Different algebraic structures on complements to divisors Complements to square-zero curves in projective surfaces sometimes have several non-isomorphic algebraic structures. Serre’s example is possibly the most famous illustration of this phenomenon (see f.e. [PS,Ex. 4.19]). Here is another example.
Consider a pencil of cubic curves in $\mathbb P^2$. They pass through 9 common points. The blow up $X$ of $\mathbb P^2$ in these points admits an elliptic fibration $f\colon X\to \mathbb P^1$.
Let $D$ be a smooth fiber of this fibration, or a fiber of type $I_m, m\ge 1$. One can define a logarithmic transform $X’$ of $X$ in $D$, which is an elliptic surface biholomorphic to $X$ outside of $D$ but having a fiber $D’$ of multiplicity $n$ instead of $D$ ([BHPV, Ch. V.13]).
The construction is purely complex analytic, yet, in our case $X’$ happens to have an algebraic structure. The first way to see this is to use results of [FM] (Ch. 1, Theorem 6.7 (i) and Theorem 6.12). They prove that a logarithmic transform of an elliptic surface with singular fibers is always projective up to a Shafarevich—Tate twist. The Shafarevich—Tate group of a rational elliptic surface is trivial, hence the claim. Also, one can notice that the elliptic surface $X’$ is a blow up of 9 points $\{p_1,..p_9\}$ on an elliptic curve $E$ in $\mathbb P^2$. The points must satisfy the condition that $\mathcal O_E(p_1+…+p_9)\otimes \mathcal O_{\mathbb P^2}(-3)|_E$ is an $n$-torsion line bundle.
The complements $X-D$ and $X’-D’$ are biholomorphic but the algebraic structures induced from $X$ and $X’$ respectively are not preserved under this isomorphism. In this fashion we construct countable number of algebraic structures on $X-D$.

Question 1: Let us blow up 9 points on $\mathbb P^2$ in a general position, so that there is only one elliptic curve passing through them. Let $E$ be the strict transform of this elliptic curve. Does the complement to $E$ have several algebraic structures?

And here is a more general question.

Question 2: Do you know other examples of non-isomorphic algebraic structures on complements to square-zero curves?

Answers to any of them will be greatly appreciated!
Bibliography:

*

*[PS] Peters, Steenbrink. Mixed Hodge structures.

*[BHPV] Barth, Hulek, Peters, Van de Ven. Compact complex surfaces.

*[FM] Friedman, Morgan. Smooth four-manifolds and complex surfaces.

 A: 
Do you know other examples of non-isomorphic algebraic
structures on complements to square-zero curves

The easiest example is
the twisted cotangent bundle to an elliptic curve.
This space can be realized as the moduli of
representations of fundamental group in two ways,
according to Simpson: as the de Rham moduli
and as the Betti moduli of local systems.
These two spaces are complex analytically
equivalent to $({\Bbb C}^*)^2$, but the
algebraic structure is different, as follows
from [1, Theorem 3.1 (c)]. Here Simpson proves that
any subvariety which is algebraic in both structures
is a translation of a subgroup.
Complex analytic compactifications of $({\Bbb C}^*)^2$
are classified in [2-4].

Let us blow up 9 points on P2 in a general position, so
that there is only one elliptic curve passing through
them. Let E be the strict transform of this elliptic
curve. Does the complement to E have several algebraic
structures?

I suspect it does. In [5, Theorem C-I (e)]
Enoki shows that a flat affine bundle over
an elliptic curve might have non-algebraic
compactifications, now known as Enoki surfaces.
It does not follow that the blown-up $P^2$
without an elliptic curve has two non-equivalent
algebraic compactifications, of course, but it
is a good sign that it could.
[1] Simpson, Carlos
Subspaces of moduli spaces of rank one local systems. (English summary)
Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 3, 361-401.
[2] Simha, R.R.: Algebraic varieties biholomorphic
to ${\Bbb C}^*\times {\Bbb C}^*$. Tohoku Math. J. 30, 455-461 (1978)
[3] Suzuki, M.: Compactifications of ${\Bbb C}\times{\Bbb C}^*$
and $({\Bbb C}^*)^2$, Tohoku Math. J. 31, 453-468 (1979)
[4] Ueda, T.: Compactifications of ${\Bbb C}\times{\Bbb C}^*$
and $({\Bbb C}^*)^2$, Tohoku Math. J. 31, 81-90 (1979)
[5] Ichiro Enoki,
On compactifiable complex analytic surfaces,
Inv. Math. 67, pp 189–211 (1982)
