5.2 Complex manifolds and differential formsD8E2D1A98ACF

D8E2D1A98ACF

Let $n\geq 0$ be an integer. On $\mathbb {C}^ n$ we write $z_ j=x_ j+\mathrm{i}y_ j$ for $1\leq j\leq n$, and we use the complex partial derivatives and differentials

A smooth function $f$ on an open subset of $\mathbb {C}^ n$ is holomorphic if $\partial _{\bar z_ j}f=0$ for $1\leq j\leq n$. Substituting the definitions, we find $\mathrm df=\sum _ j(\partial _{z_ j}f\, \mathrm dz_ j+\partial _{\bar z_ j}f\, \mathrm d\bar z_ j)$ for every smooth $f$. On each tangent space the forms $\mathrm dz_ j$ are complex linear and the forms $\mathrm d\bar z_ j$ are conjugate linear, and a real-linear form which is both complex linear and conjugate linear vanishes; hence $f$ is holomorphic if and only if its real derivative is complex linear at every point. By the chain rule, compositions of holomorphic maps, that is, of maps with holomorphic components, are therefore holomorphic. A holomorphic function is holomorphic in each variable separately in the sense of one-variable theory, since it satisfies the Cauchy–Riemann equations in each variable, and its derivatives $\partial _{z_ j}f$ are again holomorphic, because partial derivatives of smooth functions commute.

Let $n\geq 1$, and let $f$ be holomorphic on a neighborhood of the closed polydisc $\{ |z_ j-a_ j|\leq r,\ 1\leq j\leq n\} $, where $a\in \mathbb C^ n$ and $r>0$. Applying the one-variable Cauchy formula in each variable in turn, we obtain the Cauchy formula on a polydisc: for $|z_ j-a_ j|<r$,

For $|z_ j-a_ j|\leq s<r$, the geometric series $(\zeta _ j-z_ j)^{-1}=\sum _{k\geq 0}(z_ j-a_ j)^ k(\zeta _ j-a_ j)^{-k-1}$ converge uniformly for $|\zeta _ j-a_ j|=r$. The product of these series is a multiple series which converges absolutely and uniformly on the torus $\{ |\zeta _ j-a_ j|=r\} $; integrating it term by term, we obtain the power series expansion

with

where $(z-a)^\alpha =\prod _ j(z_ j-a_ j)^{\alpha _ j}$, $|\alpha |=\alpha _1+\cdots +\alpha _ n$, and $M$ is the maximum of $|f|$ on the torus $\{ |\zeta _ j-a_ j|=r\} $. The series converges normally on every polydisc $\{ |z_ j-a_ j|\leq s\} $ with $s<r$, since $\sum _\alpha (s/r)^{|\alpha |}=(1-s/r)^{-n}$. Differentiating the Cauchy formula under the integral sign, we obtain integral formulas for all derivatives of $f$, and in particular $\partial _{z_1}^{\alpha _1}\cdots \partial _{z_ n}^{\alpha _ n}f(a)=\alpha _1!\cdots \alpha _ n!\, c_\alpha $.

Grouping the terms of (5.2.1) by degree, we write $f(a+w)=\sum _{k\geq 0}P_ k(w)$ for $|w|<r$, where $P_ k$ is a homogeneous polynomial of degree $k$; this is the homogeneous expansion of $f$ at $a$. If $P_ k=0$ for all $k<m$, then

Indeed, with $t=|w|/r\leq \frac12$ we have $|w^\alpha |\leq |w|^{|\alpha |}$ and $t^{|\alpha |}\leq t^{m+1}2^{m+1-|\alpha |}$ for $|\alpha |\geq m+1$, so the terms of degree at least $m+1$ contribute at most $Mt^{m+1}2^{m+1}\sum _\alpha 2^{-|\alpha |}=2^{m+n+1}Mt^{m+1}$. The expansion (5.2.1) also gives the identity theorem: a holomorphic function on a domain that vanishes on a nonempty open subset vanishes identically. Indeed, the set of points at which all derivatives $\partial _{z_1}^{\alpha _1}\cdots \partial _{z_ n}^{\alpha _ n}f$ vanish is nonempty, and it is closed because these derivatives are continuous. It is also open: at each of its points $a$ all coefficients $c_\alpha $ vanish, by the formula for the derivatives at $a$, so $f$ vanishes on a polydisc around $a$. Hence this set is the whole domain.

Definition 5.2.3.42A02A1A5EFE A complex manifold of dimension $n$ is a paracompact Hausdorff space with an atlas of homeomorphisms onto open subsets of $\mathbb {C}^ n$, whose coordinate changes are holomorphic with holomorphic inverses. A map between complex manifolds is holomorphic if its coordinate expressions are holomorphic. A complex submanifold is a closed subset that is locally a coordinate plane in suitable holomorphic coordinates.

The underlying real manifold has dimension $2n$. If $F=(F_1,\ldots ,F_ m)$ is a holomorphic map from an open subset of $\mathbb C^ n$ to $\mathbb C^ m$, then $\mathrm dF_ j=\sum _ k\partial _{z_ k}F_ j\, \mathrm dz_ k$, so the real derivative of $F$ at a point $a$ is the complex-linear map given by the complex Jacobian matrix $(\partial _{z_ k}F_ j(a))_{j,k}$. Consequently the inverse function theorem has the following holomorphic version.

Proposition 5.2.4.A683A13C69D9

1. Let $U\subset \mathbb C^ n$ be open, let $F\colon U\to \mathbb C^ n$ be holomorphic, and let $a\in U$ be a point at which the complex Jacobian matrix $(\partial _{z_ k}F_ j(a))_{j,k}$ is invertible. Then there are open neighborhoods $U'\subset U$ of $a$ and $V'\subset \mathbb C^ n$ of $F(a)$ such that $F$ maps $U'$ bijectively onto $V'$ and the inverse map $V'\to U'$ is holomorphic.

2. Let $U\subset \mathbb C^ m$ be open, let $a\in U$, and let $f_1,\ldots ,f_ r$, $0\le r\le m$, be holomorphic functions on $U$ whose complex Jacobian matrix $(\partial _{z_ k}f_ j(a))_{1\le j\le r,\, 1\le k\le m}$ at $a$ has rank $r$. Then there are indices $k_1<\cdots <k_{m-r}$ such that the map $(f_1,\ldots ,f_ r,z_{k_1},\ldots ,z_{k_{m-r}})$ is biholomorphic from an open neighborhood of $a$ onto an open subset of $\mathbb C^ m$. Thus $f_1,\ldots ,f_ r$ are the first $r$ coordinates of a holomorphic chart near $a$.

3. Let $X$ be a complex manifold of dimension $m$, let $a\in X$, and let $f_1,\ldots ,f_ r$ be holomorphic functions on a neighborhood of $a$ whose complex Jacobian matrix at $a$, computed in a holomorphic chart, has rank $r$. Then $f_1,\ldots ,f_ r$ are the first $r$ coordinates of a holomorphic chart of $X$ near $a$.

Proof. (1) The real derivative of $F$ at $a$ is the complex-linear map given by the invertible matrix $(\partial _{z_ k}F_ j(a))$, so it is invertible. By the real inverse function theorem there are open neighborhoods $U'$ of $a$ and $V'$ of $F(a)$ such that $F$ maps $U'$ bijectively onto $V'$, the inverse map $G\colon V'\to U'$ is smooth, and $\mathrm dG_ w=(\mathrm dF_{G(w)})^{-1}$ for $w\in V'$. Since $F$ is holomorphic, $\mathrm dF_{G(w)}$ is complex linear, and hence so is its inverse. Thus the real derivative of $G$ is complex linear at every point, that is, $G$ is holomorphic.

(2) If $r=0$, the identity map has the required property. Otherwise some $r\times r$ minor of the matrix $(\partial _{z_ k}f_ j(a))$ is nonzero; let $k_1<\cdots <k_{m-r}$ be the indices of the columns not used by this minor. The map $\Phi =(f_1,\ldots ,f_ r,z_{k_1},\ldots ,z_{k_{m-r}})$ is holomorphic on $U$. The last $m-r$ rows of its complex Jacobian matrix at $a$ are the unit row vectors with entry $1$ in the columns $k_1,\ldots ,k_{m-r}$, so, expanding the determinant along these rows, we see that it equals the chosen minor up to sign; hence it is nonzero. By (1), $\Phi $ maps an open neighborhood of $a$ bijectively onto an open subset of $\mathbb C^ m$, with holomorphic inverse.

(3) The rank of the complex Jacobian matrix does not depend on the chart, by the chain rule, since coordinate changes have invertible complex Jacobian matrices. Let $\kappa \colon U\to \kappa (U)\subset \mathbb C^ m$ be a holomorphic chart around $a$ on which the $f_ j$ are defined. By (2), applied to the functions $f_ j\circ \kappa ^{-1}$ at $\kappa (a)$, there is a map $\Phi $ from an open neighborhood $V\subset \kappa (U)$ of $\kappa (a)$ onto an open subset of $\mathbb C^ m$, biholomorphic and with first components $f_ j\circ \kappa ^{-1}$. Then $\Phi \circ \kappa $ is a homeomorphism of $\kappa ^{-1}(V)$ onto an open subset of $\mathbb C^ m$ whose first $r$ components are $f_1,\ldots ,f_ r$. For every chart $\lambda $ of the atlas, the coordinate changes $(\Phi \circ \kappa )\circ \lambda ^{-1}=\Phi \circ (\kappa \circ \lambda ^{-1})$ and $\lambda \circ (\Phi \circ \kappa )^{-1}=(\lambda \circ \kappa ^{-1})\circ \Phi ^{-1}$ are compositions of holomorphic maps, so $\Phi \circ \kappa $ is a holomorphic chart. $\square$

A complex submanifold $Z$ of a complex manifold $X$ that is locally a coordinate plane of dimension $r$ is itself a complex manifold of dimension $r$. Indeed, the restrictions to $Z$ of the charts in which $Z$ is a coordinate plane $\mathbb C^ r\times \{ 0\} $, followed by the projection to $\mathbb C^ r$, form an atlas of $Z$ whose coordinate changes are restrictions of holomorphic maps to coordinate planes. Moreover $Z$ is paracompact, that is, every open cover of $Z$ has a locally finite open refinement: given an open cover of $Z$ by sets $Z\cap O_ i$ with $O_ i$ open in $X$, a locally finite open refinement of the cover of $X$ by the $O_ i$ and by $X\setminus Z$ restricts to a locally finite open refinement on $Z$.

Every connected component of a complex manifold $X$ is open, since manifolds are locally connected, and it has a countable basis of its topology. To see this, refine the cover of $X$ by all relatively compact coordinate balls to a locally finite open cover $(U_ a)$. Each $U_ a$ lies in a coordinate ball, hence in a single connected component, and its closure is compact, so it meets only finitely many members of the cover. Fix a member $U_{a_0}$, and call a member reachable if it is joined to $U_{a_0}$ by a finite chain of members, each meeting the next. By the finiteness just noted, there are only countably many reachable members. The union of the reachable members and the union of the remaining ones are disjoint open sets covering $X$. Hence the first union is open and closed; it contains the connected component $C$ of $U_{a_0}$, and it is contained in $C$ because each member lies in a single component. Thus $C$ is covered by countably many coordinate balls, and therefore has a countable basis.

In particular, connected manifolds and compact manifolds have a countable basis, whereas a general complex manifold may have uncountably many components, for instance the product of $\mathbb C^ n$ with an uncountable discrete space. We state the countability of a basis as a hypothesis whenever a global subsequence or another countable selection requires it.

A differential $k$-form is a smooth field of alternating $k$-linear forms on the real tangent spaces, with complex values. In a chart with real coordinates $t_1,\ldots ,t_ N$, where $N$ is the real dimension, every $k$-form is uniquely $\alpha =\sum _ J\alpha _ J\, \mathrm dt_ J$ with smooth coefficients $\alpha _ J$; here $J=(j_1<\cdots <j_ k)$ runs over the increasing multi-indices of length $k$, and $\mathrm dt_ J=\mathrm dt_{j_1}\wedge \cdots \wedge \mathrm dt_{j_ k}$. The pullback of $\alpha $ by a smooth map $F$ is defined pointwise by $(F^*\alpha )_ x(v_1,\ldots ,v_ k)=\alpha _{F(x)}(\mathrm dF_ xv_1,\ldots ,\mathrm dF_ xv_ k)$; it is multiplicative for the exterior product, and $F^*\mathrm dt_ j=\mathrm dF_ j$ for the components $F_ j=t_ j\circ F$. The exterior derivative is defined in coordinates by

It satisfies

for a $k$-form $\alpha $, every form $\gamma $ and every smooth map $F$ between open subsets of coordinate spaces. Indeed, the Leibniz rule follows from the product rule for the coefficients, the sign $(-1)^ k$ arising when $\mathrm dt_ l$ is moved past the $k$ factors of $\alpha $. Next, $\mathrm d^2(f\, \mathrm dt_ J)=\sum _{l,l'}\partial _{t_{l'}}\partial _{t_ l}f\, \mathrm dt_{l'}\wedge \mathrm dt_ l\wedge \mathrm dt_ J=0$, because the second derivatives of $f$ are symmetric in $l$ and $l'$, whereas $\mathrm dt_{l'}\wedge \mathrm dt_ l$ is antisymmetric. Finally, for a function $f$ the chain rule gives $\mathrm d(f\circ F)=\sum _ j(\partial _{t_ j}f\circ F)\, \mathrm dF_ j=F^*\mathrm df$. Since $\mathrm d(\mathrm dF_{j_1}\wedge \cdots \wedge \mathrm dF_{j_ k})=0$ by the Leibniz rule and $\mathrm d^2=0$, it follows that

Applied to coordinate changes, the last identity shows that the exterior derivative does not depend on the chart. Hence $\mathrm d$ is defined for forms on manifolds, and the rules (5.2.5) hold there, for smooth maps $F$ between manifolds.

On a complex manifold of dimension $n$, in holomorphic coordinates $z_ j=x_ j+\mathrm iy_ j$, the $1$-forms $\mathrm dz_ j$ and $\mathrm d\bar z_ j$ form a basis of the complex-valued real-linear forms on each tangent space, so every $k$-form $\alpha $ decomposes uniquely as

where, for an increasing multi-index $I=(i_1<\cdots <i_ p)$, we write $\mathrm{d}z_ I=\mathrm{d}z_{i_1}\wedge \cdots \wedge \mathrm{d}z_{i_ p}$, and similarly $\mathrm{d}\bar z_ J$. The form $\alpha ^{p,q}$ is the component of bidegree, or type, $(p,q)$ of $\alpha $. A holomorphic change of coordinates $w=w(z)$ gives $\mathrm dw_ j=\sum _ k\partial _{z_ k}w_ j\, \mathrm dz_ k$ and $\mathrm d\bar w_ j=\sum _ k\overline{\partial _{z_ k}w_ j}\, \mathrm d\bar z_ k$, so it transforms the $\mathrm dz_ j$ among themselves and the $\mathrm d\bar z_ j$ among themselves. Hence this decomposition does not depend on the chart, and, for the same reason, pullback by holomorphic maps preserves bidegrees. Since $\mathrm d(\mathrm dz_ I\wedge \mathrm d\bar z_ J)=0$ and $\mathrm df=\sum _ l(\partial _{z_ l}f\, \mathrm dz_ l+\partial _{\bar z_ l}f\, \mathrm d\bar z_ l)$, we have $\mathrm d(f\, \mathrm dz_ I\wedge \mathrm d\bar z_ J)=\sum _ l(\partial _{z_ l}f\, \mathrm dz_ l+\partial _{\bar z_ l}f\, \mathrm d\bar z_ l)\wedge \mathrm dz_ I\wedge \mathrm d\bar z_ J$. Thus, for a form $\alpha $ of type $(p,q)$, the form $\mathrm d\alpha $ is the sum of a form $\partial \alpha $ of type $(p+1,q)$ and a form $\bar\partial \alpha $ of type $(p,q+1)$. Extending $\partial $ and $\bar\partial $ linearly to all forms, we have $\mathrm{d}=\partial +\bar\partial $. Comparing types in $\mathrm d^2=0$ gives

Complex conjugation exchanges the types $(p,q)$ and $(q,p)$, and a form is real if it equals its conjugate. We orient a complex manifold by the form $\mathrm{d}x_1\wedge \mathrm{d}y_1\wedge \cdots \wedge \mathrm{d}x_ n\wedge \mathrm{d}y_ n$ in each holomorphic chart. These orientations agree on overlapping charts. Indeed, since $\frac{\mathrm i}2\mathrm dz_ j\wedge \mathrm d\bar z_ j=\mathrm dx_ j\wedge \mathrm dy_ j$ and $2$-forms commute with all forms, this form equals $c_ n\, \mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n\wedge \mathrm d\bar z_1\wedge \cdots \wedge \mathrm d\bar z_ n$ for a constant $c_ n\neq 0$ depending only on $n$. If $w$ is a second holomorphic chart, then $\mathrm dw_1\wedge \cdots \wedge \mathrm dw_ n=\det (\partial _{z_ k}w_ j)\, \mathrm dz_1\wedge \cdots \wedge \mathrm dz_ n$ by multilinearity and antisymmetry of the exterior product, and the conjugate identity holds for the $\mathrm d\bar w_ j$. Hence the form defined by $w$ is $|\det (\partial _{z_ k}w_ j)|^2>0$ times the form defined by $z$.

Let $X$ be a complex manifold of dimension $n$, and let $\alpha $ be a $2n$-form on $X$ with compact support whose coefficients are locally integrable, that is, Lebesgue integrable on compact subsets of charts. If $\operatorname{Supp}\alpha $ lies in a holomorphic chart, in which $\alpha =g\, \mathrm dx_1\wedge \mathrm dy_1\wedge \cdots \wedge \mathrm dx_ n\wedge \mathrm dy_ n$, we put $\int _ X\alpha =\int g\, \mathrm dV_{2n}$, the Lebesgue integral in the coordinates. This does not depend on the chart. Indeed, by the preceding paragraph, the coefficient $\tilde g$ of $\alpha $ in a second chart $w$ satisfies $g=(\tilde g\circ w)\, |\det (\partial _{z_ k}w_ j)|^2$, and $|\det (\partial _{z_ k}w_ j)|^2$ is the real Jacobian determinant of the coordinate change $z\mapsto w$, so the change of variables formula gives $\int g\, \mathrm dV_{2n}(z)=\int \tilde g\, \mathrm dV_{2n}(w)$. For general $\alpha $, we choose a partition of unity $(\chi _ a)$ subordinate to a cover of $X$ by charts (proposition 5.2.6 below). Only finitely many $\chi _ a$ are not identically zero on the compact set $\operatorname{Supp}\alpha $, and we put $\int _ X\alpha =\sum _ a\int _ X\chi _ a\alpha $. If $(\psi _ b)$ is a second such partition of unity, then, by the case of a single chart,

so the integral is well defined. It is linear, and $\int _ X\alpha \geq 0$ if $\alpha $ is a nonnegative multiple of the volume form at every point. The same definition applies to complex submanifolds, which are complex manifolds with their own complex orientation.

Proposition 5.2.6.DAE61F3D77A3 Let $X$ be a complex manifold of dimension $n$. Then $X$ admits smooth partitions of unity subordinate to every open cover and consisting of functions with compact support. For every compact set $K\subset X$ and every open set $U\supset K$ there is a smooth function $\chi \colon X\to [0,1]$ with compact support in $U$ that equals one on a neighborhood of $K$. If $n\geq 1$ and $\alpha $ is a smooth $(2n-1)$-form with compact support on $X$, then $\int _ X\mathrm{d}\alpha =0$.

Proof. Since the connected components of $X$ are open, it suffices to construct partitions of unity on each component; the union of the families so obtained is locally finite on $X$. So let $X$ be connected. By the preceding discussion, $X$ is covered by a sequence of coordinate balls $B_1,B_2,\ldots $ with compact closures. Put $K_1=\overline{B_1}$, and, once the compact set $K_ j$ is defined, cover it by finitely many of the balls $B_ i$ and let $K_{j+1}$ be the union of their closures and of $\overline{B_1},\ldots ,\overline{B_{j+1}}$. We obtain compact sets $K_1\subset \operatorname{Int}K_2\subset K_2\subset \operatorname{Int}K_3\subset \cdots $ with union $X$; put $K_ j=\varnothing $ for $j\leq 0$.

Now let an open cover of $X$ be given. For $j\geq 1$, the compact set $K_ j\setminus \operatorname{Int}K_{j-1}$ lies in the open set $\operatorname{Int}K_{j+1}\setminus K_{j-2}$. Cover it by finitely many coordinate balls $B'$ whose concentric enlargements $B$ have compact closures contained in a member of the given cover and in $\operatorname{Int}K_{j+1}\setminus K_{j-2}$. For each such pair choose, in the chart, a smooth function with values in $[0,1]$ and compact support in $B$ that equals one on $B'$ (lemma 2.5.1), and extend it by zero to $X$. The resulting family is locally finite, since a point of $\operatorname{Int}K_ i$ has the neighborhood $\operatorname{Int}K_ i$, which meets the supports chosen at stage $j$ only if $j\leq i+1$. Its sum is therefore smooth, and it is positive, because the balls $B'$ cover $X$. Dividing each function by this sum gives the required partition of unity.

For the cutoff function, cover $K$ by finitely many relatively compact coordinate balls with closures in $U$, and let $V$ be their union; thus $K\subset V$ and $\overline V$ is a compact subset of $U$. Let $(\chi _ a)$ be a partition of unity with compact supports subordinate to the cover $\{ V,X\setminus K\} $, and let $\chi $ be the sum of those $\chi _ a$ whose support is not contained in $X\setminus K$. Each of these has its support in $V$; by local finiteness, only finitely many supports meet the compact set $\overline V$, so $\chi $ is a finite sum, smooth, with values in $[0,1]$ and compact support in $\overline V\subset U$. The supports of the remaining $\chi _ a$ lie in $X\setminus K$, and their union $F$ is closed, as a locally finite union of closed sets. On the neighborhood $X\setminus F$ of $K$ we have $\chi =\sum _ a\chi _ a=1$.

For the last assertion, consider first a compactly supported $(2n-1)$-form $\gamma $ in a single chart. Then $\mathrm{d}\gamma $ is a sum of terms $\pm \partial _{t_ l}g\, \mathrm dt_1\wedge \cdots \wedge \mathrm dt_{2n}$, where $t_1,\ldots ,t_{2n}$ are the real coordinates $x_1,y_1,\ldots ,x_ n,y_ n$ and $g$ is smooth with compact support; the integral of each term vanishes by Fubini's theorem and the fundamental theorem of calculus on lines parallel to the $t_ l$-axis. Now let $(\chi _ a)$ be a partition of unity subordinate to a cover by charts. Only finitely many $\chi _ a$ are nonzero on the compact set $\operatorname{Supp}\alpha $, so $\alpha =\sum _ a\chi _ a\alpha $ is a finite sum of compactly supported forms in single charts, and $\int _ X\mathrm d\alpha =\sum _ a\int _ X\mathrm d(\chi _ a\alpha )=0$. $\square$

Cutoff functions also embed a compact manifold in a Euclidean space; later chapters use the resulting metric to measure distances to closed sets.

Lemma 5.2.7.98494B0CCC31 Let $X$ be a compact complex manifold. Then there are an integer $N\ge 0$ and a smooth injective map $F\colon X\to \mathbb R^ N$ which is a homeomorphism of $X$ onto $F(X)$. Consequently, $d(x,y)=|F(x)-F(y)|$ is a metric on $X$ which induces the topology of $X$.

Proof. If $X$ is empty, take $N=0$. Otherwise let $n$ be the dimension of $X$. For each $x\in X$ choose a holomorphic chart $\kappa _ x\colon U_ x\to \mathbb C^ n=\mathbb R^{2n}$ with $x\in U_ x$. By proposition 5.2.6, applied to the compact set $\{ x\} $ and the open set $U_ x$, there is a smooth function $\chi _ x\colon X\to [0,1]$ with compact support in $U_ x$ which equals one on an open neighborhood $W_ x$ of $x$. Since $X$ is compact, finitely many of the sets $W_ x$ cover $X$; we denote the corresponding charts, cutoff functions and neighborhoods by $\kappa _ i\colon U_ i\to \mathbb R^{2n}$, $\chi _ i$ and $W_ i$, $1\le i\le k$. The map $\chi _ i\kappa _ i$, defined as the product on $U_ i$ and as $0$ on $X\setminus U_ i$, is smooth: it is smooth on $U_ i$, it vanishes on $X\setminus \operatorname{Supp}\chi _ i$, which is open because the compact set $\operatorname{Supp}\chi _ i$ is closed in the Hausdorff space $X$, and these two open sets cover $X$ because $\operatorname{Supp}\chi _ i\subset U_ i$. Put $N=k+2nk$ and

\[ F=(\chi _1,\ldots ,\chi _ k,\chi _1\kappa _1,\ldots ,\chi _ k\kappa _ k)\colon X\to \mathbb R^ N, \]

a smooth map. It is injective. Indeed, let $F(x)=F(y)$, and choose $i$ with $x\in W_ i$. Then $\chi _ i(y)=\chi _ i(x)=1$, so $y\in \operatorname{Supp}\chi _ i\subset U_ i$, and $\chi _ i\kappa _ i(x)=\chi _ i\kappa _ i(y)$ gives $\kappa _ i(x)=\kappa _ i(y)$; since $\kappa _ i$ is injective on $U_ i$, we obtain $x=y$.

If $C\subset X$ is closed, then $C$ is compact, so $F(C)$ is compact and hence closed in $\mathbb R^ N$. Thus the continuous bijection $F\colon X\to F(X)$ maps closed sets to closed sets, so its inverse is continuous, and $F$ is a homeomorphism onto $F(X)$. Hence $d$, the Euclidean metric of $F(X)$ transported to $X$ by $F$, is a metric on $X$ that induces its topology. $\square$

For computations in $\mathbb {C}^ n$ we use the Euclidean Kähler form

Here $\mathrm dV_{2n}=\mathrm dx_1\wedge \mathrm dy_1\wedge \cdots \wedge \mathrm dx_ n\wedge \mathrm dy_ n$ is the Euclidean volume form, which we identify with Lebesgue measure on $\mathbb {C}^ n=\mathbb {R}^{2n}$; for $n=0$, $\mathrm dV_0$ is the unit mass on the singleton $\mathbb C^0$. The second identity holds because $(\mathrm{i}/2)\mathrm dz_ j\wedge \mathrm d\bar z_ j=\mathrm dx_ j\wedge \mathrm dy_ j$ and these $2$-forms commute and have square zero, so that in the expansion of $\beta ^ n$ only the $n!$ products containing each of them once survive, each equal to $\mathrm dV_{2n}$.

The operator $\mathrm{dd}^{\mathrm{c}}$ is normalized as in the Conventions:

Here $u_{j\bar k}=\partial _{z_ j}\partial _{\bar z_ k}u$, and $\Delta $ is the Laplacian of $\mathbb R^{2n}$; the matrix $(u_{j\bar k})$ is the complex Hessian of $u$. The same formulas define $\mathrm{dd}^{\mathrm{c}}u$ for a distribution $u$ on an open subset of $\mathbb C^ n$, with distributional derivatives $u_{j\bar k}$; then $\mathrm{dd}^{\mathrm{c}}u$ is a current in the sense of the next section. We also use the real operator

Indeed, since $\partial ^2=\bar\partial ^2=0$ and $\bar\partial \partial =-\partial \bar\partial $, we have $(\partial +\bar\partial )(\bar\partial -\partial )=\partial \bar\partial -\bar\partial \partial =2\partial \bar\partial $. Moreover $\overline{\mathrm d^{\mathrm c}\alpha }=\mathrm d^{\mathrm c}\bar\alpha $, because conjugation exchanges $\partial $ and $\bar\partial $ and changes the sign of $\mathrm{i}$; hence $\mathrm d^{\mathrm c}$ maps real forms to real forms.