The cup product on the second integral cohomology of a complex K3 surface is an even unimodular lattice of signature $(3,19)$. Here $U$ is the hyperbolic plane and $E_8(-1)$ is the negative-definite form of the E8 lattice. In the M-theory K3 duality, this lattice matches the heterotic electric charge lattice. The self-dual harmonic two-forms of a Ricci flat K3 metric span its positive three-plane.

If $D$ is a nontrivial nef divisor on a K3 surface with $D^2=0$, its complete linear system of a divisor has no fixed part. Write $D=F+M$. Nefness of $D$ and $M$ gives $D\cdot F=D\cdot M=M^2=M\cdot F=F^2=0$. But a nonzero fixed part satisfies $h^0(F)=1$, while Riemann–Roch theorem for algebraic surfaces and Serre duality would give $h^0(F)\ge 2$ if $F^2=0$. Hence $F=0$. Two movable members with no common component have intersection zero and are disjoint, so the system is basepoint-free and has Iitaka dimension one.

If a smooth curve $C$ of genus $g$ lies on a K3 surface, the adjunction formula and $K_X=0$ give

A smooth quartic surface $X\subseteq {\mathbb{P}}^3$ containing a line $L$ is a K3 surface. The pencil of planes through $L$ cuts out $L$ plus a residual plane cubic; the divisor class $H-L$ has square zero and its basepoint-free pencil defines an elliptic fibration $X\to {\mathbb{P}}^1$.

A primary Hopf surface is a quotient of ${\mathbb{C}}^2\setminus \{0\}$ by an infinite cyclic group generated by a suitable contracting holomorphic map; reversing the generator gives the expanding dilation description. The scalar dilation $z\mapsto 2z$ gives a compact surface diffeomorphic to $S^3\times S^1$ via polar coordinates and log radius. Hence $b_1=1$. The Betti and Hodge bounds for compact complex surfaces force $h^{1,0}=0$ and $h^{0,1}\ge 1$, furnishing a strict example of $h^{1,0}<h^{0,1}$.

In complex dimension at least two, a holomorphic function on a punctured coordinate ball extends holomorphically through its centre. Componentwise extension in a holomorphic local frame gives the same statement for holomorphic sections. This preserves global sections after removing finitely many points, but does not imply preservation of higher sheaf cohomology. This analytic theorem differs from the set-theoretic Hartogs theorem.

An atlas of charts into ${\mathbb{C}}^n$ whose transition maps are biholomorphic. Their real differentials commute with multiplication by $i$, giving an almost complex structure induced by a complex atlas.

A complex manifold is Stein if its global holomorphic functions separate points, give local coordinates, and make it holomorphically convex: the holomorphic hull of every compact set is compact. Closed complex submanifolds of affine complex space are examples, including products of copies of $\mathbb{C}$ and ${\mathbb{C}}^{\ast }$. Cartan theorem B makes these spaces useful for acyclic covers in sheaf cohomology.

For a complex Lie group, holomorphic principal bundles over a Stein manifold are classified up to holomorphic isomorphism by their topological bundle class. In particular, a topologically trivial holomorphic principal bundle over a Stein base has a global holomorphic trivialization. This supplies global complex gauges for a flat partial connection on contractible complex affine space; it does not impose prescribed behaviour at infinity.

On a Stein manifold $X$, every coherent analytic sheaf $\mathcal{F}$ has $H^q(X,\mathcal{F})=0$ for $q>0$. In particular this applies to the sheaf of holomorphic functions. If every nonempty finite intersection in an open cover is Stein, these vanishing results and the acyclic cover theorem compute the corresponding sheaf cohomology from its Čech cochain complex.

The structure sheaf assigns to an open set its ring of holomorphic functions. Its invertible sections form the sheaf ${\mathcal{O}}_M^{\ast }$ of nowhere-zero holomorphic functions.

This sheaf consists of holomorphic functions vanishing at $p$ on opens containing $p$, and all holomorphic functions on opens not containing $p$. Evaluation gives $0\to {\mathcal{I}}_p\to {\mathcal{O}}_M\to {\mathbb{C}}_p\to 0$, with a skyscraper sheaf as quotient. On an elliptic curve, $H^0({\mathcal{I}}_p)=0$, $H^1({\mathcal{I}}_p)\cong \mathbb{C}$, and all higher cohomology vanishes.

A polydisc is a product of open discs in $\mathbb{C}$. It is a contractible complex manifold, with vanishing positive-degree de Rham cohomology and vanishing Dolbeault cohomology in positive antiholomorphic degree.

A holomorphic coordinate chart on a complex manifold is a biholomorphism from an open subset of the manifold to an open subset of ${\mathbb{C}}^n$. Its component functions are holomorphic coordinates.

A holomorphic embedding is a holomorphic map that is also a smooth embedding. Its image is a complex submanifold and the map is a biholomorphism onto that image.

The holomorphic tangent bundle of a complex manifold $X$ is locally spanned in holomorphic coordinates by $\partial /\partial z^1,\dots ,\partial /\partial z^n$. Under a holomorphic coordinate change $w=w(z)$, the chain rule giveswhose coefficient matrix is holomorphic; these frames therefore define a holomorphic vector bundle.

A real connection on a vector bundle on $TM$ with $\nabla J=0$ restricts after complexification to a connection $D$ on the holomorphic tangent bundle. Conversely $D$ and its conjugate determine a unique such real connection. In holomorphic coordinates, compatibility with the holomorphic structure means $D_{{\partial }_{{\bar{z}}^i}}{\partial }_{z^j}=0$. Zero mixed torsion tensor says that this derivative equals ${\nabla }_{{\partial }_{z^j}}{\partial }_{{\bar{z}}^i}$; the two belong to opposite eigentypes, so both vanish. Reality proves the reverse implication. Preserving $J$ alone does not imply this condition.

The dual of the holomorphic tangent bundle is a holomorphic vector bundle with local frames $d{z}^1,\dots ,d{z}^n$. Coordinate changes have holomorphic invertible Jacobian matrices, so these frames have holomorphic transition functions. Its holomorphic sections are exactly the holomorphic one-forms.

In a holomorphic chart $z^j=x^j+i{y}^j$, defineThe derivative of every holomorphic transition map is complex linear and therefore commutes with this $J$, so the chartwise definitions glue to the intrinsic almost complex structure of the complex manifold.

The complexified cotangent bundle of a complex manifold splits asA differential form has type $(p,q)$ when it is locally a sum of terms containing $p$ factors $d{z}^j$ and $q$ factors $d{\bar{z}}^k$.

A differential form of type (p, q) with equal holomorphic and antiholomorphic degrees is real when it is fixed by complex conjugation. For such a form, $\overline{\bar{\partial }\eta }=\partial \eta$, so $\bar{\partial }\eta =0$ implies $d\eta =0$. The converse follows from the distinct types of $\partial \eta$ and $\bar{\partial }\eta$.

Complex conjugation interchanges the bidegrees of complex differential forms:It also interchanges the two type components of the exterior derivative, so $\bar{\partial }\bar{\alpha }=\overline{\partial \alpha }$ and $\partial \bar{\alpha }=\overline{\bar{\partial }\alpha }$.

With the convention $d^c=i(\bar{\partial }-\partial )$ and with $J$ acting on forms by applying the complex structure to every argument,Indeed, $J$ acts on a $(p,q)$-form by $i^{p-q}$; conjugating $d=\partial +\bar{\partial }$ therefore multiplies its two type components by $-i$ and $i$, respectively.

A holomorphic vector field is a holomorphic section of $T^{1,0}X$. In holomorphic coordinates it has the form ${\sum }_j{f}^j(z)\partial /\partial z^j$ with holomorphic coefficient functions $f^j$.

On a complex vector space with linear coordinates, the Euler vector field is $E={\sum }_j{z}_j{\partial }_{z_j}$. It generates scalar dilations, and the Euler homogeneous function theorem says $E(f)=kf$ for a homogeneous function of degree $k$. Contracting a top form with $E$ gives a horizontal form for the quotient by nonzero scalings; descent also requires the appropriate scaling invariance.

A complex-linear endomorphism $A$ of ${\mathbb{C}}^{n+1}$ induces a holomorphic vector field on Complex projective space by differentiating the projective transformations $[z]\mapsto [e^{tA}z]$. Its zeroes are precisely the projective eigenlines of $A$; a diagonalizable $A$ with distinct eigenvalues therefore gives exactly $n+1$ zeroes.

A complex submanifold is locally the common zero set of holomorphic coordinates, equivalently a smooth submanifold whose tangent spaces are complex linear.

A closed analytic subset of a complex manifold is nonsingular when every point has a neighbourhood on which the subset is the zero set of a holomorphic map of full rank along that zero set. The holomorphic implicit function theorem then makes it a local coordinate subspace. This includes a closed complex submanifold, whereas an arbitrary open piece of a complex submanifold need not be globally closed in its ambient manifold.

Let $Y$ be a embedded submanifold of a complex manifold. If its tangent spaces are invariant under the almost complex structure induced by a complex atlas, choose ambient holomorphic coordinates making its tangent space at one point ${\mathbb{C}}^k\times 0$. The first $k$ coordinates restrict to a local real diffeomorphism by the inverse function theorem, so $Y$ is locally the graph of a smooth function $F$. Invariance of the tangent graph gives $dF(iv)=idF(v)$, the Cauchy-Riemann equations. Thus $F$ is holomorphic and these graph coordinates give the induced complex submanifold structure. This proves the result directly, without needing the general integrability theorem as an unproved step.

For a complex submanifold $Y\subseteq X$, the holomorphic normal bundle is the quotient holomorphic vector bundle

The derivative of a section defining a regular zero locus patches intrinsically along that locus because changes of trivialization introduce derivative terms multiplied by the vanishing section. Its tangent kernel and surjectivity give $N_V\cong E{|}_V$. Combining this with adjunction for a smooth submanifold gives $K_V=(K_M\otimes \det E){|}_V$.

For a smooth complex analytic hypersurface $Y\subset X$, its holomorphic normal bundle is naturally the restriction of its holomorphic line bundle associated to a divisor:If reduced local defining functions satisfy $f_i=g_{ij}{f}_j$, then $d{f}_i{|}_Y=g_{ij}d{f}_j{|}_Y$. Pairing a normal vector with $d{f}_i$ and using the reciprocal transition functions of $\mathcal{O}(Y)$ gives the isomorphism.

The dual of the tangent-normal sequence is the short exact sequence of holomorphic vector bundles

For a complex submanifold $Y\subset X$, the holomorphic conormal bundle is the annihilator of $T^{1,0}Y$ inside ${\Omega }_X^1{|}_Y$, or equivalently the dual of the holomorphic normal bundle. For a smooth analytic hypersurface with local equation $f$, it is generated by $df{|}_Y$. It is also the locally free analytic sheaf ${\mathcal{I}}_Y/{\mathcal{I}}_Y^2$.

For reduced local equations $f_j=u_{ji}{f}_i$ with $u_{ji}$ a holomorphic unit, the holomorphic line bundle associated to a divisor $[-Y]$ has local generators $f_i$ and transition $u_{ji}$. Differentiation along $Y$ gives $d{f}_j{|}_Y=u_{ji}{|}_{Y}d{f}_i{|}_Y$. Sending the restricted local generator to $d{f}_i{|}_Y$ gives the conormal isomorphism. The term $f_{i}d{u}_{ji}$ disappears only after restricting to $Y$.

A complex analytic hypersurface in an $n$-dimensional complex manifold is a closed analytic subset of pure complex codimension one. It is irreducible when it is not the union of two proper closed analytic subsets.

A local defining function for a complex analytic hypersurface $Y$ near $x$ is a holomorphic function $f$ on a neighbourhood $U$ such that $Y\cap U=\{f=0\}$. It may be chosen reduced, with each local irreducible factor occurring once. The local ring ${\mathcal{O}}_{X,x}$ of a complex manifold is a regular local ring and hence a unique factorization domain; each height-one prime of the hypersurface germ is principal, and the product of generators for its finitely many local branches gives $f$.

The local ring of a complex manifold is the unique factorization domain of convergent power series in local coordinates. A reduced local defining function of a complex analytic hypersurface has each local irreducible factor once. The analytic zero-set ideal is the radical of its principal ideal, and reducedness makes that principal ideal radical. Thus two reduced equations for the same germ generate the same ideal and differ by a unit. A globally irreducible hypersurface may have several local branches; reducedness includes each once. Without reducedness, $f$ and $f^2$ have the same zero set and need not differ by a unit.

A divisor on a complex manifold is a locally finite formal sumwhere the $Y$ are irreducible complex analytic hypersurfaces and $n_Y\in \mathbb{Z}$. Products of powers of local defining functions give local meromorphic equations $f_{\alpha }$ for $D$ whose ratios $f_{\alpha }/f_{\beta }$ are nowhere-vanishing holomorphic functions.

At a generic smooth point of an irreducible complex analytic hypersurface $Y$, a nonzero meromorphic function has the form $f=t^{m}u$, where $t$ is a reduced transverse local equation and $u$ is a holomorphic unit. The integer $m$ is its order along $Y$: positive for a zero and negative for a pole. The unique factorization domain property of the local ring of a complex manifold makes this independent of the equation and propagates it along $Y$. These orders are the coefficients of the principal divisor on a complex manifold of $f$.

The divisor on a complex manifold of a nonzero global meromorphic function records its zero and pole orders along irreducible analytic hypersurfaces. It is called principal. Its holomorphic line bundle associated to a divisor is trivial: choosing the same global meromorphic function as the local equation on every open set gives identity transition functions. In particular, two hyperplanes in Complex projective space have isomorphic associated bundles, since the quotient of their homogeneous linear equations is a global meromorphic function with their difference as divisor.

If $f_{\alpha }$ are local meromorphic equations for a divisor on a complex manifold $D$, glue holomorphic frames $e_{\alpha }$ byThe resulting holomorphic line bundle is denoted $[D]$. The expressions $f_{\alpha }{e}_{\alpha }$ glue to its canonical meromorphic section, whose divisor is $D$.

If $f_i$ and $g_i$ are local meromorphic equations for two divisors on a complex manifold, then $f_i{g}_i$ defines their sum. With local frames satisfying $e_j=(f_i/f_j)e_i$, the transition functions of a vector bundle for the tensor product are the products of the two ratios, exactly those for $f_i{g}_i$. Multiplication of local frames therefore gives the displayed holomorphic line bundle isomorphism.

The sheaf ${\mathcal{K}}_X$ assigns to an open subset of a complex manifold its meromorphic functions. Its subsheaf ${\mathcal{K}}_X^{\ast }$ consists of the nonzero meromorphic functions, while ${\mathcal{O}}_X^{\ast }$ consists of the nowhere-zero holomorphic functions.

The quotient sheaf ${\mathcal{K}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$ records the zero and pole orders of local nonzero meromorphic functions. Each of its global sections is naturally a divisor on a complex manifold on $X$.

The connecting homomorphism ofsends a divisor on a complex manifold $D$ to its holomorphic line bundle associated to a divisor $\mathcal{O}(D)$ in the Picard group. Its kernel consists exactly of the principal divisors of global nonzero meromorphic functions.

At a point $x$ of a Riemann surface, the principal part of a meromorphic function is the negative-degree part of its Laurent series in a local coordinate. It is equivalently a germ in the quotient ${\mathcal{K}}_x/{\mathcal{O}}_x$.

The principal part at a pole is the finite sum of negative-degree terms of its Laurent series. Prescribing such parts is the problem solved by the Mittag-Leffler theorem.

On a Riemann surface, the quotient sheaf $\mathcal{P}={\mathcal{M}}_X/{\mathcal{O}}_X$ records the finite negative Laurent series tails of meromorphic germs. At $x$, a local coordinate $t$ identifies its stalk with ${⨁}_{m\ge 1}\mathbb{C}{t}^{-m}$. As a sheaf it is the direct sum of sheaves ${⨁}_x(i_x{)}_{\ast }{\mathcal{P}}_x$. Global sections prescribe locally finite families of principal parts, possibly infinite on a noncompact surface. The connecting homomorphism to $H^1(X,{\mathcal{O}}_X)$ is exactly the obstruction to a global meromorphic function with those parts and no extra poles.

The Mittag-Leffler problem asks for a global meromorphic function with prescribed principal parts at a discrete collection of points. For a finite collection, those principal parts define a Čech cocycle for the sheaf $\mathcal{O}$; vanishing of its class in ${\check{H}}^1(X,\mathcal{O})$ is exactly the condition that the problem has a solution.

The complexified cotangent bundle decomposes into $(1,0)$ and $(0,1)$ parts. The Dolbeault operator is the type-$(0,1)$ component of the exterior derivative, $\bar{\partial }:{\Omega }^{p,q}\to {\Omega }^{p,q+1}$, and integrability gives ${\bar{\partial }}^2=0$.

Together with its $(1,0)$ counterpart, the Dolbeault operator decomposes the exterior derivative on complex differential forms.

Using the bundle-valued conjugate-linear Hodge star, the formal adjoint of the holomorphic bundle's Dolbeault operator is ${\bar{\partial }}_E^{\ast }=-{\ast }_{E^{\ast }}{\bar{\partial }}_{E^{\ast }}{\ast }_E$. It lowers antiholomorphic degree. Stokes applied to the coefficient-contracted form $\psi \wedge {\ast }_E\eta$ proves the integrated adjoint identity on a compact complex manifold.

The conjugate Dolbeault operator is the type-$(1,0)$ component of the exterior derivative: $\partial :{\mathcal{A}}^{p,q}\to {\mathcal{A}}^{p+1,q}$. Complex conjugation interchanges $\partial$ and $\bar{\partial }$. On a complex manifold, $d=\partial +\bar{\partial }$, ${\partial }^2={\bar{\partial }}^2=0$, and $\partial \bar{\partial }+\bar{\partial }\partial =0$.

On a sufficiently small polydisc in a complex manifold, every $\bar{\partial }$-closed $(p,q)$-form with $q>0$ is locally $\bar{\partial }$-exact. A coefficientwise one-variable integral operator and induction over the coordinates prove the result.

For a $\bar{\partial }$-closed positive-antiholomorphic-degree differential form of type (p, q) on a polydisc, write $\phi =d{\bar{z}}_j\wedge \alpha +\beta$ with no $d{\bar{z}}_j$ in $\alpha ,\beta$. Apply the Cauchy-Green operator in coordinate $j$ to the coefficients of $\alpha$. Subtracting its Dolbeault operator derivative removes every $d{\bar{z}}_j$ term. Closedness then makes the remaining coefficients holomorphic in $z_j$. Eliminating successive coordinates preserves holomorphicity in earlier coordinates and produces a local primitive after finitely many steps.

On a polydisc, a $\partial$-closed differential form of type (p, q) with $p>0$ is $\partial$-exact. Apply the Dolbeault-Poincaré lemma to its complex conjugate, whose antiholomorphic degree is $p$.

On a disc $D\subset \mathbb{C}$, the Cauchy-Green operatoris a right inverse to $\partial /\partial \bar{z}$ in the interior: ${\partial }_{\bar{z}}Tg=g$. Applying it to coefficients gives the local homotopy operator used in the Dolbeault-Poincaré lemma.

Dolbeault cohomology is

The Dolbeault theorem identifies this group with first Čech cohomology of the trivial holomorphic line bundle. On the two standard affine charts, a holomorphic overlap function has a Laurent series. Its nonnegative powers extend to the chart at zero, and its negative powers extend to the chart at infinity. Every overlap cocycle is therefore a coboundary. Consequently the Dolbeault operator from functions to $(0,1)$-forms is surjective.

For a smooth $(0,1)$-form on the complex projective line, solve its Dolbeault operator equation on a disc about zero and a disc about infinity. The difference of the primitives is a holomorphic function on their annular overlap. Split its Laurent series into a nonnegative-power part extending to the finite disc and a negative-power part extending holomorphically across infinity. Subtract the first part from the finite primitive and add the second to the infinite primitive. The results agree and define a global primitive, without Hodge theory.

On a Stein manifold, every smooth $\bar{\partial }$-closed $(0,q)$-form with $q>0$ is globally $\bar{\partial }$-exact. The same holds componentwise for forms valued in a finite-dimensional constant vector space. Complex conjugation yields the corresponding global $\partial$ primitive for a closed $(1,0)$-form, as used in the complex potential reduction of anti-self-dual Yang-Mills.

For $X={\mathbb{P}}^1\times \mathbb{C}$, the two standard affine charts and their intersection form an acyclic cover for the sheaves of holomorphic forms. Their Čech cohomology gives $H_{\bar{\partial }}^{0,0}=\mathcal{O}(\mathbb{C})$ and $H_{\bar{\partial }}^{2,1}=\mathcal{O}(\mathbb{C})$, with all other groups for $p\in \{0,2\}$ zero. For the top-form sheaf, the transition $d(1/z)=-z^{-2}dz$ removes every Laurent series power except $z^{-1}$ from the first cohomology quotient; its coefficient may be an arbitrary entire function on the affine factor.

This is the cohomology of ${\mathcal{A}}^{p,\bullet }(X,E)$ with the bundle Dolbeault operator. The Dolbeault theorem identifies it with $H^q(X,{\Omega }_X^p\otimes \mathcal{O}(E))$. For compact Hermitian $X$, Dolbeault Hodge decomposition gives unique harmonic representatives.

Chern curvature defines $\alpha (E)=[\Theta ]\in H^1(M,{\Omega }_M^1\otimes \mathrm{End}E)$. If two metrics give connections differing by $a$, then ${\Theta }_1-{\Theta }_0={\bar{\partial }}_{\mathrm{End}E}a$. Thus this vector-bundle curvature class depends on the holomorphic bundle rather than the chosen metric.

In frames with $e^{(\beta )}=g_{\alpha \beta }{e}^{(\alpha )}$ and input-index connection matrices, the Chern connection has ${\theta }_{\beta }-g_{\alpha \beta }{\theta }_{\alpha }{g}_{\alpha \beta }^{-1}={\sigma }_{\alpha \beta }$. These holomorphic one-forms, interpreted in the beta frame, satisfy the transported Čech cocycle condition because $g_{\alpha \gamma }=g_{\beta \gamma }{g}_{\alpha \beta }$. Local primitives of curvature differ by this cocycle. A holomorphic frame change $e_{\alpha }^{\prime }=u_{\alpha }{e}_{\alpha }$ adds the Čech coboundary of $u_{\alpha }^{-1}\partial u_{\alpha }$, so the resulting curvature Dolbeault class is intrinsic.

Composition of endomorphisms and wedge product of differential forms give the class $[{\Theta }^k]\in H^k(M,{\Omega }_M^k\otimes \mathrm{End}E)$. Even total degree of vector-bundle curvature and the noncommutative telescoping identity give ${\Theta }_1^k-{\Theta }_0^k=\bar{\partial }{\sum }_{j=0}^{k-1}{\Theta }_1^{j}a{\Theta }_0^{k-1-j}$ whenever ${\Theta }_1-{\Theta }_0=\bar{\partial }a$. Thus all powers are metric independent; no commutativity of endomorphisms is assumed.

For a complex torus $X=V/\Lambda$ of complex dimension $g$, constant forms give a natural isomorphism $H_{\bar{\partial }}^{p,q}(X)\cong {\bigwedge }^p{V}^{\ast }\otimes {\bigwedge }^q\overline{V^{\ast }}$. A flat Kähler metric and the Dolbeault Hodge decomposition prove that these constant forms are precisely the harmonic representatives. Thus $h^{p,q}=\left(\genfrac{}{}{0ex}{}{g}{p}\right)\left(\genfrac{}{}{0ex}{}{g}{q}\right)$ for $0\le p,q\le g$, and zero otherwise.