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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.

On a complex manifold, a sheaf of modules over the sheaf of holomorphic functions is coherent analytic if it is locally finitely generated and the kernel of every morphism ${\mathcal{O}}^r\to \mathcal{F}$ is locally finitely generated. The structure sheaf and locally free finite-rank sheaves are examples. Cartan theorem B gives vanishing of their higher sheaf cohomology on a Stein manifold.

For $X={\mathbb{C}}^2\setminus \{0\}$, the cover $\mathbb{C}\times {\mathbb{C}}^{\ast }$ and ${\mathbb{C}}^{\ast }\times \mathbb{C}$ is acyclic for ${\mathcal{O}}_X$. The first Čech cohomology quotient is represented uniquely by normally convergent doubly negative Laurent series. Under $w_i=1/z_i$ it is $w_1{w}_2\mathcal{O}({\mathbb{C}}_w^2)$, the entire functions divisible by both coordinates. This is an infinite-dimensional analytic vector space; restricting representatives to finite Laurent polynomials loses classes. The class of $1/(z_1{z}_2)$ is nonzero, so $X$ is not a Stein manifold.

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.

Write $w=x^1+i{x}^2$, $z=x^3+i{x}^4$. The Euclidean metric is ${\sum }_a(d{x}^a{)}^2$, and the printed four-form is $4d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4$, so it specifies the usual positive orientation. The normalized metric volume form is one quarter of that expression. The self-dual frame in complex Euclidean coordinates isThese are real and have the required complex combinations. For the Hodge star operator with this orientation,and applying $\ast$ again gives the reverse relations. Thus $\ast {\omega }_i={\omega }_i$. They are linearly independent, while the $+1$ eigenspace of $\ast$ on 2-forms has dimension three, proving they span ${\Lambda }_{+}^2$. The TeX aid incorrectly reads the subscript as $1$.

The ASDYM equations require the self-dual projection of the gauge field strength to vanish. Orthogonality to ${\omega }_1,{\omega }_2$ sets its $(2,0)$ and $(0,2)$ parts to zero; orthogonality to ${\omega }_3$ removes the trace of its $(1,1)$ part. Explicitly, if $F={\sum }_{a<b}{f}_{ab}d{x}^a\wedge d{x}^b$, these conditions are $f_{13}-f_{24}=0$, $f_{14}+f_{23}=0$ and $f_{12}+f_{34}=0$. Sinceand the conjugate equation supplies the other complex component for a real curvature form of a connection, the equivalent system is

To obtain the complex potential reduction of anti-self-dual Yang-Mills, set $D_w={\partial }_w+A_w$, $D_z={\partial }_z+A_z$. The first equation is the integrability condition $[D_w,D_z]=0$. Locally it allows an invertible complex matrix $s$ satisfying ${\partial }_{w}s=-A_{w}s$ and ${\partial }_{z}s=-A_{z}s$. The Yang-Mills gauge transformation $A\mapsto s^{-1}As+s^{-1}ds$ consequently gives $A_w=A_z=0$. This is a complex gauge; a real compact gauge group alone generally cannot implement it.

In this gauge the second equation becomesThus the one-form $A_{\bar{w}}dz-A_{\bar{z}}dw$ is closed with respect to the exterior derivative in the $(w,z)$ directions. The local complex version of the Poincare lemma gives a potential $K$ such thatBecause the gauge transformation is complex, $K$ is generally valued in the complexification of a Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$; the printed $\mathfrak{g}$ must be understood in that sense. The elementary reduction is local, and the transformed fields retain a reality condition inherited from the original real connection. For the usual compact matrix gauge groups and smooth fields on all of ${\mathbb{R}}^4={\mathbb{C}}^2$, the gauge and potential can also be chosen globally if no condition at infinity is imposed. The flat partial connection defines a holomorphic principal bundle on the conjugate complex space. That base is a contractible Stein manifold, so the Oka-Grauert principle gives a global trivialization and hence a global complex gauge. After that trivialization, the conjugate of Stein vanishing for the Dolbeault cohomology of functions gives a global primitive $K$, component by component in ${\mathfrak{g}}_{\mathbb{C}}$. Prescribed framing or decay at infinity requires a separate compatibility check and is not automatically preserved by this gauge.

Substitute the potential into the remaining ASDYM equations component:Therefore all three ASDYM equations reduce to the single ASDYM potential equationThe sign follows directly from $A_{\bar{z}}=-K_w$; changing a potential convention would change the displayed commutator sign. Conversely, this equation and the displayed gauge reconstruction make all three curvature conditions vanish, subject to the inherited reality condition when a real gauge field is required.

The two opens are $U_1=\mathbb{C}\times {\mathbb{C}}^{\ast }$ and $U_2={\mathbb{C}}^{\ast }\times \mathbb{C}$; their intersection is $({\mathbb{C}}^{\ast }{)}^2$. There is just one degree-one term in the Čech cochain complex, and no degree-two term. The Čech coboundary sends $(g_1,g_2)$ to $g_2-g_1$, soThe denominator denotes the sum of the restricted function spaces.

To make the quotient explicit, expand a holomorphic function on the intersection in a normally convergent two-variable Laurent series,The terms with $m\ge 0$ extend to $U_1$. Of the remaining terms, those with $n\ge 0$ extend to $U_2$. Both subseries converge normally on their stated domains, by the coefficient bounds from the Cauchy integral formula. The unique remaining representative is the doubly negative partNo nonzero series of this form lies in the denominator, since a function on $U_1$ has no negative $z_1$ exponents and a function on $U_2$ has no negative $z_2$ exponents.

Writing $w_i=z_i^{-1}$, these representatives are exactly $w_1{w}_{2}F(w_1,w_2)$ with $F$ an entire holomorphic function on ${\mathbb{C}}^2$. To see that $F$ is entire, integrate for the coefficients on arbitrarily small product circles: $|a_{-i,-j}|\le M(r_1,r_2)r_1^i{r}_2^j$. Choosing $r_i{R}_i<1$ gives absolute convergence for $|w_i|\le R_i$, for every finite pair $R_i$. Conversely every such entire $F$ supplies a normally convergent representative on the intersection. Therefore the quotient consists of convergent doubly negative Laurent series, not merely finite Laurent polynomials:This gives the holomorphic first cohomology of punctured complex two-space; for instance $1/(z_1{z}_2)$ represents a nonzero class.

For the comparison with sheaf cohomology, $U_1,U_2$ and their intersection are Stein manifolds. They can be realized as closed complex submanifolds of affine complex spaces by adding equations $z_i{w}_i=1$ for their nonzero coordinates. Cartan theorem B makes their higher cohomology with coefficients in the sheaf of holomorphic functions vanish. Thus the cover is acyclic for ${\mathcal{O}}_X$, and the acyclic cover theorem givesThe individual cover members are Stein; their union has the nonzero cohomology just computed.

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.