# Complex surfaces with many algebraic structures

Anna Abasheva, Rodion Déev

###### Abstract

We find new examples of complex surfaces with countably many non-isomorphic
algebraic structures. Here is one such example: take an elliptic curve $E$ in
$\mathbb{P}^{2}$ and blow up nine general points on $E$. Then the complement
$M$ of the strict transform of $E$ in the blow-up has countably many algebraic
structures. Moreover, each algebraic structure comes from an embedding of $M$
into a blow-up of $\mathbb{P}^{2}$ in nine points lying on an elliptic curve
$F\not\simeq E$. We classify algebraic structures on $M$ using a Hopf
transform: a way of constructing a new surface by cutting out an elliptic
curve and pasting a different one. Next, we introduce the notion of an
analytic K-theory of varieties. Manipulations with the example above lead us
to prove that classes of all elliptic curves in this K-theory coincide. To put
in another way, all motivic measures on complex algebraic varieties that take
equal values on biholomorphic varieties do not distinguish elliptic curves.

###### Contents

  1. 1 Introduction
  2. 2 Hopf transforms
    1. 2.1 Analytic compactifications
    2. 2.2 Hopf surfaces
    3. 2.3 Secondary Hopf surfaces
    4. 2.4 Hopf duality and analytic cobordance
    5. 2.5 Hopf transforms
  3. 3 Surfaces with square-zero elliptic curves
    1. 3.1 Kodaira dimension
    2. 3.2 Rational surfaces
    3. 3.3 Surfaces of class VII
  4. 4 Analytic Grothendieck group

##  1 Introduction

A compact complex analytic variety $X^{an}$ has at most one algebraic
structure; that is to say, there is at most one algebraic variety $Y$ such
that $Y^{an}\cong X^{an}$. However, this is false in the non-compact world. We
do not have general techniques to describe the set of algebraic structures on
a non-compact complex variety. Available results show that a variety can have
infinitely many of algebraic structures. We briefly review examples in
dimension two.

  * •

Let $C$ be an affine curve of positive genus. Then $C^{an}\times\mathbb{C}$
admits uncountably many non-isomorphic affine structures [Jel15]. Indeed,
every line bundle on $C$ is analytically trivial, hence its total space is
biholomorphic to $C^{an}\times\mathbb{C}$.

  * •

The surface $\mathbb{C}^{*}\times\mathbb{C}^{*}$ is biholomorphic to the
unique non-trivial affine bundle of rank one and degree zero over any elliptic
curve [Nee89, Thm. 6.12]. Hence, it has uncountably many non-isomorphic
algebraic structures.

  * •

Let $C$ be a smooth projective curve and $p\colon X\to C$ an algebraic
elliptic surface. Choose a smooth fiber $E=p^{-1}(x)$. For every $n>0$, we can
find an elliptic fibration $p_{n}\colon X_{n}\to C$, called a log-transform of
$p$, that has a fiber of multiplicity $n$ over $x$ and is biholomorphic to $p$
over $C-\\{x\\}$. The new surfaces $X_{n}$ are algebraic unless $X\simeq
E\times C$ [FM13, Ch. I, Thm. 6.12]. Hence the surface $X-E$ has at least
countably many algebraic structures.

  * •

We can construct even more algebraic structures on the surface $X-E$ from the
previous example using Shafarevich–Tate twists. We refer the reader to [FM13,
Ch. I, Sections 1.5.1, 1.5.3] for more details.

We complement this list by the following beautiful example.

######  Theorem A (Theorem 3.2.6, Corollary 3.2.3).

  1. 1.

Pick nine points in $\mathbb{P}^{2}$ in sufficiently general position. Let $X$
be the blow-up of $\mathbb{P}^{2}$ in these points and $E$ the strict
transform of the unique elliptic curve passing through them. Then the
complement of $E$ in $X$ admits countably many algebraic structures.

  2. 2.

More generally, let $X$ be a compact rational surface containing a square-zero
elliptic curve $E$. Assume that the normal bundle to $E$ in $X$ is
sufficiently general. Then the surface $X-E$ admits countably many algebraic
structures.

In addition, in the Case 1 every algebraic structure on $X-E$ arises from a
holomorphic embedding into a blow-up $Y$ of $\mathbb{P}^{2}$ in some other
nine points. The complement of $X-E$ in $Y$ is an elliptic curve $F$ which is
remarkably not isomorphic and not even isogenous to $E$. In the algebraic
realm, this is only possible for singular curves ([BPvS23] and references
therein). Indeed, if $S$ and $S^{\prime}$ are smooth algebraic surfaces
containing smooth curves $C$ and $C^{\prime}$ respectively, then an algebraic
isomorphism between $S-C$ and $S^{\prime}-C^{\prime}$ implies an isomorphism
between $C$ and $C^{\prime}$.

Our construction in Theorem A is fundamentaly different from those in the list
above. We developed a method to manufacture new surfaces from a surface with a
square-zero elliptic curve that we call a Hopf transform (Definition 2.5.6).
It serves as a counterpart to a log-transform in the case of non-elliptic
surfaces.

We can summarize a Hopf transform as follows: take an elliptic curve on a
surface, cut it out, and paste another elliptic curve instead. Thanks to a
theorem of Arnold and Ueda (Theorem 2.5.3), a square-zero elliptic curve
$E\subset S$ whose normal bundle $L$ is sufficiently general has a holomorphic
tubular neighbourhood. That is to say, $S$ is locally biholomorphic to
$\operatorname{Tot}(L)$ along $E$. We will clarify the meaning of
“sufficiently general” in Definition 2.5.2; for now, it is enough to say that
all “sufficiently general” line bundles are non-torsion. Using Hopf surfaces,
i.e., non-Kähler surfaces of the form
$\mathbb{C}^{2}-\\{0\\}/\operatorname{diag}(\lambda,\mu)$ and their finite
quotients, we show that there exist countably many degree zero line bundles
$L^{\prime}\to F$ over different elliptic curves $F$ such that
$\operatorname{Tot}(L)^{an}-E\cong\operatorname{Tot}(L^{\prime})^{an}-F$. We
call such line bundles $L$ and $L^{\prime}$ analytically cobordant (Definition
2.4.2). A Hopf transform, which can be interpreted as a gluing of a surface
with an appropriate Hopf surface, replaces an elliptic curve $E\subset S$ with
$F$. Moreover, we are free to choose from a countable set of candidates for
$F$. We apply this surgery to a blow-up of nine points on $\mathbb{P}^{2}$ and
show that the result is again a blow-up of $\mathbb{P}^{2}$ along some other
nine points. It yields Theorem 3.2.6.

Our notion of a Hopf transform was inspired by Koike and Uehara’s construction
in [KU19]. They use the theorem of Arnold and Ueda mentioned above to glue two
open subsets of two different blow-ups of $\mathbb{P}^{2}$ in nine points into
a K3 surface.

We apply Theorem A to study the analytic Grothendieck group $K_{0}^{an}$ of
varieties (Definition 4.1.2). It is the quotient of the Grothendieck group
$K_{0}(\mathrm{Var}_{\mathbb{C}})$ of complex algebraic
varieties111$K_{0}(\mathrm{Var}_{\mathbb{C}})$ is the abelian group generated
by classes of algebraic varieties over $\mathbb{C}$ modulo scissor relations:
$[X]-[Y]=[X-Y]$, where $Y$ is a closed subvariety of $X$. by additional
relations of the form

$[X]=[Y]\>\>\>\text{ if }\>\>\>X^{an}\simeq Y^{an}.$

While $K_{0}(\mathrm{Var}_{\mathbb{C}})$ has attracted a considerable amount
of attention (see [Bit04, KT19, NS19] among many others), the notion of
$K_{0}^{an}$ is, to our knowledge, new. The group $K_{0}^{an}$ is strikingly
different from its algebraic counterpart as the theorem below illustrates.

######  Theorem B (Theorem 4.1.3).

All elliptic curves have the same classes in $K_{0}^{an}$.

In contrast, non-isomorphic curves have different classes in
$K_{0}(\mathrm{Var}_{\mathbb{C}})$ 4.1.4.

This paper is organized as follows. In Section 2, we prove that the total
space of a non-torsion degree-zero line bundle on an elliptic curve has
countably many compactifications by different elliptic curves (Theorem 2.3.2).
Then we introduce the notions of Hopf duality and analytic cobordance
(Defintion 2.4.2), and give the definition of Hopf transforms (Definition
2.5.6). In Section 3, we partially classify surfaces to which Hopf transforms
apply, that is, surfaces with square-zero elliptic curves with non-torsion
normal bundle. With this classification in mind, we prove Theorem A, the main
result of our paper. Finally, in Section 4, we introduce the analytic
Grothendieck group and prove Theorem B. Towards the end, we pose several
intriguing questions about the analytic Grothendieck group.

##  2 Hopf transforms

###  2.1 Analytic compactifications

######  Definition 2.1.1.

Let $M$ be a non-compact complex surface. A pair $(S,C)$ of a smooth compact
complex surface $S$ and (possibly singular) curve $C\subset S$ is called an
analytic compactification or just a compactification of $M$ if $S-C$ is
biholomorphic to $M$. A compactification is called minimal if none of the
curves in the support of $C$ are exceptional.

The following classification theorem is due to Enoki [Eno82, Theorem C-I0].

######  Theorem 2.1.2.

Let $\operatorname{Tot}(L)$ be the total space of a line bundle $L$ of degree
zero over an elliptic curve $E$. Then every minimal compactification $(S,C)$
of $M$ is of the following form:

  1. 1.

$S\cong\mathbb{P}({\cal O}\oplus L)$ is a $\mathbb{P}^{1}$-bundle over an
elliptic curve; $C$ is its section, and $(C)^{2}=0$.

  2. 2.

$S$ is a Hopf surface (Definition 2.2.1), and $C$ is an elliptic curve.

If $L$ is non-torsion, the set of Hopf surfaces compactifying
$\operatorname{Tot}(L)$ is countable, as we will see in Theorem 2.3.2. For
now, let us note that all such compactifications contain at least two elliptic
curves.

###  2.2 Hopf surfaces

######  Definition 2.2.1.

[Kod66, Sect. 10] A Hopf surface is a compact complex surface whose universal
cover is $\mathbb{C}^{2}-\\{0\\}$. A Hopf surface is called primary if its
fundamental group is $\mathbb{Z}$, otherwise it is called secondary.

#### 2.2.2.

All Hopf surfaces are non-Kähler. By [Kod66, Sect. 10, p. 695] every primary
Hopf surface $X$ is biholomorphic to a quotient
$\mathbb{C}^{2}-\\{0\\}/\Gamma$ where $\Gamma$ is either a linear diagonal
operator or of the form

$\Gamma(x,y)=(\alpha^{n}x+y^{n},\alpha y).$

In the second case, a Hopf surface has only one irreducible curve: the image
of the $x$-axis. Therefore, such surfaces do not arise as compactifications of
$\operatorname{Tot}(L)$.

#### 2.2.3.

Let us study the case when $\Gamma=\operatorname{diag}(\lambda,\mu)$. In order
for $\mathbb{C}^{2}-\\{0\\}/\Gamma$ to be compact, the absolute values of the
eigenvalues must be either both greater than one or both less than one. By
replacing $\Gamma$ with $\Gamma^{-1}$ if necessary, we may assume the latter.

######  Definition 2.2.4.

A diagonal Hopf surface is
$\mathbb{C}^{2}-\\{0\\}/\operatorname{diag}(\lambda,\mu)$ for $|\lambda|<1$,
$|\mu|<1$. We will denote it by $H(\lambda,\mu)$

#### 2.2.5.

One can show that two Hopf surfaces $H(\lambda,\mu)$ and
$H(\lambda^{\prime},\mu^{\prime})$ are isomorphic if and only if
$\lambda=\lambda^{\prime}$ and $\mu=\mu^{\prime}$ (up to a permutation). All
automorphisms of $H(\lambda,\mu)$ are induced by a linear diagonal
automorphism of $\mathbb{C}^{2}$.

#### 2.2.6.

If $\lambda^{n}=\mu^{m}$ for some non-zero integers $n$ and $m$ then
$H(\lambda,\mu)$ is an elliptic surface with smooth but possibly non-reduced
fibers [BHPVdV15, Ch. V, Prop. 18.2]. All elliptic curves in $X$ are fibers,
hence their normal bundles are torsion. Such a surface cannot be a
compactification of $\operatorname{Tot}(L)$ for a non-torsion line bundle $L$.

When $\lambda^{n}\neq\mu^{m}$ for any pair of non-zero integers the surface
$H(\lambda,\mu)$ contains exactly two irreducible curves [BHPVdV15, Ch. V,
Prop. 18.2]. They are the images of the axes under the quotient map and are
isomorphic to $E_{\lambda}:=\mathbb{C}^{*}/\lambda$ and
$E_{\mu}:=\mathbb{C}^{*}/\mu$. They have both square zero.

The complement to $E_{\mu}$ in $H(\lambda,\mu)$ is
$\mathbb{C}^{*}\times\mathbb{C}/\operatorname{diag}(\lambda,\mu)$. It is the
total space of a line bundle over $E_{\lambda}$, and the map to $E_{\lambda}$
is the projection to the first coordinate.

#### 2.2.7.

Every elliptic curve can be written as $\mathbb{C}/\mathbb{Z}+\mathbb{Z}\tau$
for $\operatorname{Im}(\tau)>0$. The parameter $\tau$ is defined modulo the
$\operatorname{SL}_{2}(\mathbb{Z})$-action. The map $x\mapsto e^{2\pi ix}$
induces an isomorphism

$\mathbb{C}/\mathbb{Z}+\mathbb{Z}\tau\stackrel{{\scriptstyle\sim}}{{\to}}\mathbb{C}^{*}/q^{\tau}.$

Here we use the shorthand $q^{\tau}$ for $e^{2\pi i\tau}$. The total space of
every line bundle of degree zero over $\mathbb{C}/\mathbb{Z}+\mathbb{Z}\tau$
is of the form

$\mathrm{L}(\tau,A,B)=\mathbb{C}^{2}/\sim:\>\>\>\>\>\>(x,y)\sim(x+1,Ay);\>(x,y)\sim(x+\tau,By),$
(1)

where $A,B\in U(1)$. These numbers are uniquely defined once we fix $\tau$.
Similarly, the total space of a line bundle of degree zero over
$E_{\lambda}=\mathbb{C}^{*}/\lambda$ can be written as

$\mathcal{L}(\lambda,\mu)=\mathbb{C}^{*}\times\mathbb{C}/\operatorname{diag}(\lambda,\mu),$
(2)

where $\lambda,\mu\in\mathbb{C}^{*}$.

#### 2.2.8.

The class of $\mathcal{L}(\lambda,\mu)$ in
$\operatorname{Pic}^{0}(E_{\lambda})\simeq E_{\lambda}$ is
$\mu\pmod{\lambda^{\mathbb{Z}}}$ [Kod66, Sect. 10, p. 696]. In particular,
line bundles $\mathcal{L}(\lambda,\mu)$ and
$\mathcal{L}(\lambda,\mu^{\prime})$ are isomorphic if and only if
$\mu^{\prime}=\mu\lambda^{n}$.

#### 2.2.9.

A line bundle on an elliptic curve can be represented as
$\mathrm{L}(\tau,A,B)$ or $\mathcal{L}(\lambda,\mu)$. Let us figure out how to
switch between the two descriptions. Write $A=q^{u}$ for a number
$u\in\mathbb{R}$. Then we have a linear isomorphism of line bundles

$\alpha_{\tau,u,B}\colon\mathrm{L}(\tau,q^{u},B)\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{L}(q^{\tau},q^{-u\tau}B)$

given by

$\alpha_{\tau,u,B}(x,y)=(q^{x},q^{-ux}y).$ (3)

Note that $|q^{\tau}|<1$ and $|q^{-u\tau}B|<1$ if $u$ is chosen to be
negative. Given a line bundle $\mathcal{L}(q^{\tau},\mu)$, we can find
corresponding $u$ and $B$. Indeed, write $\mu=q^{\sigma}$ and $B=q^{v}$. The
equation

$\sigma=u\tau+v$

determines $u$ and $v$ uniquely since $u$ and $v$ are real numbers. Different
choice of $\sigma$ does not change $B=q^{v}$.

#### 2.2.10.

The isomorphism (3) depends only on the choice of $\tau$ and $u$ such that
$q^{u}=A$. Let us see what happens if we change $\tau$. There is a linear
isomorphism of line bundles

$\beta_{\Gamma}\colon\mathrm{L}(\tau,q^{u},q^{v})\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{L}\left(\frac{k\tau+l}{m\tau+n},q^{mv+nu},q^{kv+lu}\right)$

for $\begin{pmatrix}k&l\\\
m&n\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{Z})$. The map (3) identifies
$\mathrm{L}(\tau,q^{u},q^{v})$ with
$\mathcal{L}(\lambda^{\prime},\mu^{\prime})$, where

$\displaystyle\lambda^{\prime}=q^{\frac{k\tau+l}{m\tau+n}},$
$\displaystyle\mu^{\prime}=q^{-(mv+nu+r)\frac{k\tau+l}{m\tau+n}+kv+lu},$

and $r\in\mathbb{Z}$. Using that $nk-ml=1$, we can rewrite the exponent in
$\mu^{\prime}$ as

$-(mv+nu+r)\frac{k\tau+l}{m\tau+n}+kv+lu=\frac{v-u\tau+r(k\tau+l)}{m\tau+n}.$

The elliptic curve in the complement to $\operatorname{Tot}(L)$ in
$H(\lambda^{\prime},\mu^{\prime})$ is

$E_{\mu^{\prime}}\simeq\mathbb{C}/\mathbb{Z}+\mathbb{Z}\cdot\left(\frac{v-u\tau+r(k\tau+l)}{m\tau+n}\right)$

If $L$ is generic, different choices of $\begin{pmatrix}k&l\\\
m&n\end{pmatrix}$ and $r$ lead to different elliptic curves
$E_{\mu^{\prime}}$.

#### 2.2.11.

Consider a translation $t_{s}\colon x\mapsto x+s$ on the curve
$\mathbb{C}/\mathbb{Z}+\mathbb{Z}\tau$. The line bundle $\mathrm{L}(\tau,A,B)$
has degree zero, hence it is isomorphic to its pullback along the translation.
The isomorphism can be written explicitly as

$\displaystyle t_{s}^{*}\colon\mathrm{L}(\tau,A,B)\to\mathrm{L}(\tau,A,B)$
$\displaystyle(x,y)\mapsto(x+s,y).$

When we write $\mathrm{L}(\tau,A,B)$ as $\mathcal{L}(q^{\tau},q^{-u\tau}B)$ as
in 2.2.9, the translation turns into the map

$\displaystyle
t_{s}^{*}\colon\mathcal{L}(q^{\tau},q^{-u\tau}B)\to\mathcal{L}(q^{\tau},q^{-u\tau}B)$
$\displaystyle(x,y)\mapsto(q^{s},q^{-us}).$

This map extends to the automorphism $\operatorname{diag}(q^{s},q^{-us})$ of
the Hopf surface $H(q^{\tau},q^{-u\tau}B)$.

#### 2.2.12.

Consider the automorphism $f$ of $E$ sending $x$ to $-x$. The line bundle
$f^{*}\mathrm{L}(\tau,A,B)$ is isomorphic to $\mathrm{L}(\tau,A^{-1},B^{-1})$.
In general, an automorphism $f$ of $E$ fixing $0$ induces a linear
biholomorphism

$f^{*}\colon\mathrm{L}(\tau,A,B)\stackrel{{\scriptstyle\sim}}{{\to}}\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime}),$

where the pair of numbers $(A^{\prime\prime},B^{\prime\prime})$ does not
necessarily coincide with $(A,B)$.

#### 2.2.13.

Every automorphism of $\mathcal{L}(\lambda,\mu)$ as a line bundle is the
fiberwise multiplication by a constant $c$. It extends to the automorphism
$\operatorname{diag}(1,c)$ of the Hopf surface $H(\lambda,\mu)$.

######  Proposition 2.2.14.

Let $L$ be a degree-zero non-torsion line bundle on an elliptic curve $E$.
Then the set of equivalence classes of open embeddings
$\operatorname{Tot}(L)\stackrel{{\scriptstyle\iota}}{{\to}}H$ into a primary
Hopf surface is countable. Here, we call two embeddings
$\iota\colon\operatorname{Tot}(L)\to H$ and
$\iota^{\prime}\colon\operatorname{Tot}(L)\to H^{\prime}$ equivalent if there
is an automorphism $g$ of $\operatorname{Tot}(L)$ and a biholomorphism
$h\colon H\to H^{\prime}$ such that the following diagram commutes:

${\operatorname{Tot}(L)}$${H}$${\operatorname{Tot}(L)}$${H^{\prime}}$$\scriptstyle{\iota}$$\scriptstyle{g}$$\scriptstyle{h}$$\scriptstyle{\iota^{\prime}}$

######  Proof: .

Suppose $E=X/\mathbb{Z}+\mathbb{Z}\tau$ and $L$ is such that
$\operatorname{Tot}(L)=\mathrm{L}(\tau,A,B)$. Consider an open embedding
$\iota\colon\mathrm{L}(\tau,A,V)\to H$. By 2.2.2 and 2.2.6, the surface $H$
must be of the form $H(\lambda,\mu)$, and $\mathrm{L}(\tau,A,B)$ is mapped
onto $H(\lambda,\mu)-E_{\mu}\simeq\mathcal{L}(\lambda,\mu)$. We saw in 2.2.9
that there exists a biholomorphism

$\alpha_{\tau^{\prime},u^{\prime},B^{\prime}}\colon\mathrm{L}(\tau^{\prime},A^{\prime},B^{\prime})\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{L}(\lambda,\mu)$

for some $\tau^{\prime}\in\mathbb{C}$ such that
$\mathrm{Im}~{}\tau^{\prime}>0$ and $A^{\prime},B^{\prime}\in U(1)$.

The elliptic curves $E=\mathbb{C}/\mathbb{Z}+\mathbb{Z}\tau$ and
$\mathbb{C}/\mathbb{Z}+\mathbb{Z}\tau^{\prime}$ must be isomorphic, hence
$\tau=\Gamma\cdot\tau^{\prime}$ for a matrix
$\Gamma\in\operatorname{SL}_{2}(\mathbb{Z})$. In particular, the set of
possible $\tau^{\prime}$ is countable. The line bundle
$\mathrm{L}(\tau^{\prime},A^{\prime},B^{\prime})$ is linearly isomorphic to
$\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})$ for some
$A^{\prime\prime},B^{\prime\prime}\in U(1)$ (2.2.10). Therefore, the embedding
$\mathrm{L}(\tau,A,B)\stackrel{{\scriptstyle\iota}}{{\to}}H(\lambda,\mu)$ is
the composition of the following maps:

${{\mathrm{L}(\tau,A,B)}}$${{\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})}}$${{\mathrm{L}(\tau,A^{\prime},B^{\prime})}}$${{\mathcal{L}(\lambda,\mu)}}$${{H(\lambda,\mu)},}$$\scriptstyle{\varphi}$$\scriptstyle{\beta_{\Gamma}}$$\scriptstyle{\alpha_{\tau^{\prime},u^{\prime},B^{\prime}}}$

where
$\mathrm{L}(\tau,A,B)\stackrel{{\scriptstyle\varphi}}{{\to}}\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})$
is some biholomorphism.

It remains to classify biholomorphisms
$\mathrm{L}(\tau,A,B)\stackrel{{\scriptstyle\varphi}}{{\to}}\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})$
modulo equivalence. Let $f$ be the restriction of $\varphi$ to the zero
section $E\subset\mathrm{L}(\tau,A,B)$. It is a composition of a translation
and an automorphism preserving zero. Two biholomorphisms
$\mathrm{L}(\tau,A,B)\to\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})$
that differ by a translation are equivalent because a translation extends to
an automorphism of $H(\lambda,\mu)$ (2.2.11). Hence, we may assume that
$f\colon E\to E$ preserves zero. We have an isomorphism of line bundles
$\mathrm{L}(\tau,A,B)\simeq
f^{*}\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})$ as in 2.2.12. The set
of automorphisms fixing zero is finite, hence the number of pairs
$(A^{\prime\prime},B^{\prime\prime})$ such that $\mathrm{L}(\tau,A,B)$ is
isomorphic to $\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})$ is also
finite. We have shown that every embedding
$\mathrm{L}(\tau,A,B)\stackrel{{\scriptstyle\iota}}{{\to}}H(\lambda,\mu)$ is
equivalent to the composition

${{L(\tau,A,B)}}$${{\mathrm{L}(\tau,A,B)}}$${{\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})}}$${{\mathrm{L}(\tau,A^{\prime},B^{\prime})}}$${{H(\lambda,\mu)},}$$\scriptstyle{\psi}$$\scriptstyle{f^{*}}$$\scriptstyle{\beta_{\Gamma}}$$\scriptstyle{\alpha_{\tau^{\prime},u^{\prime},B^{\prime}}}$

where $\psi$ is a biholomorphism of $\mathrm{L}(\tau,A,B)$ identical on $E$.
We will see later in Lemma 2.5.9 that every such biholomorphism is
multiplication by a constant. It extends to an automorphism of
$H(\lambda,\mu)$ by 2.2.13. Therefore, every embedding is equivalent to the
composition of morphisms

${{\mathrm{L}(\tau,A,B)}}$${{\mathrm{L}(\tau,A^{\prime\prime},B^{\prime\prime})}}$${{\mathrm{L}(\tau,A^{\prime},B^{\prime})}}$${{H(\lambda,\mu)}}$$\scriptstyle{f^{*}}$$\scriptstyle{\beta_{\Gamma}}$$\scriptstyle{\alpha_{\tau^{\prime},u^{\prime},B^{\prime}}}$

Such embedding depends only on the choice of an automorphism $f\colon E\to E$
fixing zero (finite number of choices), $\tau^{\prime}$ in the
$\operatorname{SL}_{2}(\mathbb{Z})$-orbit of $\tau$ (countable number of
choices) and $u^{\prime}<0$ such that $A^{\prime}=q^{u^{\prime}}$ (countable
number of choices). Our claim follows. ∎

###  2.3 Secondary Hopf surfaces

#### 2.3.1.

We have just classified primary Hopf surfaces that arise as compactifications
of $\operatorname{Tot}(L)$. We will focus in this subsection on
compactifications that are secondary Hopf surfaces. Every secondary Hopf
surface is a quotient of a primary Hopf surface by a free action of a finite
group [Kod66, Sect.10, p.695]. Assume that $S$ is a secondary Hopf surface
containing an elliptic curve with non-torsion normal bundle. Then $S$ is a
quotient of a diagonal Hopf surface $H(\lambda,\mu)$ by an operator
$\operatorname{diag}(q^{1/n},q^{r/n})$, where $q^{1/n}$ and $q^{r/n}$ are
primitive $n$-roots of unity [Kod66, Sect. 10, Thm. 32].

######  Theorem 2.3.2.

Let $L$ be a degree-zero non-torsion line bundle on an elliptic curve $E$.
Then the set of equivalence classes of open embeddings
$\operatorname{Tot}(L)\stackrel{{\scriptstyle\iota}}{{\to}}H$ of
$\operatorname{Tot}(L)$ into a Hopf surface (primary or secondary) is
countable.

######  Proof: .

We know the claim for embeddings into primary Hopf surfaces (Proposition
2.2.14), hence it is enough to assume that $H$ is a secondary Hopf surface. By
2.3.1, every secondary Hopf surface compactifying $\operatorname{Tot}(L)$ must
be of the form $H/\operatorname{diag}(q^{1/n},q^{r/n})$ for $(r,n)=1$, where
$H$ is a diagonal Hopf surface. Given an embedding of $\operatorname{Tot}(L)$
into $H/\operatorname{diag}(q^{1/n},q^{r/n})$, we can construct an embedding
of $\operatorname{Tot}(f^{*}L)$ to $H$, where $f\colon E^{\prime}\to E$ is a
quotient by an $n$-torsion element of $E^{\prime}$. The set of possible $f$ is
countable, and so is the set of equivalence classes of embeddings of
$\operatorname{Tot}(f^{*}L)$ into a primary Hopf surface, hence the claim. ∎

###  2.4 Hopf duality and analytic cobordance

#### 2.4.1.

Let $L\to E$ be a non-trivial degree-zero line bundle. The ruled surface
$\mathbb{P}(\mathcal{O}_{E}\oplus L)$ contains two sections:
$\mathbb{P}(\mathcal{O})$ and $\mathbb{P}(L)$. The map
$\operatorname{Tot}(L)\to\mathbb{P}(\mathcal{O}\oplus L)$, $v\mapsto[1:v]$
defines a biholomorphism between $\operatorname{Tot}(L)$ and
$\mathbb{P}(\mathcal{O}\oplus L)-\mathbb{P}(L)$. Similarly,
$\mathbb{P}(\mathcal{O}\oplus L)-\mathbb{P}(\mathcal{O})$ is biholomorphic to
$\operatorname{Tot}(L^{-1})$. Hence the surface
$\mathbb{P}(\mathcal{O}_{E}\oplus L)$ is a compactification of total spaces of
dual line bundles: $\operatorname{Tot}(L)$ and $\operatorname{Tot}(L^{-1})$.
The analogous property of the diagonal primary Hopf surfaces motivates the
following definition.

######  Definition 2.4.2.

Let $L\to E$ and $L^{\prime}\to E^{\prime}$ be two degree-zero line bundles on
elliptic curves. They are called

  * •

Hopf dual if there exists a Hopf surface $H$ with two elliptic curves
$E_{\lambda}$ and $E_{\mu}$ such that $H-E_{\mu}\simeq\operatorname{Tot}(L)$
and $H-E_{\lambda}\simeq\operatorname{Tot}(L^{\prime})$;

  * •

analytically cobordant if $L^{-1}\to E$ and $L^{\prime}\to E^{\prime}$ are
either Hopf dual, or $E=E^{\prime}$ and $L=L^{\prime}$.

By Proposition 2.2.14, any non-torsion line bundle admits a countable number
of Hopf dual and analytically cobordant line bundles. The base curves of
analytically cobordant line bundles are in general non-isomorphic (2.2.10).

#### 2.4.3.

Let $L\to E$ and $L^{\prime}\to E^{\prime}$ be Hopf dual line bundles. Then
the spaces $\operatorname{Tot}(L)-0_{L}$ and
$\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$ are biholomorphic, where
$0_{\Xi}\subset\operatorname{Tot}(\Xi)$ denotes the zero section. Indeed, both
spaces are biholomorphic to $H(\lambda,\mu)-\left(E_{\lambda}\cup
E_{\mu}\right)$. The biholomorphism sends the neighborhood of the zero section
in one to the neighborhood of the infinity section in the other and vice
versa. Similarly, let $L\to E$ and $L^{\prime}\to E^{\prime}$ be analytically
cobordant line bundles. Then $\operatorname{Tot}(L)-0_{L}$ and
$\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$ are biholomorphic so that a
neighbourhood of the zero section in one maps to the neighbourhood of the zero
section in the other.

#### 2.4.4.

Of course, if $E\not\cong E^{\prime}$, the biholomorphism
$\operatorname{Tot}(L)-0_{L}\simeq\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$
cannot be extended to the zero section. To internalize it, let us understand
the behavior of fibers of the bundle under this biholomorphism. Let
$H=H(\lambda,\mu)$ be a Hopf surface. The fibers of
$H-E_{\mu}=\mathcal{L}(\lambda,\mu)\to E_{\lambda}$ are vertical lines. When
we project $\mathbb{C}^{2}-\\{0\\}$ to $H(\mu,\lambda)$, we identify vertical
lines $x=c,x=ac,x=a^{2}c$ etc. Hence, the image of a vertical line in
$H(\mu,\lambda)$ is a non-closed horn-like subspace with the curve $E_{\mu}$
in its closure.

######  Proposition 2.4.5.

Let $L\to E$ and $L^{\prime}\to E^{\prime}$ be two line bundles on elliptic
curves. Then

  1. 1.

$L$ and $L^{\prime}$ are Hopf dual if and only if there exists a
biholomorphism $\operatorname{Tot}(L)-0_{L}$ and
$\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$ sending a neighborhood of
$0_{L}$ to a neighborhood of infinity in $\operatorname{Tot}(L^{\prime})$.

  2. 2.

$L$ and $L^{\prime}$ are analytically cobordant if and only if there exists a
biholomorphism $\operatorname{Tot}(L)-0_{L}$ and
$\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$ sending a neighborhood of
$0_{L}$ to a neighborhood of $0_{L^{\prime}}$.

######  Proof: .

The second statement follows immediately from the first, so we will only prove
the first one. We saw in (2.4.3) that Hopf dual line bundles satisfy the
condition of the theorem. Conversely, suppose there is a biholomorphism
$\operatorname{Tot}(L)-0_{L}$ and
$\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$ as in the proposition. Let us
glue $\operatorname{Tot}(L)$ and $\operatorname{Tot}(L^{\prime})$ by this
isomorphism. The result is a compact Hausdorff surface compactifying
$\operatorname{Tot}(L)$. By Enoki’s theorem, it is either a ruled or a Hopf
surface, hence the claim. ∎

#### 2.4.6.

By Proposition 2.4.5, analytic cobordance is an equivalence relation on the
set of pairs $(E,L)$ where $E$ is an elliptic curve and $L$ is a line bundle
on $E$. However, if $L$ is analytically cobordant to $L^{\prime}$ and
$L^{\prime\prime}$ through primary Hopf surfaces, then $L^{\prime}$ and
$L^{\prime\prime}$ can be analytically cobordant through a secondary Hopf
surface.

#### 2.4.7.

The surface $M=\operatorname{Tot}(L)$ carries a holomorphic symplectic form
$\sigma$ with a simple pole at the zero section. Indeed,

$K_{M}=\pi^{*}K_{E}\otimes K_{M/E}=K_{M/E}=\pi^{*}L^{*}.$

The second isomorphism holds because $K_{E}$ is trivial. The pullback
$\pi^{*}L$ has the tautological section, which vanishes on $0_{L}$; thus
$K_{M}=\pi^{*}L^{*}$ has a nowhere zero section with a simple pole along
$0_{L}$.

#### 2.4.8.

Consider non-torsion analytically cobordant line bundles $L$ and $L^{\prime}$.
Let $\sigma$ and $\sigma^{\prime}$ be the holomorphic symplectic forms on
$\operatorname{Tot}(L)-0_{L}$ and
$\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$ The pullback of
$\sigma^{\prime}$ under the holomophic isomorphism
$f\colon\operatorname{Tot}(L)-0_{L}\to\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$
equals $c\sigma$ for a non-zero constant $c$. Indeed,
$f^{*}\sigma^{\prime}=g\sigma$ for a holomorphic function $g$ on
$\operatorname{Tot}(L)-0_{L}$. By [KU19, Lemma 2.2], all holomorphic functions
on $\operatorname{Tot}(L)-0_{L}$ are constant.

###  2.5 Hopf transforms

#### 2.5.1.

A tubular neighborhood of a complex submanifold is in general not
biholomorphic to a neighborhood of the zero section of its normal bundle. For
example, a smooth cubic $C\subset\mathbb{P}^{2}$ can be deformed to a non-
isomorphic curve, whereas all deformations of the zero section of
$\operatorname{Tot}(\nu_{C/\mathbb{P}^{2}})$ are isomorphic to $C$. Therefore,
neighborhoods of $C$ in $\mathbb{P}^{2}$ and in
$\operatorname{Tot}(\nu_{C/\mathbb{P}^{2}})$ cannot be biholomorphic. However,
analogues of the tubular neighbourhood theorem are known in certain
situations.

######  Definition 2.5.2.

Let $L\to E$ be a degree-zero line bundle on an elliptic curve, and $d$ an
translation-invariant metric on the Picard variety $\mathrm{Pic}^{0}(E)$.
Suppose that

$-\log d(\mathcal{O}_{E},L^{\otimes n})=O(\log n).$

Then $L$ is called Diophantine.

The Diophantine property does not depend on the choice of the metric $d$. The
set of Diophantine line bundles on $E$ is the complement to a measure-zero set
in $\mathrm{Pic^{0}}(E)$.

######  Theorem 2.5.3 (Arnold–Ueda theorem [Arn76, 4.3] [Ued82, Th. 3]).

Let $S$ be a complex surface and $E\subset S$ an elliptic curve. If the normal
bundle $\nu_{E/S}$ is non-torsion and of degree zero, then the formal
neighbourhoods of $E$ in $S$ and in $\operatorname{Tot}(\nu_{E/S})$ are
isomorphic. If $\nu_{E/S}$ is Diophantine, then this isomorphism extends to a
biholomorphism of an analytic neighborhood of $E$ in $S$ with a neighborhood
of $E$ in $\operatorname{Tot}(\nu_{E/S})$.

#### 2.5.4.

The explicit description in [Arn76, 4.3] implies that if $L\to E$ and
$L^{\prime}\to E^{\prime}$ are analytically cobordant through a primary Hopf
surface and $L$ is Diophantine, then so is $L^{\prime}$. It follows easily
from this and 2.3.1 that the Diophantine property is preserved by any analytic
cobordism.

#### 2.5.5.

Arnold–Ueda theorem fails in the case of torsion line bundles. A
counterexample is a fiber $F$ of a non-isotrivial elliptic fibraton. It has a
holomorphically trivial normal bundle, yet, its neighbors are not isomorphic
to $F$.

######  Definition 2.5.6.

Let $E\subset S$ be an elliptic curve on a surface $S$ and $L$ its normal
bundle. Assume that $\deg_{E}L=0$ and $L$ satisfies the Diophantine condition.
Let $L^{\prime}\to E^{\prime}$ be a bundle analytically cobordant to $L\to E$
(Definition 2.4.2). Pick a tubular neigbourhood of $E\subset S$ as in Theorem
2.5.3, throw away $E$, and glue the tubular neighbourhood of the zero section
of $L^{\prime}\to F$ through a holomorphic isomorphism between
$\operatorname{Tot}(L)-0_{L}$ and
$\operatorname{Tot}(L^{\prime})-0_{L^{\prime}}$. The resulting surface
$S^{\prime}$ is called the Hopf transform of $S$ in $E$ by $L^{\prime}\to
E^{\prime}$. The image of the curve $E^{\prime}$ in the Hopf transform is
called the graft.

#### 2.5.7.

The new surface $S^{\prime}$ may not be algebraic or even Kähler even if $S$
is. For example, take $S=\mathbb{P}(\mathcal{O}_{E}\oplus L)$ and embed $E$ as
one of the sections. Then $S^{\prime}$ is a Hopf surface.

#### 2.5.8.

The Hopf transform of a given surface in a given curve can be made in a
countable number of non-isomorphic ways due to Theorem 2.3.2. Nevertheless,
the result of a Hopf transform depends only on the choice of an analytically
cobordant line bundle and not on an isomorphism between a neighbourhood of $E$
in $S$ and in $\operatorname{Tot}(L)$. The proof relies on the following
Lemma.

######  Lemma 2.5.9.

Let $L$ be a non-torsion degree-zero line bundle on an elliptic curve $E$. For
$r>0$, denote by $W_{r}$ the subset of vectors in $\operatorname{Tot}(L)$ of
length less than $r$. Consider an open embedding $j\colon W_{r}\to W_{R}$
identical on $E$, where $r,R\in\mathbb{R}_{>0}\sqcup\infty$. Then $j$ is the
fiberwise multiplication by a constant $c\in\mathbb{C}^{*}$ such that $|c|\leq
R/r$.

######  Proof: .

Step 1. The space $W_{r}$ (resp. $W_{R}$) is isomorphic to $\mathbb{C}\times
B_{r}/\Lambda$ (resp. $\mathbb{C}\times B_{R}/\Lambda$) where
$\Lambda:=\mathbb{Z}+\mathbb{Z}\cdot\tau$ acts as follows:

$\gamma\cdot(x,y)=(x+\gamma,\rho(\gamma)y).$

Here $B_{r}$ (resp. $B_{R}$) is the open disk of radius $r$ (resp. $R$) and
$\rho\colon\Lambda\to U(1)$ is the monodromy representation. By the universal
property of universal covers, we can lift $j$ to an open embedding
$J\colon\mathbb{C}\times B_{r}\to\mathbb{C}\times B_{R}$. The map $J$ descends
to quotients, hence for every $\gamma\in\Lambda$ there exists
$\gamma^{\prime}\in\Lambda$ such that
$J(\gamma\cdot(x,y))=\gamma^{\prime}\cdot J(x,y)$. When $y=0$ the map $J$ is
the identity, hence $\gamma^{\prime}=\gamma$. We conclude that $J$ is
$\Lambda$-equivariant.

Step 2. Write $J$ as $(f,g)$, where $f\colon\mathbb{C}\times
B_{r}\to\mathbb{C}$ and $g\colon\mathbb{C}\times B_{r}\to B_{R}$. We will see
in this step that $g(x,y)=cy$ for a constant $c\in\mathbb{C}^{*}$. For a fixed
$y\in B_{r}$, the map $g$ is a holomorphic function from $\mathbb{C}$ to
$B_{R}$, hence constant. We conclude that $g=g(y)$. The $\Lambda$-equivariance
of $J$ implies that $g(\rho(\gamma)y)=\rho(\gamma)g(y)$. The image of
$\Lambda$ in $U(1)$ is dense because $L$ is non-torsion. Therefore, $g$ is
$U(1)$-equivariant, and our claim follows.

Step 3. By $\Lambda$-equivariance of $J$ we have that

$f(x+\gamma,\rho(\gamma)y)=f(x,y)+\gamma.$ (4)

Let as differentiate the equation (4) $k$ times with respect to $y$. We obtain

$\rho^{k}(\gamma)\frac{\partial^{k}f}{\partial
y^{k}}(x+\gamma,\rho(\gamma)y)=\frac{\partial^{k}f}{\partial y^{k}}(x,y)$

Set $y=0$. The function $\left|\frac{\partial^{k}f}{\partial
y^{k}}(x,0)\right|$ is invariant under shifts by $\gamma\in\Lambda$, hence
bounded. We conclude that $\frac{\partial^{k}f}{\partial y^{k}}(x,0)$ is
constant. The representation $\rho^{k}$ is non-trivial for every $k$, hence
$\frac{\partial^{k}f}{\partial y^{k}}(x,0)$ vanishes. We conclude that $f$
does not depend on $y$. Hence $f(x,y)=f(x,0)=x$. We showed that
$J(x,y)=(x,cy)$, hence the claim. ∎

######  Corollary 2.5.10.

Let $E\subset S$ be as in Definition 2.5.6. Fix a line bundle $L^{\prime}$
analytically cobordant to $L\to E$ (Definition 2.4.2). Then the Hopf transform
of $S$ in $E$ by $L^{\prime}$ is well-defined. Namely, consider two
biholomorphisms $\varphi_{1}\colon
U_{1}\stackrel{{\scriptstyle\sim}}{{\to}}W_{1}$ and $\varphi_{2}\colon
U_{2}\stackrel{{\scriptstyle\sim}}{{\to}}W_{2}$ between neighborhoods $U_{1}$
and $U_{2}$ of $E$ in $S$ and neighborhoods $W_{1}$ and $W_{2}$ of $E$ in
$\operatorname{Tot}(L)$. Assume that $\varphi_{1}|_{E}=\varphi_{2}|_{E}$. Then
the Hopf transforms $S_{1}$ and $S_{2}$ of $S$ obtained through $\varphi_{1}$
and $\varphi_{2}$ respectively are biholomorphic.

######  Proof: .

The Hopf transforms $S_{i}$, $i=1,2$, will not change if we shrink $U_{i}$.
Hence we may assume that $U_{1}\subset U_{2}$ and $\varphi_{1}(U_{1})=W_{r}$,
$\varphi_{2}(U_{2})=W_{R}$ for some $r,R\in\mathbb{R}_{>0}$. The map
$j:=\varphi_{2}|_{U_{2}}\circ\varphi_{1}^{-1}$ is an open embedding $W_{r}\to
W_{R}$. Lemma 2.5.9 implies that $j$ is the multiplication by a constant
$c\in\mathbb{C}^{\times}$. Embed $\operatorname{Tot}(L^{-1})$ into a Hopf
surface $H(\lambda,\mu)$ such that
$H(\lambda,\mu)-E_{\lambda}\simeq\operatorname{Tot}(L^{-1})$ and
$H(\lambda,\mu)-E_{\mu}\simeq\operatorname{Tot}(L^{\prime})$. The
biholomorphism $j$ extends to the automorphism $\operatorname{diag}(1,c)$ of
$H(\lambda,\mu)$ by 2.2.13. This automorphism of $H(\lambda,\mu)$ induces a
biholomorphism $S_{1}\to S_{2}$. ∎

#### 2.5.11.

We only define the Hopf transform in elliptic curves with Diophantine normal
bundle. The same definition works for any square-zero elliptic curve with a
holomorphic tubular neighbourhood.

An equivalent formulation of Enoki’s theorem (Theorem 2.1.2) is that every
compactification of $\operatorname{Tot}(L)$ is a Hopf transform of
$\mathbb{P}(\mathcal{O}\oplus L)$ in the infinity section. This statement
generalizes partially to other complex surfaces.

######  Theorem 2.5.12.

Suppose $M$ is a complex surface realizable as the complement to a square-zero
elliptic curve $E$ with a Diophantine normal bundle $L$ in a compact surface
$S$. Then every minimal analytic compactification of $M$ is a Hopf transform
of $S$ in $E$.

######  Proof: .

The Diophantine condition implies that the neighborhood of infinity in $X$ is
biholomorphic to a neighborhood of the zero section in $\operatorname{Tot}(L)$
with the zero section removed (Theorem 2.5.3). Hence, every partial
compactification of $\operatorname{Tot}(L)-0_{L}$ near the zero section
produces a compactification of $M$ and vice versa. By Theorem 2.3.2 and
Enoki’s theorem, every partial compactification of
$\operatorname{Tot}(L)-0_{L}$ is a Hopf transform at $0_{L}$. ∎

######  Corollary 2.5.13.

A Hopf transform of a Hopf transform in its graft is either a Hopf transform,
or the initial surface. Inverse of a Hopf transform is a Hopf transform.

######  Proof: .

The composition of Hopf transforms of $S$ in $E$ is a compactification of
$S-E$, hence a Hopf transform (Theorem 2.5.12). ∎

######  Corollary 2.5.14.

Let $S$ be a projective surface with a square-zero elliptic curve $E$ with a
Diophantine normal bundle. Suppose that each Hopf transform of $S$ in $E$ is
projective. Then the set of algebraic structures on $M:=S-E$ is countable.

######  Proof: .

By Theorem 2.5.12, every compactification of $M$ is a Hopf transform of $S$ in
$E$. Thus every compactification induces an algebraic structure on $M$.
Conversely, every algebraic structure is induced from a compactification.
Therefore, the set of compactifications $(S^{\prime},E^{\prime})$ of $M$ is
countable (Theorem 2.3.2). Moreover, the set of possible $E^{\prime}$ is also
countable. So, it is enough to prove that two compactifications
$(S^{\prime},E^{\prime})$ and $(S^{\prime\prime},E^{\prime\prime})$ such that
$E^{\prime}\not\simeq E^{\prime\prime}$ induce distinct algebraic structures
on $M$. An algebraic isomorphism between $\varphi\colon
S^{\prime}-E^{\prime}\to S^{\prime\prime}-E^{\prime\prime}$ would induce a
birational morphism $\varphi\colon S^{\prime}\mathbin{\leavevmode\hbox
to15.28pt{\vbox to7.15pt{\pgfpicture\makeatletter\hbox{\hskip
2.07999pt\lower-2.39998pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,3.0pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{ }{}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{ }{}{{ {\pgfsys@beginscope
\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{}
\pgfsys@moveto{-2.07999pt}{2.39998pt}\pgfsys@curveto{-1.69998pt}{0.95998pt}{-0.85318pt}{0.28pt}{0.0pt}{0.0pt}\pgfsys@curveto{-0.85318pt}{-0.28pt}{-1.69998pt}{-0.95998pt}{-2.07999pt}{-2.39998pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}\pgfsys@moveto{0.0pt}{2.15277pt}\pgfsys@lineto{12.60002pt}{2.15277pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.80002pt}{2.15277pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}S^{\prime\prime}$. A
birational map of smooth surfaces cannot contract a non-rational curve, hence
$E^{\prime}\simeq\varphi(E^{\prime})\simeq E^{\prime\prime}$, contradiction. ∎

##  3 Surfaces with square-zero elliptic curves

###  3.1 Kodaira dimension

#### 3.1.1.

Let $S$ be a complex surface (not necessarily projective) containing a square-
zero elliptic curve $E$. Then its Kodaira dimension $\kappa(S)$ is at most
one. Indeed, suppose $\kappa(S)=2$. Then the map $\varphi\colon
S\to\mathbb{P}^{N}$ induced by the linear system $|nK_{S}|$ is birational onto
its image for sufficiently large $n$. Every square-zero elliptic curve $E$
satisfies $K_{S}\cdot E=0$. Hence the map $\varphi$ must contract $E$. Yet, a
birational map of surfaces cannot contract a square-zero curve.

######  Proposition 3.1.2.

Let $S$ be a surface with square-zero elliptic curve $E\subset S$ whose normal
bundle $\nu_{E/S}$ is non-torsion. Then $\kappa(S)=-\infty$.

######  Proof: .

Observe that $\nu_{E/S}\simeq\mathcal{O}_{E}(-K_{S}|_{E})$. We consider two
cases.

Case 1: $\kappa=0$. Every minimal surface with $\kappa=0$ has torsion
canonical bundle, hence the statement is trivial in this case. If $S$ is non-
minimal, then $K_{S}=\sum E_{i}+D$, where $E_{i}$’s are exceptional curves and
$D$ is a torsion divisor. We get that $K_{S}|_{E}$ is either torsion or has
positive degree for every curve $E\subset S$.

Case 2: $\kappa=1$. The linear system $|nK_{S}|$ for $n\gg 0$ induces a
morphism $\varphi\colon S\to\mathbb{P}^{N}$ whose image is a curve. It is an
elliptic fibration. Elliptic curves in the fibers of $\varphi$ have torsion
normal bundle, so we can consider only horizontal curves. For every horizontal
curve $C$, we have $nK_{S}|_{C}=(\varphi|_{C})^{*}H$, where $H$ is the
hyperplane section. Therefore, $K_{S}\cdot C$ is positive. ∎

#### 3.1.3.

A surface of negative Kodaira dimension is one of the following:

  * •

rational;

  * •

birational to a ruled surface;

  * •

of class VII, i.e., a non-Kähler surface with $\kappa=-\infty$ and $b_{1}=1$.

If a blow-up of a ruled surface $S$ contains an elliptic curve, then $S$ is
ruled over an elliptic curve.

######  Proposition 3.1.4.

Let $E\subset S$ be an anti-canonical elliptic curve with a degree-zero non-
torsion normal bundle $\nu_{E/S}$. Assume $b_{1}(S)=0$. Then $S$ is a blow-up
of a length nine subscheme in $\mathbb{P}^{2}$.

######  Proof: .

The canonical class of $S$ is anti-effective, hence all plurigenera
$p_{n}:=h^{0}(K_{S}^{n})$ of $S$ vanish. Since $b_{1}(S)=0$, the irregularity
$q=h^{1}(\mathcal{O}_{Y})$ vanishes as well. Castelnuovo theorem implies that
$S$ is rational, in particular, projective [BHPVdV15, VI (3.4)]. Its minimal
model is either $\mathbb{P}^{2}$ or a Hirzebruch surface
$\mathbb{F}_{n}=\mathbb{P}\left(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-n)\right)$.
The image of $E$ in a minimal model is an irreducible anti-canonical curve:
indeed, $E$ is anti-canonical, so it intersects each $(-1)$-curve transversely
at one point. Since $K_{S}^{2}=0$, $S$ is either a blow-up of nine points in
$\mathbb{P}^{2}$ or of eight points in $\mathbb{F}_{n}$.

The Hirzebruch surface $\mathbb{F}_{n}$ contains a section $C_{n}$ of square
$-n$. Projection formula yields $-K_{\mathbb{F}_{n}}\cdot C=2-n$, thus for
$n>2$ an anti-canonical curve contains $C$ and is reducible. Therefore, any
minimal model of $S$ is $\mathbb{P}^{2}$,
$\mathbb{F}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}$, or $\mathbb{F}_{2}$. An
anti-canonical curve in $\mathbb{F}_{2}$ does not intersect $C_{2}$. A blow-up
of a point away from $C_{2}$ is isomorphic to a blow-up of a length-two
subscheme in $\mathbb{P}^{2}$. A blowup of any point on
$\mathbb{F}_{0}=\mathbb{P}^{1}\times\mathbb{P}^{1}$ is isomorphic to a blowup
of two points on $\mathbb{P}^{2}$. Thus $S$ is a blow-up of $\mathbb{P}^{2}$.
∎

###  3.2 Rational surfaces

Let us start with the following lemma.

######  Lemma 3.2.1.

Let $S$ be a surface, and $C\subset S$ a smooth curve. Then the natural map
$H_{1}(S-C,\mathbb{Q})\to H_{1}(S,\mathbb{Q})$ is surjective. If $S$ is
Kähler, then it is also injective.

######  Proof: .

Look at the long exact sequence of cohomology associated with the
decomposition $S=C\sqcup(S-C)$:

$\dots\to H^{2}(S,\mathbb{Q})\to H^{2}(C,\mathbb{Q})\to
H^{3}_{c}(S-C,\mathbb{Q})\to H^{3}(S,\mathbb{Q})\to H^{3}(C,\mathbb{Q})=0.$

If $S$ is Kähler, the class of $C$ in $H_{2}(S)$ is non-trivial. Therefore the
map $H^{2}(S,\mathbb{Q})\to H^{2}(C,\mathbb{Q})$ is surjective, and
$H_{1}(S-C,\mathbb{Q})\simeq H^{3}_{c}(S-C,\mathbb{Q})\simeq
H^{3}(S,\mathbb{Q})\simeq H_{1}(S,\mathbb{Q})$. ∎

The next statement follows easily from Lemma 3.2.1, and we omit its proof.

######  Corollary 3.2.2.

  1. 1.

Let $S$ be a rational surface, $C\subset S$ a smooth curve. Then
$h_{1}(S-C)=0$.

  2. 2.

Let $S$ be a blow-up of a ruled surface over an elliptic curve, $C\subset S$ a
smooth curve. Then $h_{1}(S-C)=2$.

######  Corollary 3.2.3.

  1. 1.

A Hopf transform of a rational surface is rational.

  2. 2.

A Hopf transform of a surface birational to a ruled surface is birational
either to a ruled surface or a Hopf surface.

######  Proof: .

Let $S$ be a rational surface. Lemma 3.2.1 implies that $h_{1}(S-C)=0$.
Consider a Hopf transform $(S^{\prime},C^{\prime})$ of $S$ in $C$. The space
$S^{\prime}-C^{\prime}$ is biholomorphic to $S-C$, hence
$h_{1}(S^{\prime}-C^{\prime})=h_{1}(S-C)$. Lemma 3.2.1 implies that
$h_{1}(S^{\prime}-C^{\prime})\geqslant h_{1}(S^{\prime})$. Thus a Hopf
transform of a rational surface satisfies $h_{1}(S^{\prime})=0$. The
classification in 3.1.3 implies that $S^{\prime}$ is rational.

A Hopf transform of a blow-up of a ruled surface cannot be rational by the
previous statement. By 3.1.3 its Hopf transform is either a blow-up of a ruled
surface or a Hopf surface. ∎

#### 3.2.4.

Choose an elliptic curve $E$ in $\mathbb{P}^{2}$ and nine points
$p_{1},...p_{9}\in E$ (points are allowed to collide). Denote the hyperplane
section of $\mathbb{P}^{2}$ as $H$. Assume that the line bundle
$\mathcal{O}_{E}(3H-p_{1}-...-p_{9})$ satisfies the Diophantine condition
(Definition 2.5.2). This line bundle is isomorphic to the normal bundle to the
strict transform of $E$ in the blow-up $X$ in $p_{1},...p_{9}$. Therefore, a
Hopf transform $Y$ of $X$ in $E$ is well-defined.

######  Theorem 3.2.5.

Any Hopf transform $Y$ of $X$ is the blow-up of $\mathbb{P}^{2}$ in a length
nine subscheme $\gamma$. The graft is the strict preimage in $Y$ of the
elliptic curve $F\subset\mathbb{P}^{2}$ passing through $\gamma$.

######  Proof: .

Step 1. The surface $(X,E)$ is log Calabi–Yau, that is, $K_{X}+E=0$. This
implies existence of a meromorphic symplectic form on $X$ with a simple pole
along $E\subset X$. We know from 2.4.8 that a Hopf transform is a gluing along
a holomorphic symplectomorphism, hence $Y$ carries a symplectic form with a
simple pole along $F\subset Y$. Thus $K_{Y}+F=0$.

Step 2. By Corollary 3.2.3, $Y$ is rational. Moreover, its anti-canonical
divisor is effective. By Proposition 3.1.4, $Y$ is a blowup of a length-nine
subscheme in $\mathbb{P}^{2}$, and $F\subset Y$ is the strict transform of the
only plane cubic passing through it. ∎

The following question remains open:

######  Problem 1.

Determine the nine points $q_{1},...q_{9}\in\mathbb{P}^{2}$ such that
$Y\simeq\operatorname{Bl}_{q_{1},..q_{9}}(\mathbb{P}^{2})$.

######  Theorem 3.2.6.

Let $X$ be the blowup of $\mathbb{P}^{2}$ in a very generic length-nine
subscheme $\gamma$, and $E\subset X$ the strict transform of the plane cubic
passing through $\gamma$. Then the analytification of $X-E$ admits countably
many algebraic structures.

######  Proof: .

The claim follows from Corollary 2.5.14 because every Hopf transform of $X$ is
projective.

∎

#### 3.2.7.

There are rational surfaces containing a not anti-canonical elliptic curve of
square zero. Let $Q\subset\mathbb{P}^{2}$ be a plane quartic with two nodes.
Blow them up; the strict transform $\tilde{Q}$ has square $8$. By blowing up
eight more points on $\tilde{Q}$, one gets a square-zero elliptic curve on
$\mathbb{P}^{2}$ blown up in ten points. Notice that ‘having a node at a given
point’ is a codimension three condition, thus plane quartics with two nodes at
$p,q\in\mathbb{P}^{2}$ form an eight-dimensional space. Thus, a generic tuple
of ten points with two distinguished ones determines a unique plane quartic
with two nodes through it.

#### 3.2.8.

By [KU19, Lemma 2.2], punctured neighbourhood of a square-zero elliptic curve
with non-torsion normal bundle carries no nonconstant functions. Hence none of
the surfaces $X-E$ from Theorem 3.2.6 is Stein.

###  3.3 Surfaces of class VII

All surfaces of class VII that contain an elliptic curve with a non-torsion
degree-zero normal bundle are Hopf surfaces, as we prove below.

######  Proposition 3.3.1.

Let $S$ be a surface of class VII and $E\subset S$ a smooth elliptic curve
with non-torsion normal bundle $\nu_{E/S}$ of degree zero. Then $S$ is a
primary Hopf surface $H(\lambda,\mu)$ with $\lambda^{n}\neq\mu^{m}$ for any
$n,m\in\mathbb{Z}$, or a secondary Hopf surface $H(\lambda,\mu)/\mu_{n}$ from
2.3.1, or a blow-up of such a surface away from $E$.

######  Proof: .

Every curve on a smooth non-Kähler surface has a non-positive square
[BHPVdV15, Ch. IV, Thm. 2.14]. Applying this fact to a minimal model of $S$,
we obtain that a curve of square zero does not intersect exceptional curves.

If $E\subset S$ is a nonsingular square-zero curve on a minimal surface $S$ of
class VII, then $S$ is a Hopf surface [Eno81, Proposition 4.12]. Hopf surfaces
that contain an elliptic curve with non-torsion normal bundle were classified
in 2.2.6 and 2.3.1. ∎

##  4 Analytic Grothendieck group

######  Definition 4.1.1.

The Grothendieck group $K_{0}(\mathrm{Var}_{\mathbb{C}})$ of complex algebraic
varieties is the abelian group generated by classes of $\mathbb{C}$-varieties
modulo scissor relations:

$[X-Y]=[X]-[Y],$

where $Y\subset X$ is a closed algebraic subvariety.222It is actually a ring
with multiplication $[U]\cdot[V]=\left[{U}\times_{k}{V}\right]$, but we do not
use it.

We introduce an interesting quotient of this group.

######  Definition 4.1.2.

The analytic Grothendieck group of varieties $K_{0}^{an}$ is the quotient of
$K_{0}(\mathrm{Var}_{\mathbb{C}})$ by differences $[U]-[V]$ where $U$ and $V$
are biholomorphic varieties.

######  Proposition 4.1.3.

The classes of any two elliptic curves in $K_{0}^{an}$ are equal.

######  Proof: .

Step 1. Suppose that $E,F$ are two elliptic curves and $L\to E$,
$L^{\prime}\to F$ are two analytically cobordant Diophantine line bundles.
Then $[E]=[F]\in K_{0}^{an}$. Indeed, let $X$ be a blow-up of nine points in
$\mathbb{P}^{2}$ such that the cubic passing through them is isomorphic to
$E$, and the normal bundle of its strict transform is isomorphic to $L$. Even
in $K_{0}(\mathrm{Var}_{\mathbb{C}})$, one has
$[X]=\mathbb{L}^{2}+10\mathbb{L}+[\mathrm{pt}]$, where $\mathbb{L}$ is the
class of the affine line. By Theorem 3.2.5, the Hopf transform $Y$ of $X$ in
$E$ by $L^{\prime}\to F$ is also a blow-up of $\mathbb{P}^{2}$ in a length
nine subscheme. Thus $[Y]=\mathbb{L}^{2}+10\mathbb{L}+1=[X]$. It follows from
the relations $[X]=[E]+[X-E]$ and $[Y]=[F]+[Y-F]$ that

$[E]^{an}=[X]^{an}-[X-E]^{an}=[Y]^{an}-[Y-F]^{an}=[F]^{an}\in K_{0}^{an}.$

Step 2. For any pair of elliptic curves $E$, $F$, there exists a primary Hopf
surface $H=H_{E,F}$ containing $E$ and $F$. For a fixed $E$, the set of curves
$F$ for which the normal bundle $\nu_{E/H_{E,F}}$ is not Diophantine is meagre
in the moduli space of elliptic curves. Thus for arbitrary two curves $E$,
$E^{\prime}$, the complements of the corresponding meagre sets have nonempty
intersection. Let $F$ be any curve from this intersection. Step 1 implies that
$[E]^{an}=[F]^{an}$ and $[F]^{an}=[E^{\prime}]^{an}$, hence the claim. ∎

#### 4.1.4.

Although the classes of elliptic curves in $K_{0}^{an}$ coincide, the classes
of different elliptic curves in $K_{0}(\mathrm{Var}_{\mathbb{C}})$ are
distinct, as follows from Franziska Bittner’s description of
$K_{0}(\mathrm{Var}_{\mathbb{C}})$ in [Bit04]. She proved that
$K_{0}(\mathrm{Var}_{\mathbb{C}})$ is generated by classes of smooth
projective varieties modulo relations

$[X]-[Y]=[Bl_{Y}X]-[E],$ (5)

where $Y$ is a smooth subvariety of a smooth projective variety $X$ and
$E\subset Bl_{Y}X$ is the exceptional divisor of the blow-up of $X$ in $Y$.
This result is very powerful as it enables us to construct motivic measures
(i.e., maps from $K_{0}(\mathrm{Var}_{\mathbb{C}})$ to an abelian group) by
checking only the relation (5). For example, consider the functional on smooth
projective varieties sending a variety to its stable birationality class. This
functional descends to a motivic measure

$K_{0}(\mathrm{Var}_{\mathbb{C}})\to\mathbb{Z}[\mathrm{SB}]$

with values in the free abelian group generated by stable birationality
classes. Indeed, $Bl_{Y}X$ is birational to $X$ and $E$ is stably birational
to $Y$, so the relation (5) is preserved. We see that non-stably birational
varieties have different classes in $K_{0}(\mathrm{Var}_{\mathbb{C}})$. Two
curves are stably birational if and only if they are isomorphic [Har13, V Ex.
2.1]. In particular, non-isomorphic elliptic curves have different classes in
$K_{0}(\mathrm{Var}_{\mathbb{C}})$.

#### 4.1.5.

Topological Euler characteristic is a homomorphism $K_{0}^{an}\to\mathbb{Z}$.
However, we were not able to find more interesting analytic motivic measures.
This motivates the following

######  Problem 2.

Is the class of an elliptic curve in $K_{0}^{an}$ trivial? If so, is
$K_{0}^{an}$ isomorphic to $\mathbb{Z}$?

##### Acknowledgements.

We are grateful to Piotr Achinger, to whom we owe the idea of Section 4. Many
thanks to Robert Friedman, Michał Kapustka, Takayuki Koike, Aleksandr Petrov,
and Giulia Saccà for insightful discussions, and to Nathan Chen, Andrés
Fernández Herrero, and Morena Porzio for reading an early draft of the paper.

## Bibliography

  * [Arn76] Vladimir I. Arnold.  Bifurcations of invariant manifolds of differential equations and normal forms in neighborhoods of elliptic curves.  Funct. Anal. Appl., 10(4):249–259, 1976.
  * [BHPVdV15] Wolf Barth, Klaus Hulek, Chris Peters, and Antonius Van de Ven.  Compact complex surfaces, volume 4.  Springer, 2015.
  * [Bit04] Franziska Bittner.  The universal Euler characteristic for varieties of characteristic zero.  Compositio Mathematica, 140(4):1011–1032, 2004.
  * [BPvS23] Jérémy Blanc, Pierre-Marie Poloni, and Immanuel van Santen.  Complements of hypersurfaces in projective spaces.  2023\.  arXiv:2301.13040.
  * [Eno81] Ichiro Enoki.  Surfaces of class VII0 with curves.  Tôhoku Math. Journ., 33:453–492, 1981.
  * [Eno82] Ichiro Enoki.  On compactifiable complex analytic surfaces.  Inventiones mathematicæ, 67:189–212, 1982.
  * [FM13] Robert Friedman and John W Morgan.  Smooth four-manifolds and complex surfaces, volume 27.  Springer Science & Business Media, 2013.
  * [Har13] Robin Hartshorne.  Algebraic geometry, volume 52.  Springer Science & Business Media, 2013.
  * [Jel15] Zbigniew Jelonek.  Simple examples of affine manifolds with infinitely many exotic models.  Adv. Math., 284:112–121, 2015.
  * [Kod66] Kunihiko Kodaira.  On the Structure of Compact Complex Analytic Surfaces, II.  American Journal of Mathematics, 88(3):682–721, 1966.
  * [KT19] Maxim Kontsevich and Yuri Tschinkel.  Specialization of birational types.  Inventiones mathematicae, 217:415–432, 2019.
  * [KU19] Takayuki Koike and Takato Uehara.  A gluing construction of K3 surfaces.  2019\.  arXiv:1903.01444.
  * [Nee89] Amnon Neeman.  Ueda Theory: Theorems and Problems, volume 4.  American Mathematical Soc., 1989.
  * [NS19] Johannes Nicaise and Evgeny Shinder.  The motivic nearby fiber and degeneration of stable rationality.  Inventiones mathematicae, 217:377–413, 2019.
  * [Ued82] Tetsuo Ueda.  On the neighborhood of a compact complex curve with topologically trivial normal bundle.  J. Math. Kyoto Univ., 22(4):583–607, 1982.

Anna Abasheva
Columbia University,
Department of Mathematics,
2990 Broadway,
New York, NY, USA
anabasheva(at)math.columbia.edu

Rodion Déev
Institute of Mathematics,
Polish Academy of Sciences,
Śniadeckich 8,
Warsaw, Poland
rdeev(at)impan.pl