25A311AD092F
Blowing up an ideal replaces its zero set by the space of directions of its generators, and makes the pulled-back ideal locally principal. We construct blowups of complex spaces along coherent ideals, prove their universal property, and compute blowups of manifolds along submanifolds in explicit charts; these charts give the transformation rules for ideals of submanifolds needed in the next sections. At the end of the section we state the existence theorems for resolutions and principalizations and deduce that log resolutions exist.
Definition 9.6.1.757D038DF549 Let $X$ be a complex space and let $\mathfrak a\subset \mathcal O_ X$ be a coherent ideal sheaf that is not identically zero on any branch of $(X,x)$, for every $x\in X$. On an open set where $\mathfrak a$ is generated by $f_0,\ldots ,f_ r$, the blowup $\operatorname {Bl}_{\mathfrak a}X$ is the closure, with its reduced structure, of the graph
\[ \bigl \{ \bigl (x,[f_0(x):\cdots :f_ r(x)]\bigr ):x\in X\setminus V(\mathfrak a)\bigr \} \subset X\times \mathbb P^ r. \]
Here $\mathbb P^ r$ is the space of nonzero vectors in $\mathbb C^{r+1}$ modulo multiplication by nonzero scalars, with the charts $\{ \xi _ i\neq 0\} $ and the affine coordinates $\xi _ j/\xi _ i$. It is compact, because every point has a representative on the compact unit sphere and the quotient map is continuous. The projection to $X$ is denoted by $\pi $.
An ideal sheaf on a complex space is invertible if it is locally generated by one nonzerodivisor. For a holomorphic map $g\colon Y\to X$ we write $\mathfrak a\mathcal O_ Y$ for the ideal sheaf generated by the pullbacks $f\circ g$ of local sections $f$ of $\mathfrak a$.
Proposition 9.6.2.1CAC94BB1C9D The blowup is a complex space, independent of the chosen generators up to canonical isomorphism. Its projection $\pi $ is proper and bimeromorphic, and $\mathfrak a\mathcal O_{\operatorname {Bl}_{\mathfrak a}X}$ is an invertible ideal. If $g\colon Y\to X$ is a holomorphic map from a complex space $Y$ such that $\mathfrak a\mathcal O_ Y$ is invertible, then $g$ has a unique holomorphic lift $\widetilde g\colon Y\to \operatorname {Bl}_{\mathfrak a}X$, that is, $\pi \circ \widetilde g=g$.
Proof.
In $X\times \mathbb P^ r$ consider the analytic set $Z$ defined by the equations of $X$ and by $f_ i\xi _ j=f_ j\xi _ i$ for all $i,j$; in the chart $\{ \xi _ i\neq 0\} $ these are the holomorphic equations $f_ j=f_ i\cdot (\xi _ j/\xi _ i)$ and $f_ k\cdot (\xi _ j/\xi _ i)=f_ j\cdot (\xi _ k/\xi _ i)$. Over a point $x\notin V(\mathfrak a)$, the only point of $Z$ is $(x,[f(x)])$, because $\xi $ must be proportional to the nonzero vector $f(x)$. Thus the graph is the part of $Z$ outside the analytic set $\pi ^{-1}(V(\mathfrak a))$, and by lemma 9.3.7 its closure is analytic; near each point it is the union of the branches of $Z$ not contained in $\pi ^{-1}(V(\mathfrak a))$. It is closed in $X\times \mathbb P^ r$, and $\mathbb P^ r$ is compact, so $\pi $ is proper. Over $X\setminus V(\mathfrak a)$ the blowup is the graph, and $\pi $ is an isomorphism there, with inverse $x\mapsto (x,[f(x)])$, which is holomorphic because on $\{ f_ i\neq 0\} $ the affine coordinates $f_ j/f_ i$ are holomorphic. The set $V(\mathfrak a)$ is nowhere dense in $X$, by corollary 9.3.6, and the graph is dense in the blowup; so $\pi $ is bimeromorphic.
On the chart $\{ \xi _ i\neq 0\} $ of the blowup, the relations $f_ j\circ \pi =(f_ i\circ \pi )\cdot (\xi _ j/\xi _ i)$ hold on the graph, hence everywhere, so the pulled-back ideal is generated by $f_ i\circ \pi $ there. This generator is not identically zero on any branch of the blowup at any point of the chart, because the graph points of the chart, at which $f_ i\circ \pi \neq 0$, are dense in every such branch. Hence it is a nonzerodivisor, and the pulled-back ideal is invertible.
Let $g\colon Y\to X$ be as in the statement. Near a point of $Y$, let $d$ be a nonzerodivisor generating $\mathfrak a\mathcal O_ Y$, and write $f_ i\circ g=db_ i$. Writing $d=\sum _ ic_ i\, (f_ i\circ g)=d\sum _ ic_ ib_ i$ and cancelling the nonzerodivisor $d$, we get $\sum _ ic_ ib_ i=1$; so the $b_ i$ have no common zero, and $y\mapsto (g(y),[b_0(y):\cdots :b_ r(y)])$ is a holomorphic map into $X\times \mathbb P^ r$. Where $d\neq 0$, some $f_ i(g(y))$ is nonzero, and this point is the graph point over $g(y)$; since $\{ d=0\} $ is nowhere dense, the whole image lies in the closure of the graph. Replacing $d$ by $ud$ for a unit $u$ replaces $b$ by $u^{-1}b$ and does not change the projective point, so these local lifts glue. A lift of $g$ is uniquely determined over $Y\setminus g^{-1}(V(\mathfrak a))=Y\setminus \{ d=0\} $, where $\pi $ is an isomorphism, hence everywhere by density.
Finally, the blowups defined by two families of generators both pull back $\mathfrak a$ to an invertible ideal. By the universal property, each maps to the other over $X$, and both compositions are the identity by uniqueness. These canonical isomorphisms also glue the local constructions on overlaps, compatibly on triple overlaps by the same uniqueness. The glued space is Hausdorff: points with different images in $X$ are separated by the inverse images of disjoint neighborhoods, and points with the same image lie in one of the local constructions, which are Hausdorff. Since $X$ has a countable basis, countably many of the open sets over which the local constructions are made cover $X$, and each local construction, an analytic subset of the product of a local model with $\mathbb P^ r$, has a countable basis; hence so has the glued space. Being locally compact and Hausdorff with a countable basis, it is paracompact.
$\square$
An effective Cartier divisor $D$ on a complex space $X$ is given by an invertible ideal sheaf, written $\mathcal O_ X(-D)$; a local generator is a local equation of $D$, determined up to a unit. We write $\mathcal O_ X(D)$ for the inverse invertible sheaf, generated locally by the reciprocal of a local equation. Addition of effective Cartier divisors corresponds to multiplication of their ideals. If $g\colon Y\to X$ is holomorphic and the local equations of $D$ composed with $g$ are nonzerodivisors, they define the pullback $g^*D$, with $\mathcal O_ Y(-g^*D)=\mathcal O_ X(-D)\mathcal O_ Y$; restriction to an analytic subset is a special case.
On a complex manifold, an effective divisor has simple normal crossings support, and is then called a simple normal crossings divisor, if each irreducible component of its support is smooth and near each point there are holomorphic coordinates in which the support is $\{ z_1\cdots z_ k=0\} $. Its multiplicities may be arbitrary positive integers; the condition concerns only the support. A submanifold $C$ has normal crossings with such a divisor if near each point of $C$ there are coordinates in which the support of the divisor is a union of coordinate hyperplanes and $C$ is a coordinate subspace.
For a proper bimeromorphic map between complex manifolds, an effective divisor is called exceptional if each irreducible component of its support has image of codimension at least two; these images are analytic by theorem 9.7.2, proved in the next section.
Proposition 9.6.3.865FE5EC97E6 Let $C$ be a closed submanifold of a complex manifold $M$, of positive codimension at each of its points, and let $\pi \colon M'\to M$ be the blowup of $M$ along $\mathcal I_ C$. Then $M'$ is a complex manifold, the inverse image $E=\pi ^{-1}(C)$ is a smooth hypersurface with $\mathcal I_ C\mathcal O_{M'}=\mathcal O_{M'}(-E)$, and the fiber of $\pi $ over a point of $C$ at which $C$ has codimension $r$ is $\mathbb P^{r-1}$. If $C$ is a hypersurface, then $\pi $ is the identity and $E=C$. Explicitly, let $(z,w)$, $w=(w_1,\ldots ,w_ r)$, be holomorphic coordinates on an open set $U\subset M$ with $C\cap U=\{ w=0\} $. Then $\pi ^{-1}(U)$ is covered by $r$ charts, one for each pivot $i\in \{ 1,\ldots ,r\} $, with coordinates $(z,t,v)$, $v=(v_ j)_{j\neq i}$, in which
E62F169B3F9F
9.6.4
\begin{equation} \pi (z,t,v)=(z,t\tilde v),\qquad \tilde v_ i=1,\quad \tilde v_ j=v_ j\ (j\neq i), \end{equation}
and $E=\{ t=0\} $; the Jacobian determinant of $\pi $ in these coordinates is $\pm t^{r-1}$.
Let $W\subset M$ be a closed submanifold containing $C$, such that $C$ has positive codimension in $W$ at each of its points. Then the strict transform $W'$ of $W$, the closure of $\pi ^{-1}(W\setminus C)$, is the blowup $\operatorname {Bl}_ CW$ of $W$ along $\mathcal I_ C\mathcal O_ W$, and
4C48E8A5FA42
9.6.5
\begin{equation} \mathcal I_ W\mathcal O_{M'}=\mathcal O_{M'}(-E)\mathcal I_{W'}. \end{equation}
In holomorphic coordinates $(z,w)$, $w=(w_1,\ldots ,w_ s)$, on an open set $U\subset M$ with $W\cap U=\{ w=0\} $ and $C\cap U=\{ w=0,\ z_1=\cdots =z_ k=0\} $, which exist near every point of $C$, the center is generated by $z_1,\ldots ,z_ k,w_1,\ldots ,w_ s$, and the strict transform $W'$ meets only the charts whose pivot is one of $z_1,\ldots ,z_ k$. The chart with pivot $z_ i$ has coordinates $(z'',t,u,v)$, where $z''$ consists of the coordinates $z_ l$ with $l>k$, and is given by $z_ i=t$, $z_ l=tu_ l$ for $l\leq k$, $l\neq i$, and $w_ j=tv_ j$; in it, $W'=\{ v=0\} $. If instead $W$ is a closed submanifold disjoint from $C$, the strict transform $W'=\pi ^{-1}(W)$ maps isomorphically onto $W$, and $\mathcal I_ W\mathcal O_{M'}=\mathcal I_{W'}$.
Proof.
Let $(z,w)$ be coordinates on an open set $U$ as in the first part of the statement; such coordinates exist near every point of $C$. On $U$ the ideal $\mathcal I_ C$ is generated by $w_1,\ldots ,w_ r$, by lemma 9.3.12. Let $Z\subset U\times \mathbb P^{r-1}$ be the analytic set defined by the equations $w_ j\xi _ i=w_ i\xi _ j$. Over a point of $U\setminus C$, its only point is $((z,w),[w])$, because $\xi $ must be proportional to the nonzero vector $w$; so the part of $Z$ over $U\setminus C$ is the graph of definition 9.6.1, whose closure is the blowup of $U$. In the chart $\{ \xi _ i\neq 0\} $, the equations $w_ j\xi _ i=w_ i\xi _ j$ say that $w_ j=w_ iv_ j$ with $v_ j=\xi _ j/\xi _ i$, $j\neq i$; hence this part of $Z$ is the graph of the map $(z,t,v)\mapsto (z,w)$ with $w_ i=t$ and $w_ j=tv_ j$, $j\neq i$. It is a manifold with coordinates $(z,t,v)$, in which the graph part is $\{ t\neq 0\} $, a dense subset. Hence these charts cover the blowup, $M'$ is a manifold, and $\mathcal I_ C\mathcal O_{M'}$ is generated by $t$ in each chart. Thus $E=\{ t=0\} $ is a smooth hypersurface.
With $w_ i$ listed first among the coordinates $w$, the Jacobian matrix of (9.6.4) is block lower triangular with diagonal blocks $I$, $1$ and $tI_{r-1}$, because $\partial w_ i/\partial t=1$, $\partial w_ j/\partial t=v_ j$ and $\partial w_ j/\partial v_ l=t\delta _{jl}$ for $j,l\neq i$, while $z$ does not depend on $(t,v)$ and $w$ does not depend on $z$; so the Jacobian determinant of $\pi $ is $\pm t^{r-1}$. The fiber over a point $(z,0)$ is obtained by gluing the coordinate charts $v\in \mathbb C^{r-1}$, that is, it is $\mathbb P^{r-1}$. If $r=1$, there is one chart, and the coordinates and the map are unchanged.
For the second part, choose coordinates $(z,w)$ near a point of $C$ with $W=\{ w_1=\cdots =w_ s=0\} $ and $C=\{ w_1=\cdots =w_ s=z_1=\cdots =z_ k=0\} $, where $k\geq 1$ is the codimension of $C$ in $W$. Such coordinates exist: take coordinates $(x,w)$ with $W=\{ w=0\} $, then holomorphic coordinates $z=\phi (x)$ on $W$ near the point in which $C$ is a coordinate plane, and use $(\phi (x),w)$, whose Jacobian matrix is block triangular and invertible, so that these are holomorphic coordinates near the point by proposition 5.2.4(1). The center is now generated by $z_1,\ldots ,z_ k,w_1,\ldots ,w_ s$. In a chart whose pivot is $z_ i$, we have $z_ i=t$, $z_ l=tu_ l$ for $l\neq i$, $l\leq k$, and $w_ j=tv_ j$; here $\pi ^{-1}(W\setminus C)=\{ v=0,\ t\neq 0\} $, so $W'=\{ v=0\} $ in this chart, and
\[ \mathcal I_ W\mathcal O_{M'}=(tv_1,\ldots ,tv_ s)=(t)(v_1,\ldots ,v_ s)=\mathcal O_{M'}(-E)\mathcal I_{W'}. \]
In a chart whose pivot is $w_ i$, we have $w_ i=t$, so $\pi ^{-1}(W)$ lies in $\{ t=0\} =E$. Moreover, $W'$ does not meet this chart, because the points of $\pi ^{-1}(W\setminus C)$, and therefore those of its closure, have projective coordinates $\xi $ whose components corresponding to $w_1,\ldots ,w_ s$ vanish. The pulled-back ideal contains $t$ and is therefore $(t)=\mathcal O_{M'}(-E)$, which proves (9.6.5) in this chart as well. In the charts of the first kind, $W'$ has the coordinates $(t,u)$ together with the remaining $z$-coordinates, and $\pi |_{W'}$ is given by the formulas of the blowup of $W$ along $C$; so $W'=\operatorname {Bl}_ CW$. Finally, if $C$ is disjoint from $W$, then $\pi $ is an isomorphism over the neighborhood $M\setminus C$ of $W$.
$\square$
We shall use the following two existence theorems, which we quote without proof. They are the analytic versions of [Theorems 1.1 and 1.3, pp. 611 and 614; Definitions 2.1–2.2, pp. 618–619, BM08]; that these results hold for complex analytic spaces is stated on p. 611 there. Further references are given in subsection 25.1.21.
Theorem 9.6.6 (Resolution of singularities).4CEF0578EFE0 Every reduced complex space $X$ has a proper bimeromorphic map $\pi \colon M\to X$ from a complex manifold $M$ that is a composite of blowups with smooth centers and an isomorphism over $X_{\mathrm{reg}}$. The divisor formed by the inverse images in $M$ of the blowup centers has simple normal crossings. The sequence of blowups is finite when $X$ is compact, and finite over each relatively compact open subset of $X$ in general.
Theorem 9.6.7 (Principalization).24330D25741D Let $M$ be a complex manifold, let $\mathfrak a\subset \mathcal O_ M$ be a coherent ideal that is not identically zero on any connected component of $M$, and let $D$ be a simple normal crossings divisor on $M$, possibly empty. Then there is a sequence of blowups $\sigma _ i\colon M_ i\to M_{i-1}$, $i\geq 1$, with $M_0=M$ and smooth centers $C_ i\subset M_{i-1}$, finite over each relatively compact open subset of $M$, with the following properties. Put $\mathfrak a_0=\mathfrak a$ and $E_ i=\sigma _ i^{-1}(C_ i)$, and, for $i\geq 0$, call the union of the strict transforms of $D,E_1,\ldots ,E_ i$ in $M_ i$ the accumulated divisor on $M_ i$. Each center $C_ i$ is contained in $V(\mathfrak a_{i-1})$ and has normal crossings with the accumulated divisor on $M_{i-1}$; the ideal
012823948235
9.6.8
\begin{equation} \mathfrak a_ i=\mathcal O_{M_ i}(E_ i)\, \mathfrak a_{i-1}\mathcal O_{M_ i} \end{equation}
is holomorphic; and over each relatively compact open subset of $M$ the last of these ideals is the unit ideal. The accumulated divisors have simple normal crossings, and the composite is an isomorphism off $V(\mathfrak a)$.
In this notation, theorem 9.6.7 is the theorem for marked ideals with mark $1$ and $N=M$. The cosupport of $\mathfrak a_ i$ is $V(\mathfrak a_ i)$, and the transform (9.6.8) divides by a single exceptional factor at each step, even where $\mathfrak a_{i-1}$ vanishes to higher order along the center. If $M$ is compact, the sequence is finite, say with $t$ steps, and iterating (9.6.8) gives
\[ \mathfrak a\mathcal O_{M_ t}=\mathcal O_{M_ t}(-H), \qquad H=\sum _{i=1}^ t(\sigma _{i+1}\circ \cdots \circ \sigma _ t)^*E_ i, \]
where the empty composition for $i=t$ is the identity, so that the last term is $E_ t$; the same holds over every relatively compact open subset of $M$.
The divisor $H$ is effective, and its support has simple normal crossings. Indeed, for a blowup $\sigma $ with center $C$ and exceptional divisor $E$, and a closed set $D$, we have $\sigma ^{-1}(D)\subset D'\cup E$, where $D'$ is the strict transform, the closure of $\sigma ^{-1}(D\setminus C)$. By induction, the inverse image in $M_ t$ of each $E_ i$ is contained in the accumulated divisor on $M_ t$. The support of $H$ is the union of these inverse images, the zero set of a local equation of $H$, which is not identically zero on any component of $M_ t$; so it is empty or pure of codimension one, by proposition 9.5.6. By corollary 9.3.17(4) and (3), each of its irreducible components is contained in, and hence equal to, an irreducible component of the accumulated divisor. Thus the support of $H$ is a union of some components of a divisor whose support has simple normal crossings.
Near a point, in coordinates in which the support of the accumulated divisor is $\{ z_1\cdots z_ k=0\} $, the germ of each of its components through the point is the germ of a connected submanifold of codimension one, hence irreducible by lemma 9.3.12. It is contained in the union of the germs of the hyperplanes $\{ z_ l=0\} $, so it is contained in one of them and, by proposition 9.3.9, equal to it. Hence the support of $H$ is near this point the union of some of these hyperplanes, and it has simple normal crossings.
Let $\mathfrak a$ be a coherent ideal on a complex space $X$, not identically zero on any branch of any germ of $X$. A log resolution of $\mathfrak a$ is a proper bimeromorphic map $\pi \colon Y\to X$ from a complex manifold $Y$ that is a composite of blowups with smooth centers, finite over each relatively compact open subset of $X$, such that $\mathfrak a\mathcal O_ Y=\mathcal O_ Y(-D)$ for an effective divisor $D$ on $Y$ whose support has simple normal crossings.
Log resolutions exist. Let $\rho \colon M\to X$ be a resolution given by theorem 9.6.6. The zero set of the pulled-back ideal $\mathfrak a\mathcal O_ M$ is $\rho ^{-1}(V(\mathfrak a))$, which is nowhere dense, because $\rho $ is bimeromorphic and $V(\mathfrak a)$ is nowhere dense in $X$ (lemma 9.4.1); so $\mathfrak a\mathcal O_ M$ is not identically zero on any component of $M$. We principalize it by theorem 9.6.7, taking as the given simple normal crossings divisor the exceptional divisor $D_\rho $ of $\rho $, formed by the inverse images of its blowup centers, and obtain $\sigma \colon Y\to M$ with $\mathfrak a\mathcal O_ Y=\mathcal O_ Y(-D)$, $D=H$.
The argument given above for the support of $H$ applies also to $\sigma ^{-1}(D_\rho )$. This set lies in the accumulated divisor by the same induction, and it is the zero set of the pullback of a local equation of $D_\rho $. This pullback is not identically zero on any component of $Y$: the map $\sigma $ is an isomorphism off the support of $D$, which is the nowhere dense set $\sigma ^{-1}(V(\mathfrak a\mathcal O_ M))$, so $\sigma $ is bimeromorphic and $\sigma ^{-1}(D_\rho )$ is nowhere dense by lemma 9.4.1. Hence the support of $D$ together with $\sigma ^{-1}(D_\rho )$ is a union of irreducible components of the accumulated divisor, and it has simple normal crossings. The composite $\pi =\rho \circ \sigma $ is proper, and it is bimeromorphic: it is an isomorphism over $X_{\mathrm{reg}}\setminus V(\mathfrak a)$, the set $\sigma ^{-1}(\rho ^{-1}(V(\mathfrak a)))$ is the support of $D$, a hypersurface, and $\sigma ^{-1}(\rho ^{-1}(X_{\mathrm{sing}}))$ is nowhere dense, because $\rho ^{-1}(X_{\mathrm{sing}})$ is nowhere dense and $\sigma $ is an isomorphism off the support of $D$.
When $X$ is a manifold, the centers of a log resolution of $\mathfrak a$ are closed submanifolds of positive codimension at each of their points, because the ideal of a center must not vanish identically on any component (definition 9.6.1). No resolution step is needed then: theorem 9.6.7 with the empty divisor gives a log resolution that is an isomorphism off $V(\mathfrak a)$. These are the log resolutions used in section 6.6, section 6.13; in section 6.14, a log resolution of the ideal in the type of a function with analytic singularities is also called a log resolution of that function.
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