Lemma 10.7.8189AC91F2DE3

Proof. The question is local. We show that the given function $f$ satisfies $\bar\partial f=0$ in the sense of distributions near each point of $A$, by a one-variable argument on complex lines. Near a point of $A$, the set $A$ is the common zero set of finitely many holomorphic functions on a connected coordinate neighborhood, and one of them, $b$, does not vanish identically, since otherwise $A$ would contain a nonempty open set. The lowest-order nonzero homogeneous term $P$ of $b$ at the point is a nonzero polynomial, so its zero set is closed and nowhere dense; hence we may choose linearly independent vectors $v_1,\ldots ,v_ n$ with $P(v_ k)\neq 0$, choosing each $v_ k$ outside the union of this zero set and the span of $v_1,\ldots ,v_{k-1}$. We use linear coordinates centered at the point in which the $v_ k$ are the coordinate unit vectors. Then $t\mapsto b(tv_ k)$ has a zero of finite order at $t=0$, so by the Weierstrass preparation theorem (theorem 5.5.2), on a sufficiently small polydisc, $b$ is a unit times a monic polynomial in the $k$th coordinate with holomorphic coefficients in the others, for each $k$. Hence every complex line parallel to a coordinate axis meets the zero set of $b$ in this polydisc in finitely many points.

By Fubini's theorem, $f$ is square integrable on almost every such line. On such a line, near each of the finitely many punctures, the Laurent expansion of $f$ has no terms of negative order: by orthogonality of the powers on circles, a term of order $-j$ with $j\geq 1$ would make the integral of $|f|^2$ diverge like $\int _0^ rt^{1-2j}\, \mathrm dt$. Hence the restriction of $f$ to almost every line extends holomorphically, and its $\bar\partial $-derivative along the line vanishes as a distribution. Integrating against compactly supported test functions and using Fubini's theorem, we find that every derivative $\partial f/\partial \bar z_ k$ vanishes as a distribution on the polydisc. A distribution with $\bar\partial f=0$ satisfies $\Delta f=4\sum _ k\partial _{z_ k}\partial _{\bar z_ k}f=0$, so it is smooth by Weyl's lemma (theorem 3.1.11) and therefore holomorphic. This holomorphic function agrees with $f$ on the complement of $A$, and the local extensions agree on overlaps. For sections of a line bundle, apply the result in local holomorphic frames. $\square$