Theorem 9.3.20D73EBFFE10D2

Proof. Working on each connected component of $M$ separately, we may assume that $M$ is connected, of dimension $n$.

Step 1: supports of high codimension. Suppose first that, near a point, the support of $R$ lies in a real submanifold $\{ g_1=\cdots =g_ k=0\} $ with independent differentials $\mathrm dg_1,\ldots ,\mathrm dg_ k$, $k\geq 3$. Since $R$ has measure coefficients and $g_ j$ vanishes on its support, $g_ jR=0$; since $R$ is closed, $\mathrm dg_ j\wedge R=\mathrm d(g_ jR)=0$. Take real coordinates $x$ whose first three are $g_1,g_2,g_3$. For $l\leq 3$ and $l\notin \{ a,b\} $, the coefficient of $\mathrm dx_ l\wedge \mathrm dx_ a\wedge \mathrm dx_ b$ in $\mathrm dx_ l\wedge R$ is, up to sign, the coefficient of $\mathrm dx_ a\wedge \mathrm dx_ b$ in $R$, so this coefficient vanishes. Since no pair $\{ a,b\} $ contains all three indices $1,2,3$, every coefficient of $R$ vanishes, and $R=0$ near the point. Now let $A$ have codimension at least two. Near a point of $A_{\mathrm{reg}}$, the set $A$ is a complex submanifold of real codimension at least four, so $R$ vanishes there, and the support of $R$ lies in $A_{\mathrm{sing}}$. This is analytic of smaller dimension near each point by corollary 9.3.13(1), and again of codimension at least two; repeating the argument, we find after finitely many steps near each point that $R=0$. This proves the last assertion.

Step 2: regular points of codimension one. Let $y\in A_{\mathrm{reg}}$ be a point where $A$ is a complex hypersurface. Choose holomorphic coordinates $(\zeta ,w)$ near $y$ with $A=\{ \zeta =0\} $, write $\zeta =\xi +\mathrm i\eta $, and let $t\in \mathbb R^{2n-2}$ be the real coordinates of $w$. The identities $\mathrm d\xi \wedge R=\mathrm d\eta \wedge R=0$ of Step 1 force $R=\mu \, \mathrm d\xi \wedge \mathrm d\eta $ for a real measure $\mu $ supported on $\{ \xi =\eta =0\} $, so $\mu =\delta _0(\xi ,\eta )\otimes \sigma (t)$ for a real measure $\sigma $. Since $R$ is closed, $\mathrm dR=\sum _ l(\partial \mu /\partial t_ l)\, \mathrm dt_ l\wedge \mathrm d\xi \wedge \mathrm d\eta $ vanishes, the derivatives in $\xi $ and $\eta $ disappearing after the exterior product with $\mathrm d\xi \wedge \mathrm d\eta $; so $\partial \mu /\partial t_ l=0$ as distributions. Testing this against functions $\chi (\xi ,\eta )\psi (t)$ with $\chi =1$ near $0$, we find that all distributional derivatives $\partial \sigma /\partial t_ l$ vanish. On a smaller ball in the $t$-variables, the convolutions of $\sigma $ with a smoothing kernel therefore have zero gradient and are constant; integrating them against a fixed test function of integral one, we see that these constants converge as the kernel shrinks, and that $\sigma $ is a constant multiple $c$ of Lebesgue measure. Hence $R=c[\{ \zeta =0\} ]$ near $y$, with $c$ real. (If $n=1$, $\sigma $ is a measure on a point and the same conclusion holds.)

Step 3: conclusion. Let $D=Z_\alpha $ be an irreducible component of codimension one, the closure of the connected component $S_\alpha $ of $A_{\mathrm{reg}}$. By Step 2, near each point of $S_\alpha $ we have $R=c[D]$ for a real constant $c$, and this constant is locally constant on the connected manifold $S_\alpha $; we call it $c_ D$. By corollary 9.3.16 the sum $\sum _ Dc_ D[D]$ is locally finite, and by lemma 9.3.19 it is a closed real current with measure coefficients. The difference $R'=R-\sum _ Dc_ D[D]$ vanishes near every point of $A_{\mathrm{reg}}$: near a point of $S_\alpha $ with $Z_\alpha $ of codimension one by Step 2, since the other components do not meet $S_\alpha $; and near a point of $A_{\mathrm{reg}}$ where $A$ has codimension at least two by Step 1, since no component of codimension one passes through such a point. Thus the support of $R'$ lies in $A_{\mathrm{sing}}$.

We show that $A_{\mathrm{sing}}$ has codimension at least two; then $R'=0$ by Step 1. A branch of $A$ of dimension $n$ at a point $y$ would be a subgerm of the germ $(M,y)$, which is irreducible of dimension $n$ by lemma 9.3.12, hence equal to it by proposition 9.3.9; then $A$ would contain a neighborhood of $y$, which is impossible because $A$ is nowhere dense. So the branches of $A$ have dimension at most $n-1$, and $A_{\mathrm{sing}}$, which has smaller dimension than each branch (corollary 9.3.13(1)), has codimension at least two. The coefficients are unique, because near a point of $S_\alpha $ the current $R$ determines $c_ D$; and if $R\geq 0$, then $c_ D\geq 0$, because $[D]$ is positive and nonzero near such a point. $\square$