# Deformations of varieties of general type

János Kollár<EMAIL_ADDRESS>
###### Abstract.

We prove that small deformations of a projective variety of general type are
also projective varieties of general type, with the same plurigenera.

Our aim is to prove the following.

######  Theorem 1.

Let $g:X\to S$ be a flat, proper morphism of complex analytic spaces. Fix a
point $0\in S$ and assume that the fiber $X_{0}$ is projective, of general
type, and with canonical singularities. Then there is an open neighborhood
$0\in U\subset S$ such that

  1. (1.1)

the plurigenera of $X_{s}$ are independent of $s\in U$ for every $r$, and

  2. (1.2)

the fibers $X_{s}$ are projective for every $s\in U$.

Here the $r$th plurigenus of $X_{s}$ is $h^{0}(Y_{s},\omega_{Y_{s}}^{r})$,
where $Y_{s}\to X_{s}$ is any resolution of $X_{s}$. By [Nak04, VI.5.2] (see
also (10.2)) $X_{s}$ has canonical singularities, so this is the same as
$h^{0}(X_{s},\omega_{X_{s}}^{[r]})$, where $\omega_{X_{s}}^{[r]}$ denotes the
double dual of the $r$th tensor power $\omega_{X_{s}}^{\otimes r}$.

Comments 1.3. Many cases of this have been proved, but I believe that the
general result is new, even for $X_{0}$ smooth and $S$ a disc.

For smooth surfaces proofs are given in [KS58, Iit69], and for 3-folds with
terminal singularities in [KM92, 12.5.1]. If $g$ is assumed projective, then
of course all fibers are projective, and deformation invariance of plurigenera
was proved by [Siu98] for $X_{0}$ smooth, and by [Nak04, Chap.VI] when $X_{0}$
has canonical singularities. However, frequently $g$ is not projective; see
Example 4 for some smooth, 2-dimensional examples. Many projective varieties
have deformations that are not projective, not even algebraic in any sense; K3
and elliptic surfaces furnish the best known examples.

In Example 3 we construct a deformation of a projective surface with a
quotient singularity and ample canonical class, whose general fibers are non-
algebraic, smooth surfaces of Kodaira dimension 0. Thus canonical is likely
the largest class of singularities where Theorem 1 holds. See also Example 5
for surfaces with simple elliptic singularities.

The projectivity of $X_{0}$ is essential in our proof, but (1.1) should hold
whenever $X_{0}$ is a proper algebraic space of general type with canonical
singularities. Such results are proved in [RT20], provided one assumes that
either $X_{0}$ is smooth and all fibers are Moishezon, or almost all fibers
are of general type.

Our main technical result says that the Minimal Model Program works for
$g:X\to S$. For $\dim X_{0}=2$ and $X_{0}$ smooth, this goes back to [KS58].
For $\dim X_{0}=3$ and terminal singularities, this was proved in [KM92,
12.4.4]. The next result extends these to all dimensions.

######  Theorem 2.

Let $g:X\to S$ be a flat, proper morphism of reduced, complex analytic spaces.
Fix a point $0\in S$ and assume that $X_{0}$ is projective and has canonical
singularities. Then every sequence of MMP-steps $X_{0}=X_{0}^{0}\dasharrow
X_{0}^{1}\dasharrow X_{0}^{2}\dasharrow\cdots$ (see Definition 7) extends to a
sequence of MMP-steps

$X=X^{0}\dasharrow X^{1}\dasharrow X^{2}\dasharrow\cdots,$

over some open neighborhood $0\in U\subset S$.

The proof is given in Paragraph 8 when $S$ is a disc ${\mathbb{D}}$, and in
Paragraph 12 in general. The assumption that $X_{0}$ has canonical
singularities is necessary, as shown by semistable 3-fold flips [KM92].
Extending MMP steps from divisors with canonical singularities is also studied
in [AK19].

If $X_{0}$ is of general type, then a suitable MMP for $X_{0}$ terminates with
a minimal model $X_{0}^{\rm m}$ by [BCHM10], which then extends to $g^{\rm
m}:X^{\rm m}_{U}\to U$ by Theorem 2. For minimal models of varieties of
general type, deformation invariance of plurigenera is easy, leading to a
proof of (1.1) in Paragraph 13. This also implies that all fibers are
bimeromorphic to a projective variety.

If $X_{0}$ is smooth, then it is Kähler, and the $X_{s}$ are also Kähler by
[KS58]. A Kähler variety that is bimeromorphic to an algebraic variety is
projective by [Moi66].

However, there are families of surfaces with simple elliptic singularities
$g:X\to S$ such that $K_{X_{0}}$ is ample, all fibers are bimeromorphic to an
algebraic surface, yet the projective fibers correspond to a countable, dense
set on the base; see Example 5.

We use Theorem 14—taken from [Kol21b, Thm.2]—to obtain the projectivity of the
fibers and complete the proof of Theorem 1 in Paragraph 13.

##  1\. Examples and consequences

The first example shows that Theorem 1 fails very badly for surfaces with non-
canonical quotient singularities.

######  Example 3.

We give an example of a flat, proper morphism of complex analytic spaces
$g:X\to{\mathbb{D}}$, such that

  1. (3.1)

$X_{0}$ is a projective surface with a quotient singularity and ample
canonical class, yet

  2. (3.2)

$X_{s}$ is smooth, non-algebraic, and of Kodaira dimension 0 for very general
$s\in{\mathbb{D}}$.

Let us start with a K3 surface $Y_{0}\subset{\mathbb{P}}^{3}$ with a
hyperplane section $C_{0}\subset Y_{0}$ that is a rational curve with 3 nodes.
We blow up the nodes $Y^{\prime}_{0}\to Y_{0}$ and contract the birational
transform of $C_{0}$ to get a surface $\tau_{0}:Y^{\prime}_{0}\to X_{0}$. Let
$E_{1},E_{2},E_{3}\subset X_{0}$ be the images of the 3 exceptional curves of
the blow-up.

By explicit computation, we get a quotient singularity of type
${\mathbb{C}}^{2}/\frac{1}{8}(1,1)$, $(E_{i}^{2})=-\frac{1}{2}$ and
$(E_{i}\cdot E_{j})=\frac{1}{2}$ for $i\neq j$. Furthermore,
$E:=E_{1}+E_{2}+E_{3}\sim K_{X_{0}}$ and it is ample by the Nakai-Moishezon
criterion. (Note that $(E\cdot E_{i})=\frac{1}{2}$ and $X_{0}\setminus E\cong
Y_{0}\setminus C_{0}$ is affine.)

Take now a deformation $Y\to{\mathbb{D}}$ of $Y_{0}$ whose very general fibers
are non-algebraic K3 surfaces that contain no proper curves. Take 3 sections
$B_{i}\subset Y$ that pass through the 3 nodes of $C_{0}$. Blow them up and
then contract the birational transform of $C_{0}$; cf. [MR71]. In general
[MR71] says that the normalization of the resulting central fiber is $X_{0}$,
but in our case the central fiber is isomorphic to $X_{0}$ since
$R^{1}(\tau_{0})_{*}{\mathcal{O}}_{Y^{\prime}_{0}}=0$. The contraction is an
isomorphism on very general fibers since there are no curves to contract. We
get $g:X\to{\mathbb{D}}$ whose central fiber is $X_{0}$ and all other fibers
are K3 surfaces blown up at 3 points.

In general, it is very unclear which complex varieties occur as deformations
of projective varieties; see [KLS21] for some of their properties.

######  Example 4.

[Ati58] Let $S_{0}:=(g=0)\subset{\mathbb{P}}^{3}_{\mathbf{x}}$ and
$S_{1}:=(f=0)\subset{\mathbb{P}}^{3}_{\mathbf{x}}$ be surfaces of the same
degree. Assume that $S_{0}$ has only ordinary nodes, $S_{1}$ is smooth,
$\operatorname{Pic}(S_{1})$ is generated by the restriction of
${\mathcal{O}}_{{\mathbb{P}}^{3}}(1)$ and $S_{1}$ does not contain any of the
singular points of $S_{0}$. Fix $m\geq 2$ and consider

$X_{m}:=(g-t^{m}f=0)\subset{\mathbb{P}}^{1}_{\mathbf{x}}\times{\mathbb{A}}^{1}_{t}.$

The singularities are locally analytically of the form $xy+z^{2}-t^{m}=0$.
Thus $X_{m}$ is locally analytically factorial if $m$ is odd. If $m$ is even
then $X_{m}$ is factorial since the general fiber has Picard number 1, but it
is not locally analytically factorial; blowing up $(x=z-t^{m/2}=0)$ gives a
small resolution. Thus we get that

  1. (4.1)

$X_{m}$ is bimeromorphic to a proper, smooth family of projective surfaces iff
$m$ is even, but

  2. (4.2)

$X_{m}$ is not bimeromorphic to a smooth, projective family of surfaces.

######  Example 5.

Let $E\subset{\mathbb{P}}^{2}$ be a smooth cubic and take $r$ general lines
$L_{i}\subset{\mathbb{P}}^{2}$. To get $S_{0}$, blow up all singular points of
$E+\textstyle{\sum}L_{i}$ and then contract the birational transform of
$E+\textstyle{\sum}L_{i}$. A somewhat tedious computation shows that
$K_{S_{0}}$ is ample for $r\geq 6$. It has 1 simple elliptic singularity
(coming from $E$) and $r$ quotient singularities (coming from the $L_{i}$).

Deform this example by moving the $3r$ points $E\cap\textstyle{\sum}L_{i}$
into general position $p^{1}_{t},\dots,p^{3r}_{t}\in E$ and the points
$L_{i}\cap L_{j}$ into general position on ${\mathbb{P}}^{2}$. Blow up these
points and then contract the birational transform of $E$ to get the surfaces
$S_{t}$. It has only 1 simple elliptic singularity (coming from $E$).

We get a flat family of surfaces with central fiber $S_{0}$ and general fibers
$S_{t}$. Let $L$ denote the restriction of the line class on
${\mathbb{P}}^{2}$ to $E$.

It is easy to see that such a surface $S_{t}$ is non-projective if the
$p^{i}_{t}$ and $L$ are linearly independent in $\operatorname{Pic}(E)$. Thus
$S_{t}$ is not projective for very general $t$ and has Kodaira dimension 0.

The next result is the scheme-theoretic version of Theorem 1. Ideally it
should be proved by the same argument. However, some of the references we use,
especially [Nak04], are worked out for analytic spaces, not for general
schemes. So for now we proceed in a somewhat roundabout way.

######  Corollary 6.

Let $S$ be a noetherian, excellent scheme over a field of characteristic 0.
Let $g:X\to S$ be a flat, proper algebraic space. Fix a point $0\in S$ and
assume that $X_{0}$ is projective, of general type and with canonical
singularities. Then there is an open neighborhood $0\in S^{\circ}\subset S$
such that, for every $s\in S^{\circ}$,

  1. (6.1)

the plurigenera $h^{0}(X_{s},\omega_{X_{s}}^{[r]})$ are independent of $s$ for
every $r$, and

  2. (6.2)

the fiber $X_{s}$ is projective.

###### Proof.

A proper algebraic space $Y$ over a field $k$ is projective iff $Y_{K}$ is
projective over $K$ for some field extension $K\supset k$. Noetherian
induction then shows that it is enough to prove the claims for the generic
points of the completions (at the point $0\in S$) of irreducible subvarieties
$0\in T\subset S$. Since the defining equations of $\hat{T}$ and of
$X\times_{S}\hat{T}$ involve only countably many coefficients, we may assume
that the residue field is ${\mathbb{C}}$.

Consider now the local universal deformation space $\operatorname{Def}(X_{0})$
of $X_{0}$ in the complex analytic category; see [Bin87]. It is the germ of a
complex analytic space and there is a complex analytic universal family
$G:{\mathbf{X}}\to\operatorname{Def}(X_{0}).$ Since a deformation over an
Artin scheme is automatically complex analytic, we see that the formal
completion $\hat{G}:\hat{\mathbf{X}}\to\widehat{\operatorname{Def}}(X_{0})$ is
the universal formal deformation of $X_{0}$. In particular,
$X\times_{S}\hat{T}$ is the pull-back of
$\hat{G}:\hat{\mathbf{X}}\to\widehat{\operatorname{Def}}(X_{0})$ by a morphism
$\hat{T}\to\widehat{\operatorname{Def}}(X_{0})$. Thus Theorem 1 implies both
claims. ∎

##  2\. Relative MMP

See [KM98] for a general introduction to the minimal model program.

######  Definition 7 (MMP-steps and their extensions).

Let $X\to S$ be a proper morphism of complex analytic spaces with irreducible
fibers. Assume that $K_{X/S}$ is ${\mathbb{Q}}$-Cartier. By an MMP-step for
$X$ over $S$ we mean a diagram

$None$
$\begin{array}[]{lcr}X&\stackrel{{\scriptstyle\pi}}{{\dasharrow}}&X^{+}\\\
\phi\searrow&&\swarrow\phi^{+}\\\ &Z&\end{array}$

where all morphisms are bimeromorphic and proper over $S$, $-K_{X/S}$ is ample
over $Z$, $K_{X^{+}/S}$ is ample over $Z$ and $\phi^{+}$ is small (that is,
without exceptional divisors).

If $X$ is ${\mathbb{Q}}$-factorial and the relative Picard number of $X/Z$ is
1, then there are 2 possible MMP steps:

  * •

Divisorial: $\phi$ contracts a single divisor and $\phi^{+}$ is the identity.

  * •

Flipping: both $\phi$ and $\phi^{+}$ are small.

However, in general there is a more complicated possibility:

  * •

Mixed: $\phi$ contracts (possibly several) divisors and $\phi^{+}$ is small.

For our applications we only need to know that, by [KM98, 3.52], $X^{+}$
exists iff $\oplus_{r\geq 0}\ \omega_{Z/S}^{[r]}$ (which is equal to
$\oplus_{r\geq 0}\phi_{*}\omega_{X/S}^{[r]}$) is a finitely generated sheaf of
${\mathcal{O}}_{Z}$-algebras, and then

$None$ $X^{+}=\operatorname{Proj}_{Z}\oplus_{r\geq 0}\ \omega_{Z/S}^{[r]}.$

We index a sequence of MMP-steps by setting $X^{0}:=X$ and
$X^{i+1}:=(X^{i})^{+}$.

Fix a point $s\in S$ and let $X_{s}$ denote the fiber over $S$. We say that a
sequence of MMP-steps (over $S$) $X^{0}\dasharrow X^{1}\dasharrow
X^{2}\dasharrow\cdots$ extends a sequence of MMP-steps (over $s$)
$X_{s}^{0}\dasharrow X_{s}^{1}\dasharrow X_{s}^{2}\dasharrow\cdots$ if, for
every $i$,

$None$
$\begin{array}[]{rcl}X_{s}^{i}&\stackrel{{\scriptstyle\pi_{s}^{i}}}{{\dasharrow}}&\quad
X_{s}^{i+1}\\\ \phi_{s}^{i}\searrow&&\swarrow(\phi_{s}^{i})^{+}\\\
&Z_{s}^{i}&\end{array}\quad\begin{array}[]{cc}\mbox{is the fiber}\\\
\mbox{over $s$
of}\end{array}\quad\begin{array}[]{rcl}X^{i}&\stackrel{{\scriptstyle\pi^{i}}}{{\dasharrow}}&\quad
X^{i+1}\\\ \phi^{i}\searrow&&\swarrow(\phi^{i})^{+}\\\ &Z^{i}&\end{array}$

######  8Proof of Theorem 2 for $S={\mathbb{D}}$, the disc.

Since MMP-steps preserve canonical singularities, by induction it is enough to
prove the claim for one MMP step. So we drop the upper index $i$ and identify
$K_{X/{\mathbb{D}}}$ with $K_{X}$.

Let $\phi_{0}:X_{0}\to Z_{0}$ be an extremal contraction. By [MR71]111This
should be changed to [KM92, 11.4], it extends to a contraction $\phi:X\to Z$,
where $Z$ is flat over ${\mathbb{D}}$ with central fiber $Z_{0}$ since
$R^{1}(\phi_{0})_{*}{\mathcal{O}}_{X_{0}}=0$. Note that $K_{X}$ is
${\mathbb{Q}}$-Cartier by (10.1), and $\phi$ is projective since $-K_{X}$ is
$\phi$-ample.

If $\phi_{0}$ is a divisorial contraction, then $K_{Z_{0}}$ is
${\mathbb{Q}}$-Cartier, and so is $K_{Z}$ by (10.1). Thus $X^{+}=Z$.

If $\phi_{0}$ is a flipping or mixed contraction, then $K_{Z}$ is not
${\mathbb{Q}}$-Cartier. By (7.2),

$None$ $X^{+}=\operatorname{Proj}_{Z}\oplus_{r\geq 0}\ \omega_{Z}^{[r]},$

provided $\oplus_{r\geq 0}\ \omega_{Z}^{[r]}$ is a finitely generated sheaf of
${\mathcal{O}}_{Z}$-algebras. (We have identified $\omega_{Z}$ with
$\omega_{Z/{\mathbb{D}}}$.)

Functoriality works better if we twist by the line bundle
${\mathcal{O}}_{Z}(Z_{0})$ and write it as

$X^{+}=\operatorname{Proj}_{Z}\oplus_{r\geq 0}\ \omega_{Z}^{[r]}(rZ_{0}).$

Let $\tau:Y\to X$ be a projective resolution of $X$ (that is, $\tau$ is
projective) such that $Y_{0}$, the bimeromorphic transform of $X_{0}$, is also
smooth. Set $g:=\phi\circ\tau$.

The hardest part of the proof is Nakayama’s theorem (9) which gives a
surjection

$None$ $\oplus_{r\geq
0}g_{*}\omega_{Y}^{r}(rY_{0})\twoheadrightarrow\oplus_{r\geq
0}(g_{0})_{*}\omega_{Y_{0}}^{r}.$

Since $X_{0}$ has canonical singularities
$\tau_{*}\omega_{Y_{0}}^{r}=\omega_{X_{0}}^{[r]}$, and hence
$g_{*}\omega_{Y_{0}}^{r}=\omega_{Z_{0}}^{[r]}$. We also have a natural
inclusion
$g_{*}\omega_{Y}^{r}(rY_{0})\lhook\joinrel\to\omega_{Z}^{[r]}(rZ_{0})$. Thus
pushing forward (8.2) we get a surjection

$None$ $\oplus_{r\geq 0}g_{*}\omega_{Y}^{r}(rY_{0})\to\oplus_{r\geq 0}\
\omega_{Z}^{[r]}(rZ_{0})\twoheadrightarrow\oplus_{r\geq 0}\
\omega_{Z_{0}}^{[r]}.$

Note that $\oplus_{r\geq 0}\ \omega_{Z_{0}}^{[r]}$ is a finitely generated
sheaf of ${\mathcal{O}}_{Z_{0}}$-algebras, defining the MMP-step of $X_{0}\to
Z_{0}$.

Now (11) says that $\oplus_{r\geq 0}\ \omega_{Z}^{[r]}(rZ_{0})$ is also a
finitely generated sheaf of ${\mathcal{O}}_{Z}$-algebras, at least in some
neighborhood of the compact $Z_{0}$. ∎

Next we discuss various results used in the proof.

######  Theorem 9.

[Nak04, VI.3.8] Let $\pi:Y\to S$ be a projective, bimeromorphic morphism of
analytic spaces, $Y$ smooth and $S$ normal. Let $D\subset Y$ be a smooth, non-
exceptional divisor. Then the restriction map

$\pi_{*}\omega_{Y}^{m}(mD)\to\pi_{*}\omega_{D}^{m}\quad\mbox{is surjective for
$m\geq 1$.}\quad\qed$

This is a special case of [Nak04, VI.3.8] applied with $\Delta=0$ and
$L=K_{Y}+D$.

Warning. The assumptions of [Nak04, VI.3.8] are a little hard to find. They
are outlined 11 pages earlier in [Nak04, VI.2.2]. It talks about varieties,
which usually suggest algebraic varieties, but [Nak04, p.231, line 13]
explicitly states that the proofs work with analytic spaces; see also [Nak04,
p.14]. (The statements of [Nak04] allow for a boundary $\Delta$. However,
$K_{Y}+D+\Delta$ should be ${\mathbb{Q}}$-linearly equivalent to a
${\mathbb{Z}}$-divisor and $\lfloor{\Delta}\rfloor=0$ is assumed on [Nak04,
p.231]. There seem to be few cases when both of these can be satisfied.)

######  Lemma 10.

[Nak04, VI.5.2] Let $g:X\to S$ be a flat morphism of complex analytic spaces.
Assume that $X_{0}$ has a canonical singularity at a point $x\in X_{0}$. Then
there is an open neighborhood $x\in X^{*}\subset X$ such that

  1. (10.1)

$K_{X^{*}/S}$ is ${\mathbb{Q}}$-Cartier, and

  2. (10.2)

all fibers of $g|_{X^{*}}:X^{*}\to S$ have canonical singularities.

###### Proof.

(1) is proved in [Kol83, 3.2.2]; see also [Kol95, 12.7] and [Kol21a, 2.8]. The
harder part is (2), proved in [Nak04, VI.5.2]. ∎

Remark 10.3. If $S$ is smooth then $X^{*}$ has canonical singularities. By
induction, it is enough to prove this when $S={\mathbb{D}}$. Then the proof of
[Nak04, VI.5.2] shows that even the pair $(X^{*},X_{0}\cap X^{*})$ has
canonical singularities.

######  Lemma 11.

Let $\pi:X\to S$ be a proper morphism of normal, complex spaces. Let $L$ be a
line bundle on $X$ and $W\subset S$ a Zariski closed subset. Assume that
${\mathcal{O}}_{W}\otimes_{S}\bigl{(}\oplus_{r\geq 0}\pi_{*}L^{r}\bigr{)}$ is
a finitely generated sheaf of ${\mathcal{O}}_{W}$-algebras.

Then every compact subset $W^{\prime}\subset W$ has an open neighborhood
$W^{\prime}\subset U\subset S$ such that
${\mathcal{O}}_{U}\otimes_{S}\bigl{(}\oplus_{r\geq 0}\pi_{*}L^{r}\bigr{)}$ is
a finitely generated sheaf of ${\mathcal{O}}_{U}$-algebras.

###### Proof.

The question is local on $S$, so we may as well assume that $W$ is a single
point. We may also assume that
${\mathcal{O}}_{W}\otimes_{S}\bigl{(}\oplus_{r\geq 0}\pi_{*}L^{r}\bigr{)}$ is
generated by $\pi_{*}L$. After suitable blow-ups we are reduced to the case
when the base locus of $L$ is a Cartier divisor $D$. By passing to a smaller
neighborhood, we may assume that every irreducible component of $D$ intersects
$\pi^{-1}(W)$. By the Nakayama lemma, the base locus of $L^{r}$ is a subscheme
of $rD$ that agrees with it along $rD\cap\pi^{-1}(W)$. Thus $rD$ is the the
base locus of $L^{r}$ for every $r$. We may thus replace $L$ by $L(-D)$ and
assume that $L$ is globally generated.

Thus $L$ defines a morphism $X\to\operatorname{Proj}_{S}\oplus_{r\geq
0}\pi_{*}L^{r}$, let $\pi^{\prime}:X^{\prime}\to S$ be its Stein
factorization. Then $L$ is the pull-back of a line bundle $L^{\prime}$ that is
ample on $X^{\prime}\to S$ and $\oplus_{r\geq 0}\pi_{*}L^{r}=\oplus_{r\geq
0}\pi^{\prime}_{*}{L^{\prime}}^{r}$ is finitely generated. ∎

######  12Proof of Theorem 2 for general $S$.

As in Paragraph 8, it is enough to prove the claim for one MMP step, so let
$\phi_{0}:X_{0}\to Z_{0}$ be an extremal contraction and $\phi:X\to Z$ its
extension. As before, $Z$ is flat over $S$ with central fiber $Z_{0}$.

We claim that, for every $r$,

  1. (12.1)

$\omega_{Z/S}^{[r]}$ is flat over $S$, and

  2. (12.2)

$\omega_{Z/S}^{[r]}|_{Z_{0}}\cong\omega_{Z_{0}}^{[r]}$.

In the language of [Kol08] or [Kol21a, Chap.9], this says that
$\omega_{Z/S}^{[r]}$ is its own relative hull. There is an issue with precise
references here, since [Kol21a, Chap.9] is written in the algebraic setting.
However, [Kol21a, 9.72] considers hulls over the spectra of complete local
rings. Thus we get that there is a unique largest subscheme
$\hat{S}^{u}\subset\hat{S}$ (the formal completion of $S$ at $0$) such that
(1–2) hold after base change to $\hat{S}^{u}$.

By Paragraph 8 we know that (1–2) hold after base change to any disc
${\mathbb{D}}\to S$, which implies that $\hat{S}^{u}=\hat{S}$. That is, (1–2)
hold for $\hat{S}$. Since both properties are invariant under formal
completion, we are done.

Now we know that

$None$ $X^{+}:=\operatorname{Proj}_{Z}\oplus_{r\geq 0}\ \omega_{Z/S}^{[r]},$

is flat over $S$ and its central fiber is $X^{+}_{0}$. Thus it gives the
required extension of the flip of $X_{0}\to Z_{0}$. ∎

##  3\. Proof of Theorem 1

We give a proof using only the $S={\mathbb{D}}$ case of Theorem 2.

######  13.

Fix $r\geq 2$ and assume first that $S={\mathbb{D}}$. Since $X_{0}$ is of
general type, a suitable MMP for $X_{0}$ ends with a minimal model $X_{0}^{\rm
m}$, and, by Theorem 2, $X_{0}\dasharrow X_{0}^{\rm m}$ extends to a fiberwise
bimeromorphic map $X\dasharrow X^{\rm m}$. We have $g^{\rm m}:X^{\rm
m}\to{\mathbb{D}}$. (From now on, we replace ${\mathbb{D}}$ with a smaller
disc whenever necessary.) Since $K_{X_{0}^{\rm m}}$ is nef and big, the higher
cohomology groups of $\omega_{X_{0}}^{[r]}$ vanish for $r\geq 2$. Thus
$s\mapsto H^{0}(X^{\rm m}_{s},\omega_{X^{\rm m}_{s}}^{[r]})$ is locally
constant at the origin.

By (10.2) $X_{s}$ and $X^{\rm m}_{s}$ both have canonical singularities, so
they have the same plurigenera. Therefore $s\mapsto
H^{0}(X_{s},\omega_{X_{s}}^{[r]})$ is also locally constant at the origin. By
Serre duality, the deformation invariance of $H^{0}(X_{s},\omega_{X_{s}})$ is
equivalent to the deformation invariance of
$H^{n}(X_{s},{\mathcal{O}}_{X_{s}})$. In fact, all the
$H^{i}(X_{s},{\mathcal{O}}_{X_{s}})$ are deformation invariant. For this the
key idea is in [DJ74], which treats deformations of varieties with normal
crossing singularities. The method works for varieties with canonical (even
log canonical) singularities; this is worked out in [Kol21a, Sec.2.5].

For arbitrary $S$, note that $s\mapsto H^{0}(X_{s},\omega_{X_{s}}^{[r]})$ is a
constructible function on $S$, thus locally constant at $0\in S$ iff it is
locally constant on every disc ${\mathbb{D}}\to S$. Once $s\mapsto
H^{0}(X_{s},\omega_{X_{s}}^{[r]})$ is locally constant at $0\in S$, Grauert’s
theorem guarantees that $g_{*}\omega_{X/S}^{[r]}$ is locally free at $0\in S$
and commutes with base changes.

In principle it could happen that for each $r$ we need a smaller and smaller
neighborhood, but the same neighborhood works for all $r\geq 1$ by Lemma 11.

Thus the plurigenera are deformation invariant, all fibers are of general
type, and $g$ is fiberwise bimeromorphic to the relative canonical model

$X^{\rm c}:=\operatorname{Proj}_{S}\oplus_{r\geq 0}g^{\rm m}_{*}\omega_{X^{\rm
m}/S}^{[r]},$

which is projective over $S$. The projectivity of all fibers now follows from
the more precise Theorem 14. ∎

The following is a special case of [Kol21b, Thm.2].

######  Theorem 14.

Let $g:X\to S$ be a flat, proper morphism of complex analytic spaces whose
fibers have rational singularities only. Assume that $g$ is bimeromorphic to a
projective morphism $g^{\rm p}:X^{\rm p}\to S$, and $X_{0}$ is projective for
some $0\in S$.

Then there is a Zariski open neighborhood $0\in U\subset S$ and a locally
closed, Zariski stratification $S=\cup_{i}S_{i}$ such that each

$g|_{X_{i}}:X_{i}:=g^{-1}(S_{i})\to S_{i}\quad\mbox{is
projective.}\quad\hfill\qed$

##  4\. Open problems

For deformations of varieties of general type, the following should be true.

######  Conjecture 15.

Let $X_{0}$ be a projective variety of general type with canonical
singularities. Then its universal deformation space
$\operatorname{Def}(X_{0})$ has a representative ${\mathbf{X}}\to S$ where $S$
is a scheme of finite type and ${\mathbf{X}}$ is an algebraic space.

For varieties of non-general type, the following is likely true [RT20, 1.10].

######  Conjecture 16.

Let $g:X\to S$ be a flat, proper morphism of complex analytic spaces. Assume
that $X_{0}$ is projective and with canonical singularities. Then the
plurigenera $h^{0}(X_{s},\omega_{X_{s}}^{[r]})$ are independent of $s\in S$
for every $r$, in some neighborhood of $0\in S$.

Comments. One can try to follow the proof of Theorem 1. If $X_{0}$ is not of
general type, we run into several difficulties in relative dimensions $\geq
4$. MMP is not know to terminate and even if we get a minimal model, abundance
is not known. If we have a good minimal model, then we run into the following.

######  Conjecture 17.

Let $X$ be a complex space and $g:X\to S$ a flat, proper morphism. Assume that
$X_{0}$ is projective, has canonical singularities and $\omega_{X_{0}}^{[r]}$
is globally generated for some $r>0$. Then the plurigenera are locally
constant at $0\in S$.

Comments. More generally, the same may hold if $X_{0}$ is Moishezon (that is,
bimeromorphic to a projective variety), Kähler or in Fujiki’s class
$\mathcal{C}$ (that is, bimeromorphic to a compact Kähler manifold; see
[Uen83] for an introduction).

A positive answer is known in many cases. [KM92, 12.5.5] proves this if
$X_{0}$ is projective and has terminal singularities. However, the proof works
for the Moishezon and class $\mathcal{C}$ cases as well.

The projective case with canonical singularities is discussed in [Nak04,
VI.3.15–16]; I believe that the projectivity assumption is very much built
into the proof given there; see [Nak04, VI.3.11].

######  Acknowledgments.

I thank D. Abramovich, F. Campana, J.-P. Demailly, O. Fujino, A. Landesman, S.
Mori, T. Murayama, V. Tosatti, D. Villalobos-Paz, C. Voisin and C. Xu for
helpful comments and corrections. Partial financial support was provided by
the NSF under grant number DMS-1901855.

## References

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  * [KM98] by same author, _Birational geometry of algebraic varieties_ , Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original.
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Princeton University, Princeton NJ 08544-1000,


$t_{i}^{2}=\lambda\log x_{i-1,i}^{2}\log x_{i,i+1}^{2}$ held fixed. Physically
this limit is controlled by operators with large spin and is similar to the
large spin limit discussed above. To make a more concrete connection, it would
be interesting to study a similar double-scaling limit (with $\lambda$
replaced with $g_{\rm YM}^{2}$) for the correlation functions in the ${\rm
SU}(2)$ theory.

In addition, the results in [137] are given by determinants of Bessel
functions, which also show up in the analysis of the large $N_{c}$ 2d YM. One
interesting question is if the correlation functions studied in [137] can be
mapped to some observables555555The correlation function in the light-like
limit is known to be related to a null polygonal Wilson loop [138]. In the
planar limit, the null polygonal Wilson loop at finite $\lambda$ can be
studied using the so-called Pentagon decomposition [139], in which the basic
building block is the pentagon form factor which can be interpreted as a form
factor of a branch-point twist operator [140] on the two-dimensional
worldsheet. It is therefore tempting to conjecture that the relevant
observable in this limit is a branch-point twist operator in $2d$ YM. in the
effective 2d YM.

##### Combining large $N$ and large $J$ and black holes.

For the application to (standard) holography, it is important to study the
combined large $N$-large $J$ limits. One difficulty in this case is that the
semiclassical analysis cannot be easily justified. To understand this point,
let us recall the basics of the standard ’t Hooft limit. The action of
$\mathcal{N}=4$ SYM is proportional to $N/\lambda$, which is infinitely large
in the ’t Hooft limit,

$S\sim\frac{N}{\lambda}\int d^{4}x{\rm Tr}\left[\cdots\right]\,.$ (6.2)

One might therefore think that the limit can be studied using the saddle point
approximation. As is well-known, this conclusion is incorrect for the
following reason: if we approximate the path integral by its saddle point, the
saddle point action scales like

$S_{\rm saddle}\sim\frac{N}{\lambda}\left.{\rm
Tr}\left[\cdots\right]\right|_{\rm saddle}\sim\frac{N^{2}}{\lambda}\,.$ (6.3)

However, since there are $\sim N^{2}$ dynamical degrees of freedom, the one-
loop correction to the saddle point contribution also scales like

$S_{\text{1-loop}}\sim N^{2}\,.$ (6.4)

Thus, unless we take $\lambda$ to be small, the saddle-point approximation
cannot be justified.

One potential way to overcome this difficulty is to combine it with the large
$J$ limit. For instance, when the saddle point configuration scales
like565656The scaling with respect to $g_{\rm YM}$ and $J$ is fixed by the
fact that the Noether charge is quadratic in fields and proportional to
$g_{\rm YM}^{2}$; namely $J\sim\frac{1}{g_{\rm YM}^{2}}{\rm Tr}[\Phi_{\rm
cl}^{2}]$. The exponent $\alpha$ depends on how the charge is distributed
among different components of the ${\rm SU}(N)$ adjoint field. For instance,
if the semiclassical configuration only has a single non-vanishing diagonal
entry, we expect $\alpha=0$ while if it is distributed equally among $N$
diagonal entries, we expect $\alpha=1$. $g_{\rm YM}\sqrt{J}/N^{\alpha}$, (6.3)
will be enhanced to

$S_{\rm saddle}\sim\frac{N^{2}}{\lambda}\frac{g_{\rm
YM}^{2}J}{N^{2\alpha}}\,,$ (6.5)

while the scaling for $S_{\text{1-loop}}$ remains the same. By taking the
ratio $g_{\rm YM}\sqrt{J}/N^{\alpha}\sim\sqrt{\lambda J/N}/N^{\alpha}$ to be
large, we can then suppress the 1-loop correction parametrically. Of course
the estimate we gave here is rather crude and it is important to perform
detailed analysis for concrete cases.

It would be interesting to study double-scaling limits also on the gravity
side. When the ’t Hooft coupling $\lambda$ is infinite, the bulk is described
by supergravity. As we decrease $\lambda$, it receives higher derivative
corrections proportional to some powers of $1/\lambda$. On the other hand, the
effects of these higher derivative corrections are suppressed for a black hole
of a very large size. Since the size of the black hole with a large quantum
number $J$ is controlled roughly by the parameter $J/N^{2}$, the effective
strength of higher derivative corrections is parametrized by
$\frac{N^{2}}{\lambda^{\beta}J}$ with some positive exponent $\beta$. This
offers the possibility to study the limit $\frac{N^{2}}{\lambda^{\beta}J}\ll
1$ both at weak and strong couplings. At strong coupling, it is described by
supergravity as discussed above, while we can try to reach this limit from the
weak coupling by first taking the double-scaling limit in which $\lambda\to 0$
with $\tilde{\lambda}\equiv\lambda^{\beta}J/N^{2}$ fixed and then later taking
the limit $\tilde{\lambda}\to\infty$, much like what we did in this paper for
the strong coupling limit of $\lambda_{J}$. Note that there is no guarantee
that different ways of taking the limits give the same answer and more
detailed analysis is certainly necessary.

#####  $\mathcal{N}=2$ SCFT.

Much of our analysis can be generalized to study large charge operators and
their correlation functions in the double-scaling limit for general
${\mathcal{N}}=2$ SCFTs with a conformal manifold such as the superconformal
QCD (SQCD) described by ${\mathcal{N}}=2$ ${\rm SU}(N)$ SYM coupled to $2N$
fundamental hypermultiplets. Previous works on large charge limits of these
${\mathcal{N}}=2$ theories have focused on the two-point function of BPS
operators [23, 27, 26] which contain information about the extremal correlator
sector of the SCFT [28] and are nontrivial due to the reduced supersymmetry.
In particular, at strong coupling $\lambda_{J}\gg 1$, these two-point
functions exhibit an infinite tower of worldline instanton contributions
similar to what we have seen here for more general observables in the
${\mathcal{N}}=4$ SYM. Nonetheless, it would be much more interesting to
extend the ${\mathcal{N}}=2$ large charge analysis to non-BPS operators and
also higher-point functions by exploiting our methods here.

Indeed, one copy of the maximally-centrally-extended $\mathfrak{psu}(2|2)$
symmetry that played an important role in our “bootstrap” analysis of the non-
BPS large charge operator spectrum survives in general ${\mathcal{N}}=2$
theories. In this case, the superconformal R-symmetry is
${\mathfrak{u}}(1)_{R}\times\mathfrak{su}(2)_{R}$ and the relevant large
charge operators are Coulomb branch chiral primary operators
${\mathcal{O}}_{J}$ that carry charge $J>0$ under the ${\mathfrak{u}}(1)_{R}$
and are singlet under $\mathfrak{su}(2)_{R}$. The BPS condition reads
$\Delta=J$ and thus as before, the operator inserted at origin is preserved by
the superconformal generator $C\equiv{\hat{D}-\hat{J}\over 2}$. Together with
the rotation symmetry $\mathfrak{so}(4)$ and the $\mathfrak{su}(2)_{R}$
symmetry, the full superconformal subalgebra preserved by the operator
insertion ${\mathcal{O}}_{J}(0)$ is

$\displaystyle\mathfrak{su}(2)\times\mathfrak{psu}(2|2)\ltimes{\mathbb{R}}\,,$
(6.6)

compared to (3.4) for the ${\mathcal{N}}=4$ case (see around (3.3)). In the
large charge ’t Hooft limit, we expect this symmetry to be further centrally
extended, which includes a maximally-centrally-extended
$\mathfrak{psu}(2|2)\ltimes{\mathbb{R}}^{3}$ factor, that would govern the
physics of excitations on the large charge state created by
${\mathcal{O}}_{J}(0)$. It would be interesting to carry out this analysis
explicitly, for example in the case of the ${\mathcal{N}}=2$ ${\rm SU}(2)$
SQCD, to learn about the non-BPS large charge operators.

Another potentially intriguing direction is to develop a better understanding
of the large charge EFT in the large charge ’t Hooft limit which shares many
similarities with the conventional Coulomb branch EFT, as we have commented on
in the main text, in general ${\mathcal{N}}=2$ theories. It is well-known that
the latter EFT has an elegant low energy description in terms of the Seiberg-
Witten curve and the complex geometric data therein, largely thanks to the
${\mathcal{N}}=2$ Poincaré supersymmetry [122, 123]. It is thus tantalizing to
speculate a version of the Seiberg-Witten curve for the large charge EFT that
follows from the centrally extended $\mathfrak{psu}(2|2)$ symmetry.575757See
Section 3.1.4 for comments on the relation between the centrally extended
$\mathfrak{psu}(2|2)$ symmetry preserved in the large charge EFT and the
Poincaré supersymmetry that govern the Coulomb branch EFT. In particular, it
would be interesting to extract the correlation functions of large charge
Coulomb branch operators from such an EFT. We leave this investigation to
future work.

### Acknowledgement

We thank Gabriel Cuomo, Nicola Dondi, Simeon Hellerman, Zohar Komargodski,
Andrew McLeod, Domenico Orlando, Eric Perlmutter, Himanshu Raj, Susanne
Reffert, Raffaele Savelli, Kostya Zarembo, Sasha Zhiboedov and Masataka
Watanabe for helpful discussions. We in particular thank Eric Perlmutter for
communicating to us the results of [54] prior to the publication. SK thanks
the organizers and participants of the workshop “Large Charge in Les
Diablerets” in SwissMAP Research Station for an opportunity to present this
work and have stimulating discussions. The work of YW was supported in part by
NSF grant PHY-2210420 and by the Simons Junior Faculty Fellows program.

##  Appendix A Action & conventions

In this appendix, we establish our conventions for the flat space action,
which will be used for perturbative computations. We begin with the
$\mathcal{N}=4$ SYM action which we write as

$\mathcal{S}_{\mathcal{N}=4{\,\rm{SYM}}}=\frac{2}{g^{2}_{\rm{YM}}}\int
d^{4}x{\rm
Tr\,}\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}D_{\mu}\Phi_{I}D^{\mu}\Phi_{I}+\frac{1}{4}[\Phi_{I},\Phi_{J}][\Phi_{I},\Phi_{J}]+{\rm{fermions}}\right)\,,$
(A.1)

where we omit the fermionic part of the action since we will not use it
explicitly.

We express the six real scalar fields in terms of complex scalar fields in the
usual manner,

$\displaystyle\Phi_{1}$
$\displaystyle=\frac{Z+\bar{Z}}{\sqrt{2}},\quad\Phi_{2}=\frac{Z-\bar{Z}}{i\sqrt{2}},\quad\Phi_{3}=\frac{X+\bar{X}}{\sqrt{2}},\quad\Phi_{4}=\frac{X-\bar{X}}{i\sqrt{2}},\quad\Phi_{5}=\frac{Y+\bar{Y}}{\sqrt{2}},\quad\Phi_{6}=\frac{Y-\bar{Y}}{i\sqrt{2}}.$
(A.2)

Note that each of the complex scalar fields takes values on the Lie algebra of
the gauge group SU(2), with the generators of the Lie algebra taken to be
Hermitian in our conventions. This allows us to represent the fields
explicitly as follows

$\displaystyle Z=\begin{pmatrix}z^{0}&z^{+}\\\
z^{-}&-z^{0}\end{pmatrix},\quad$
$\displaystyle\bar{Z}=\begin{pmatrix}\bar{z}^{0}&\bar{z}^{-}\\\
\bar{z}^{+}&-\bar{z}^{0}\end{pmatrix},$ (A.3) $\displaystyle
X=\begin{pmatrix}x^{0}&x^{+}\\\ x^{-}&-x^{0}\end{pmatrix},\quad$
$\displaystyle\bar{X}=\begin{pmatrix}\bar{x}^{0}&\bar{x}^{-}\\\
\bar{x}^{+}&-\bar{x}^{0}\end{pmatrix},$ $\displaystyle
Y=\begin{pmatrix}y^{0}&y^{+}\\\ y^{-}&-y^{0}\end{pmatrix},\quad$
$\displaystyle\bar{Y}=\begin{pmatrix}\bar{y}^{0}&\bar{y}^{-}\\\
\bar{y}^{+}&-\bar{y}^{0}\end{pmatrix}\,.$

Similarly, the gluons, fermions, and ghost fields can be decomposed in the
same way

$A_{\mu}=\begin{pmatrix}a_{{}_{\mu}}^{0}&a_{{}_{\mu}}^{+}\\\
a_{{}_{\mu}}^{-}&-a_{{}_{\mu}}^{0}\end{pmatrix},\quad\psi=\begin{pmatrix}\psi^{0}&\psi^{+}\\\
\psi^{-}&-\psi^{0}\end{pmatrix},\quad\bar{\psi}=\begin{pmatrix}\bar{\psi}^{0}&\bar{\psi}^{-}\\\
\bar{\psi}^{+}&-\bar{\psi}^{0}\end{pmatrix},$ (A.4)
$c=\begin{pmatrix}c^{0}&c^{+}\\\
c^{-}&-c^{0}\end{pmatrix},\quad\bar{c}=\begin{pmatrix}\bar{c}^{0}&\bar{c}^{-}\\\
\bar{c}^{+}&-\bar{c}^{0}\end{pmatrix}\,.$ (A.5)

Among these fields, $Z$ and $\bar{Z}$ acquire a classical profile in the large
charge background

$Z_{\rm{cl}}=\begin{pmatrix}z_{{\rm{cl}}}^{0}&0\\\
0&-z_{{\rm{cl}}}^{0}\end{pmatrix},\quad\bar{Z}_{\rm{cl}}=\begin{pmatrix}\bar{z}_{{\rm{cl}}}^{0}&0\\\
0&-\bar{z}_{{\rm{cl}}}^{0}\end{pmatrix}\,,$ (A.6)

around which we expand the action. In order to perform perturbative
computations, we first gauge fix the action by adding to it the gauge fixing
term

$\mathcal{S}_{\rm gf}=\frac{2}{g^{2}_{\rm{YM}}}\int
d^{4}x\,\left(-\frac{1}{2}{\rm
Tr\,}\left(\partial_{\mu}A^{\mu}+i[Z,\bar{Z}_{\rm cl}]+i[\bar{Z},Z_{\rm
cl}]\right)^{2}\right)$ (A.7)

and the corresponding ghost term

$\mathcal{S}_{\rm gh}=\frac{2}{g^{2}_{\rm{YM}}}\int d^{4}x\,{\rm
Tr\,}\left(\bar{c}\left(\partial^{\mu}D_{\mu}c-[[c,Z],Z_{\rm
cl}]-[[c,\bar{Z}],Z_{\rm cl}]-[[c,\bar{Z}_{\rm cl}],Z_{\rm cl}]-[[c,Z_{\rm
cl}],\bar{Z}_{\rm cl}]\right)\right)\,,$ (A.8)

where $c,\bar{c}$ are the ghost fields. The resulting effective action can be
split as follows

$\mathcal{S}_{\mathcal{N}=4{\,\rm{SYM}}}+\mathcal{S}_{\rm gf}+\mathcal{S}_{\rm
gh}=\mathcal{S}_{\rm{kin}}+\mathcal{S}_{\rm{m,b}}+\mathcal{S}_{\rm{cubic}}+\mathcal{S}_{\rm{quartic}}+{\rm{fermions}}\,,$
(A.9)

where from now on we will omit completely the discussion about the fermionic
part. The kinetic terms piece $\mathcal{S}_{\rm{kin}}$ is given by

$\displaystyle\mathcal{S}_{\rm{kin}}=\frac{2}{g_{\rm{YM}}^{2}}\int
d^{4}x\Bigl{[}a_{\mu}^{0}\Box(a^{0})^{\mu}+a_{\mu}^{+}\Box(a^{-})^{\mu}+2\sum_{\Phi\in\\{x,y,z,c\\}}\bar{\Phi}^{0}\Box\Phi^{0}+\sum_{\begin{subarray}{c}\Phi\in\\{x,y,z,c\\}\\\
p=\pm\end{subarray}}\bar{\Phi}^{p}\Box\Phi^{p}\Bigr{]}.$ (A.10)

where $\Box=\partial_{\mu}\partial^{\mu}$. The bosonic mass part of the action
$\mathcal{S}_{\rm{m,b}}$ reads

$\mathcal{S}_{\rm{m,b}}=-\frac{16}{g^{2}_{\rm{YM}}}\int
d^{4}x\Bigl{[}|z^{0}_{\rm{cl}}|^{2}\Bigl{(}a^{+}_{\mu}(a^{-})^{\mu}+\sum_{\begin{subarray}{c}\Phi\in\\{x,y,z\\}\\\
p=\pm\end{subarray}}|\Phi^{p}|^{2}\bigr{)}+\frac{i}{2}\,\partial^{\mu}\bar{z}^{0}_{\rm{cl}}\bigl{(}z^{+}a^{-}_{\mu}-z^{-}a^{+}_{\mu}\bigr{)}+\frac{i}{2}\,\partial^{\mu}{z}^{0}_{\rm{cl}}\bigl{(}\bar{z}^{-}a^{-}_{\mu}-\bar{z}^{+}a^{+}_{\mu}\bigr{)}\Bigr{]}.$
(A.11)

We note that the classical profile of $z^{0}_{\rm{cl}}$ and
$\bar{z}^{0}_{\rm{cl}}$ induces a mixing term between the gauge fields and the
scalars $z^{\pm},\bar{z}^{\pm}$.

We now proceed with the cubic part of the action, which due to its length can
be sub-split into several parts according to the field content

$\mathcal{S}_{\rm{cubic}}=\mathcal{S}_{\rm{cubic,g}}+\mathcal{S}_{\rm{cubic,s}}+\mathcal{S}_{\rm{cubic,g,s}}+\mathcal{S}_{\rm{cubic,s,gh}}\,.$
(A.12)

Each one of these parts read as

$\mathcal{S}_{\mathrm{cubic,g}}=\frac{2}{g_{\mathrm{YM}}^{2}}\int
d^{4}x\,2i\Big{[}(a_{\mu}^{-}a_{\nu}^{0}-a_{\nu}^{-}a_{\mu}^{0})\partial^{\nu}(a^{\mu})^{+}+(a_{\nu}^{+}a_{\mu}^{0}-a_{\mu}^{+}a_{\nu}^{0})\partial^{\nu}(a^{\mu})^{-}+(a_{\mu}^{+}a_{\nu}^{-}-a_{\mu}^{-}a_{\nu}^{+})\partial^{\nu}(a^{0})^{\mu}\Big{]}\,,$
(A.13) $\displaystyle\mathcal{S}_{\rm{cubic,s}}=\frac{2}{g^{2}_{\rm{YM}}}\int
d^{4}x$ $\displaystyle
4\Biggl{[}\sum_{\Phi\in\\{x,y\\}}\Bigl{(}\bar{\Phi}^{0}\bar{z}^{0}_{\rm{cl}}(\Phi^{-}z^{+}+\Phi^{+}z^{-})+\bar{\Phi}^{0}z^{0}_{\rm{cl}}(\Phi^{-}\bar{z}^{-}+\Phi^{+}\bar{z}^{+})-2\,\bar{z}^{0}z_{\rm{cl}}^{0}(|\Phi^{+}|^{2}+|\Phi^{-}|^{2})\Bigr{)}$
(A.14)
$\displaystyle-\,\bigl{(}|z^{-}|^{2}+|z^{+}|^{2}\bigr{)}\bar{z}^{0}z_{\rm{cl}}^{0}+2\,z^{-}z^{+}\bar{z}^{0}\bar{z}^{0}_{\rm{cl}}+h.c.\Biggr{]}\,,$

where $h.c.$ means adding the conjugate of all terms of the Lagrangian where
unbarred fields become barred and vice-versa. The remaining cubic terms read

$\displaystyle\mathcal{S}_{\text{cubic,g,s}}=\frac{4}{g_{\text{YM}}^{2}}\int
d^{4}x$
$\displaystyle\Biggl{[}-2(a^{-})^{\mu}a_{\mu}^{0}(z^{+}\bar{z}_{\text{cl}}^{0}+\bar{z}^{-}z_{\text{cl}}^{0})-2(a^{+})^{\mu}a_{\mu}^{0}(z^{-}\bar{z}_{\text{cl}}^{0}+\bar{z}^{+}z_{\text{cl}}^{0})$
(A.15)
$\displaystyle-4(a^{-})^{\mu}a_{\mu}^{+}(\bar{z}^{0}z_{\text{cl}}^{0}+z^{0}\bar{z}_{\text{cl}}^{0})+\sum_{\Phi\in\\{x,y,z\\}}\biggl{(}i\partial^{\mu}\Phi^{-}(\bar{\Phi}^{-}a_{\mu}^{0}-\bar{\Phi}^{0}a_{\mu}^{+})$
$\displaystyle\quad+i\partial^{\mu}\Phi^{+}(\bar{\Phi}^{0}a_{\mu}^{-}-\bar{\Phi}^{+}a_{\mu}^{0})+i\partial^{\mu}\Phi^{0}(\bar{\Phi}^{+}a_{\mu}^{+}-\bar{\Phi}^{-}a_{\mu}^{-})+h.c.\biggr{)}\Biggr{]}\,,$
$\displaystyle\mathcal{S}_{\rm{cubic,s,gh}}=\frac{2}{g^{2}_{\rm{YM}}}\int
d^{4}x$
$\displaystyle\,\Biggl{[}\bar{z}^{0}z^{0}_{\rm{cl}}\left(2\bar{c}^{-}c^{-}+2\bar{c}^{+}c^{+}\right)+z^{0}\bar{z}^{0}_{\rm{cl}}\left(2\bar{c}^{-}c^{-}+2\bar{c}^{+}c^{+}\right)$
$\displaystyle-\bar{z}^{0}_{\rm{cl}}\left(2z^{-}\bar{c}^{-}+2z^{+}\bar{c}^{+}\right)c^{0}-z^{0}_{\rm{cl}}\left(2\bar{c}^{+}\bar{z}^{-}+2\bar{c}^{-}\bar{z}^{+}\right)c^{0}\Biggr{]}\,.$

Finally, we display explicitly the quartic part of the action which we write
as

$\mathcal{S}_{\rm{quartic}}=\mathcal{S}_{\rm{quartic,g}}+\mathcal{S}_{\rm{quartic,g,s}}+\mathcal{S}_{\rm{quartic,s}}$
(A.16)

where

$\displaystyle\mathcal{S}_{\rm{quartic,g}}=\frac{1}{g^{2}_{\rm{YM}}}\int
d^{4}x$
$\displaystyle\Biggl{[}4(a^{0})^{\mu}(a^{0})^{\nu}\left(a^{+}_{\mu}a^{-}_{\nu}+a^{-}_{\mu}a^{+}_{\nu}\right)-2\,a^{-}_{\nu}(a^{+})^{\nu}\left(a^{-}_{\mu}(a^{+})^{\mu}+2a^{0}_{\mu}(a^{0})^{\mu}\right)$
(A.17)
$\displaystyle-4\,a^{-}_{\mu}(a^{+})^{\mu}a^{0}_{\nu}(a^{0})^{\nu}+a^{+}_{\mu}\left(a^{+}\right)^{\mu}\left(a^{-}\right)^{\nu}a^{-}_{\nu}+a^{-}_{\mu}(a^{-})^{\mu}a^{+}_{\nu}(a^{+})^{\nu}\Biggr{]}\,,$

$\displaystyle\mathcal{S}_{\rm{quartic,g,s}}=\frac{2}{g^{2}_{\rm{YM}}}\sum_{\Phi\in\\{x,y,z\\}}\int
d^{4}x$
$\displaystyle\Biggl{[}\,4\,a^{-}_{\mu}(a^{0})^{\mu}\left(\Phi^{+}\bar{\Phi}^{0}+\bar{\Phi}^{-}\Phi^{0}\right)+2\,a^{-}_{\mu}\left(a^{-}\right)^{\mu}\left(\Phi^{+}\bar{\Phi}^{-}\right)$
(A.18)
$\displaystyle+4\,a^{+}_{\mu}(a^{0})^{\mu}\left(\Phi^{-}\bar{\Phi}^{0}+\Phi^{0}\bar{\Phi}^{+}\right)+2\,a^{+}_{\mu}\left(a^{+}\right)^{\mu}\left(\Phi^{-}\bar{\Phi}^{+}\right)$
$\displaystyle-2\,(a^{-}_{\mu}(a^{+})^{\mu}+2a^{0}_{\mu}(a^{0})^{\mu})\left(\Phi^{-}\bar{\Phi}^{-}+\Phi^{+}\bar{\Phi}^{+}\right)-8\,a^{-}_{\mu}(a^{+})^{\mu}\left(\Phi^{0}\bar{\Phi}^{0}\right)\Biggr{]}\,,$
$\displaystyle\mathcal{S}_{\rm{quartic,s}}=\frac{2}{g^{2}_{\rm{YM}}}\int
d^{4}x$ $\displaystyle\sum_{\begin{subarray}{c}\Phi\in\\{x,y,z\\}\\\
\Psi\in\\{x,y,z\\}\end{subarray}}\Biggl{[}-4\bar{\Psi}^{-}\bar{\Phi}^{+}\Psi^{0}\Phi^{0}+4\Psi^{-}\bar{\Phi}^{-}\Phi^{0}\bar{\Psi}^{0}-8\Psi^{-}\bar{\Psi}^{-}\Phi^{0}\bar{\Phi}^{0}$
$\displaystyle+4\Psi^{+}\bar{\Phi}^{+}\Phi^{0}\bar{\Psi}^{0}-8\Psi^{+}\bar{\Psi}^{+}\Phi^{0}\bar{\Phi}^{0}-\Psi^{-}\Phi^{-}\bar{\Psi}^{-}\bar{\Phi}^{-}-\Psi^{+}\Phi^{+}\bar{\Psi}^{+}\bar{\Phi}^{+}$
$\displaystyle-2\Psi^{+}\Phi^{-}\bar{\Psi}^{+}\bar{\Phi}^{-}+4\Psi^{-}\Phi^{+}\bar{\Psi}^{0}\bar{\Phi}^{0}+4\Psi^{+}\Phi^{-}\bar{\Psi}^{-}\bar{\Phi}^{+}\Biggr{]}\,.$

##  Appendix B Large charge background and scalar propagators

In this section, we aim to find the propagators for the scalar fields in the
effective action. We consider a two point function of local operators with a
very large charge, which we denote as

$O_{J}(x)=(z^{0}(x))^{J}\,,\quad\bar{O}_{J}(x)=(\bar{z}^{0}(x))^{J}\,.$ (B.1)

In the limit when $J\rightarrow\infty$, the insertion of such operators in the
path integral source a non-trivial profile for $z^{0}$ and $\bar{z}^{0}$ which
can be obtained as a saddle point of the two-point function given by

$\langle\bar{O}_{J}(x_{1})O_{J}(x_{2})\rangle\simeq
e^{S_{\rm{eff}}^{\ast}}\quad{\rm{as}}\quad
J\rightarrow\infty\;,\;g_{\rm{YM}}\rightarrow
0\quad{\rm{with}}\quad\lambda_{J}=g^{2}_{\rm{YM}}J/2\;{\rm{fixed}}\,,$ (B.2)

where the corresponding effective action is given by

$S_{\rm{eff}}^{\ast}=S_{\rm{eff}}\Big{\rvert}_{z^{0},\,\bar{z}^{0}\rightarrow
z^{0}_{\rm{cl}},\,\bar{z}^{0}_{\rm{cl}}}\;\;{\rm
with}\;\;S_{\rm{eff}}=\frac{2J}{\lambda_{J}}\int d^{4}x\,\left[\bar{z}^{0}\Box
z^{0}+\frac{\lambda_{J}}{2}\left(\log(\bar{z}^{0})\,\delta(x-x_{1})+\log(z^{0})\,\delta(x-x_{2})\right)\right]\,.$
(B.3)

In this expression, $z^{0}_{\rm{cl}},\,\bar{z}^{0}_{\rm{cl}}$ are the
solutions of the saddle point equations that follow from the action
$S_{\rm{eff}}$. The explicit solutions for $z^{0}_{\rm{cl}}(x)$ and
$\bar{z}^{0}_{\rm{cl}}(x)$ are given by

$z^{0}_{\rm{cl}}(x)=\frac{e^{i\phi}\,|x_{1}-x_{2}|}{2\pi|x-x_{1}|^{2}}\,\sqrt{\frac{\lambda_{J}}{2}}\,,\quad\bar{z}^{0}_{\rm{cl}}(x)=\frac{e^{-i\phi}\,|x_{1}-x_{2}|}{2\pi|x-x_{2}|^{2}}\,\sqrt{\frac{\lambda_{J}}{2}}\,.$
(B.4)

From (A.11), we find that the scalars $z^{\pm},x^{\pm}$ and $y^{\pm}$ have
masses given by

$m^{2}(x)=8|z_{\rm{cl}}^{0}|^{2}=\frac{|x_{1}-x_{2}|^{2}}{\pi^{2}|x-x_{1}|^{2}|x-x_{2}|^{2}}\,\lambda_{J}\,.$
(B.5)

The quadratic part of the action for each of these scalars, which we
generically denote by $\Phi$, is then given by

$\mathcal{S}_{\rm{quad},\Phi}=\frac{2}{g^{2}_{\rm{YM}}}\int
d^{4}x\left(\bar{\Phi}\Box\Phi-m^{2}(x)|\Phi|^{2}\right)\,.$ (B.6)

The scalar propagator is obtained by solving the equation

$\quad(-\Box_{x}+m^{2}(x))\,G(x,y)=\,\delta^{(4)}(x-y)\qquad{\rm{with}}\quad\,G(x,y):=\frac{2}{g^{2}_{\rm{YM}}}\langle\bar{\Phi}(x)\Phi(y)\rangle\,.$
(B.7)

The solution to this equation has already been worked out in [9]. One
considers an expansion in powers the ’t Hooft coupling $\lambda_{J}$

$G(x,y)=\sum_{k=0}^{\infty}g_{k}(x,y)\,,\quad
g_{k}(x,y)\sim\mathcal{O}(\lambda_{J}^{k})\,.$ (B.8)

The leading order term satisfies the massless propagator equation

$-\Box_{x}g_{0}=\delta^{(4)}(x-y)\quad\Rightarrow\quad
g_{0}=\frac{1}{4\pi^{2}}\frac{1}{|x-y|^{2}}\,,$ (B.9)

whereas the remaining terms obey

$-\Box_{x}g_{k+1}+m^{2}g_{k}=0\,,$ (B.10)

where we have omitted the arguments for simplicity. It is simple to obtain
iteratively the perturbative solution at any order $k$

$g_{k}(x,y)=(-1)^{k}\left(\prod_{n=1}^{k}\int
d^{4}z_{n}\,m^{2}(z_{n})g_{0}(z_{n},z_{n+1})\right)g_{0}(x,z_{1})\,,$ (B.11)

where we defined $z_{k+1}\equiv y$. More explicitly, this is equivalent to

$g_{k}(x,y)=\left(-4g^{2}\right)^{k}\frac{(x_{1}-x_{2})^{2}}{4\pi^{2}(x-x_{1})^{2}(y-x_{2})^{2}}\,F^{(k)}(z,\bar{z})$
(B.12)

where $F^{(k)}(z,\bar{z})$ stems from a ladder type integral and can be
pictorially represented by:

$\displaystyle\vbox{\hbox{
\includegraphics[scale={1.2}]{ladder}}}=\frac{\pi^{2k}}{(y-x)^{2}(x_{1}-x_{2})^{2k}}(1-z)(1-\bar{z})F^{(k)}(z,\bar{z})$

Here the integration points are represented by the black dots while the white
dots are unintegrated external points. The conformal cross-ratios $z,\bar{z}$
are given by

$z\bar{z}\equiv{(y-x_{1})^{2}(x-x_{2})^{2}\over(x-x_{1})^{2}(y-x_{2})^{2}}\,,\quad(1-z)(1-\bar{z})\equiv{(x_{1}-x_{2})^{2}(x-y)^{2}\over(x-x_{1})^{2}(y-x_{2})^{2}}\,.$
(B.13)

With the explicit form of the ladder integrals [141, 142], it was found in
[47] the resumed expression giving the two-point function (B.8)

$\langle\bar{\Phi}(x)\Phi(y)\rangle=\frac{g^{2}_{\rm{YM}}}{4\pi^{2}|x-y|^{2}}\int_{0}^{\infty}dt\frac{\sqrt{\frac{z\bar{z}}{(1-z)^{2}(1-\bar{z})^{2}}}\sinh(t-\frac{1}{2}\log{z\bar{z}})\,J_{0}\left(\sqrt{\frac{\lambda_{J}}{\pi^{2}}\,t\,(t-\log
z\bar{z})}\right)}{\left(1-\frac{1+z\bar{z}}{(1-z)(1-\bar{z})}+2\sqrt{\frac{z\bar{z}}{(1-z)^{2}(1-\bar{z})^{2}}}\cosh\left(t-\frac{1}{2}\log{z\bar{z}}\right)\right)^{2}}$
(B.14)

where $J_{k}$ denotes the Bessel function. The ladder integrals can be
represented in an alternative form that is more useful for analyzing the
strong coupling behavior of correlation functions [47, 94, 49]. It is given by

$\langle\bar{\Phi}(x)\Phi(y)\rangle=\frac{g^{2}_{\rm{YM}}}{8\pi^{2}(x-y)^{2}}\sum_{n=0}^{\infty}\sum_{a=1}^{\infty}(-4g^{2})^{n}\frac{(1-z)(1-\bar{z})}{z-\bar{z}}\int_{-\infty}^{\infty}\frac{du}{2\pi}\,\frac{2a\sin(a\varphi)e^{-2iu\sigma}}{\left(u^{2}+\frac{a^{2}}{4}\right)^{n+1}}$
(B.15)

where $e^{i\varphi}=\sqrt{z/\bar{z}}$ and $e^{-\sigma}=\sqrt{z\bar{z}}$. Since
this type of integrals are common in integrability approaches to correlation
functions [49], we will refer to it as the integrability-like representation
of the ladder integral.

Of particular importance are the following limits of this formula. When
$\delta^{2}:=|x-x_{1}|^{2}\rightarrow 0$ we obtain [9]

$\langle\bar{\Phi}(x_{1})\Phi(y)\rangle=\frac{4g^{2}}{J}\frac{1}{|x_{1}-y|^{2}\sqrt{1+16g^{2}}}\left(\frac{\delta|x_{2}-y|}{|x_{1}-x_{2}||x_{1}-y|}\right)^{\sqrt{1+16g^{2}}-1}$
(B.16)

and when $|y-x_{2}|^{2}\rightarrow 0$

$\langle\bar{\Phi}(x)\Phi(x_{2})\rangle=\frac{4g^{2}}{J}\frac{1}{|x_{2}-x|^{2}\sqrt{1+16g^{2}}}\left(\frac{\delta|x_{1}-x|}{|x_{1}-x_{2}||x_{2}-x|}\right)^{\sqrt{1+16g^{2}}-1}\,.$
(B.17)

##  Appendix C Perturbative computation

In this appendix, we set up the perturbative computation of one-loop diagrams
from the effective action derived in the Appendix A around the background
field determined in Appendix B. We will be computing the diagrams listed
below:

$\displaystyle\scalebox{0.88}{$G_{1}:=$}\vbox{\hbox{
\includegraphics[scale={0.9}]{4scalarvertex}}}\quad\quad\scalebox{0.88}{$G_{2}:=$}\vbox{\hbox{
\includegraphics[scale={0.9}]{4scalarvertex2}}}\quad\quad\scalebox{0.88}{$G^{\pm}_{3}:=$}\vbox{\hbox{
\includegraphics[scale={0.9}]{gluonexchange}}}\,.$

The double line corresponds to the massive scalar propagator and the coiled
line represents the propagator of the massless mode of the gluon field.

##### Diagram $G_{1}$.

After factoring out the tree level contribution and using the part of the
action denoted by $\mathcal{S}_{\rm{quartic,s}}$ in (A.18), we obtain the
following one-loop result:

$G_{1}=-\frac{32g^{2}}{J}\frac{1}{1+16g^{2}}\,\pi^{2}\,(2\pi)^{4}\,|x_{1}-x_{2}|^{4}X_{1122}\,,$
(C.1)

which is independent of the sign of $y$. We consider the simultaneous point-
splitting $x_{3}^{\mu}=x_{1}^{\mu}+\epsilon^{\mu}$ and
$x_{4}^{\mu}=x_{2}^{\mu}+\epsilon^{\mu}$ with $\epsilon^{\mu}\rightarrow 0$ of
the integral $X_{1234}$ given by

$X_{1234}=\int\frac{d^{4}z}{(2\pi)^{8}}\frac{1}{|x_{1}-z|^{2}|x_{2}-z|^{2}|x_{3}-z|^{2}|x_{4}-z|^{2}},.$
(C.2)

The result is given by [98]

$X_{1122}\equiv\lim_{\begin{subarray}{c}x_{3}\rightarrow x_{1}\\\
x_{4}\rightarrow
x_{2}\end{subarray}}X_{1234}=\frac{1}{8\pi^{2}}\left(1-\log\left(\frac{\epsilon^{2}}{|x_{1}-x_{2}|^{2}}\right)\right)\times\frac{1}{(2\pi)^{4}|x_{1}-x_{2}|^{4}}.$
(C.3)

We conclude that the logarithmic divergence of such a diagram is

$G_{1}\big{\rvert}_{\log}=\frac{1}{J}\frac{4g^{2}}{1+16g^{2}}\simeq\frac{1}{J}\left(4g^{2}-64g^{4}+\dots\right).$
(C.4)

##### Diagram $G_{2}$.

The diagram $G_{2}$ is similar to $G_{1}$ except with a different
combinatorial factor. We get

$G_{2}=\frac{64g^{2}}{J}\frac{1}{1+16g^{2}}\,\pi^{2}\,(2\pi)^{4}\,|x_{1}-x_{2}|^{4}X_{1122}\,.$
(C.5)

This gives the following logarithmic divergence:

$G_{2}\big{\rvert}_{\log}=-\frac{1}{J}\frac{8g^{2}}{1+16g^{2}}\simeq\frac{1}{J}\left(-8g^{2}+128g^{4}+\dots\right).$
(C.6)

##### Diagram $G_{3}^{\pm}$.

The gluon exchange diagram is derived from the part of the action denoted by
$\mathcal{S}_{\rm{quartic,g,s}}$ in (A.18) and produces the following result

$G_{3}^{\pm}=\mp\frac{64g^{2}}{J}\frac{1}{1+16g^{2}}(2\pi)^{6}|x_{1}-x_{2}|^{4}\lim_{\begin{subarray}{c}x_{3}\rightarrow
x_{1}\\\ x_{4}\rightarrow
x_{2}\end{subarray}}\partial_{2}\cdot\partial_{4}H_{12,34}$ (C.7)

with the two-fold integral $H_{12,34}$ given by

$H_{12,34}=\int\frac{d^{4}zd^{4}w}{(2\pi)^{10}}\frac{1}{|x_{1}-z|^{2}|x_{2}-z|^{2}|w-z|^{2}|x_{3}-w|^{2}|x_{4}-w|^{2}}\,.$
(C.8)

We use the result for its derivatives that can be found for example in [98]
and reads

$\lim_{\begin{subarray}{c}x_{3}\rightarrow x_{1}\\\ x_{4}\rightarrow
x_{2}\end{subarray}}\partial_{2}\cdot\partial_{4}H_{12,34}=\frac{1}{16\pi^{2}}\left(2-\log\left(\frac{\epsilon^{2}}{|x_{1}-x_{2}|^{2}}\right)\right)\times\frac{1}{(2\pi)^{4}|x_{1}-x_{2}|^{4}}\,.$
(C.9)

When extracting the logarithmic divergence we now obtain

$G_{3}^{\pm}\big{\rvert}_{\log}=\pm\frac{1}{J}\frac{16g^{2}}{1+16g^{2}}\simeq\pm\frac{1}{J}\left(16g^{2}-256g^{4}+\dots\right)\,.$
(C.10)

This concludes the perturbative one-loop computation.

[1]ReferencesdraftLC

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