Algebraic models of the Euclidean plane

We introduce a new invariant, the real (logarithmic)-Kodaira dimension, that allows to distinguish smooth real algebraic surfaces up to birational diffeomorphism. As an application, we construct infinite families of smooth rational real algebraic surfaces with trivial homology groups, whose real loci are diffeomorphic to $\mathbb{R}^2$, but which are pairwise not birationally diffeomorphic. There are thus infinitely many non-trivial models of the euclidean plane, contrary to the compact case.


Introduction
A real quasi-projective algebraic variety X can be viewed as a complex quasi-projective algebraic variety endowed with an anti-regular involution, or equivalently as a locally closed subscheme of P n C which is defined over R. We can then speak about the set X(R) of real points of X (real locus). If X is smooth, this set is naturally endowed with the structure of differential real manifold, and X is said to be an algebraic model of this differential manifold. Two models X 1 and X 2 of the same manifold are said to be equivalent if there exists a diffeomorphism X 1 (R) → X 2 (R) which comes from a birational map ϕ : X 1 X 2 , such that ϕ and ϕ −1 are defined at each point of X 1 (R) and X 2 (R) respectively. Such a map is called a birational diffeomorphism. In general a manifold can admit plenty of different models. For example, the hypersurfaces of P 3 R given by the equations x 2n + y 2n + z 2n − t 2n = 0, n ≥ 1, provide infinitely many models of the sphere S 2 which are pairwise not birational. Nevertheless, if one restricts to the simplest ones, namely the rational models, then for smooth compact manifolds of dimension at most 2, the model is then unique. In dimension 1, we obtain only P 1 and in dimension 2 this is the following result of Biswas and Huisman: Theorem. [1,Corollary 8.1] A compact connected real manifold of dimension 2 admits a rational model if and only if it is non-orientable or diffeomorphic to S 2 or S 1 × S 1 . Moreover, this model is unique, up to birational diffeomorphism.
In the non-compact case, the real locus of the real affine algebraic variety A 2 R provides an obvious rational algebraic model of the Euclidean plane R 2 endowed with its standard structure of differential manifold. It is easy to find plenty of other rational models of R 2 : we can choose for instance the complement in P 2 R of a smooth irreducible real curve Γ ⊆ P 2 R of odd degree d ≥ 3 such that Γ (R) is an oval equivalent to a line by a diffeomorphism of RP 2 . It is thus natural to restrict the study of such models to the smaller class of "Fake real planes", introduced in [5] as being smooth algebraic surfaces S defined over R, non isomorphic to A 2 R but whose real locus is diffeomorphic to R 2 and whose complexifications S C have "minimal topology" in the sense that they are Q-acyclic topological manifolds, that is, topological manifolds whose singular homology groups with rational coefficientsH i (S C ; Q) are all trivial.
By general results [7,9,10,5] all these surfaces are affine and rational. A partial classification of them as real algebraic varieties was given in [5], according to their usual Kodaira dimension. Families of fake real planes of each Kodaira dimension κ ∈ {−∞, 0, 1, 2} birationally diffeomorphic to A 2 R were constructed in [6]. The existence of fake real planes non birationally diffeomorphic to A 2 R was left open. Here we show that R 2 admits algebraic models non birationally diffeomorphic to A 2 R of every Kodaira dimension κ = 0, 1, 2, answering the main question of [5] : Theorem 1. There are infinitely many rational models S of the plane R 2 up to birational diffeomorphism, all having trivial reduced homology groupsH i (S C ; Q). Such models exist for every κ = 0, 1, 2, and moreover, for κ = 1, 2, there exist infinitely many models S up to birational diffeomorphism for which S C is even topologically contractible.
In order to prove this result, we define a notion of real Kodaira dimension κ R (S) (Definition 2.4), which has the property to be smaller than or equal to the classical one κ(S), and can be computed in a very similar way (see Definition 2.4 and Remark 2.5). Moreover, we have equality κ(S) = κ R (S) in the natural case where S admits a smooth projective completion V with SNC boundary B = V \S consisting only of real curves isomorphic to P 1 R , and intersecting only at real points. The main new noteworthy feature of κ R (S) is that it is invariant under birational diffeomorphisms (Corollary 2.12), contrary to κ(S) (Example 2.6).
We establish the following result, from which Theorem 1 directly follows.

Theorem 2.
For each l ∈ {0, 1, 2}, there is a smooth affine surface S, algebraic model of the plane R 2 , with trivial rational homology groupsH i (S C ; Q) and κ R (S) = κ(S) = l. Moreover, for l ∈ {1, 2}, we can find infinitely many such S with topologically contractible complexifications S C , up to birational diffeomorphism.
As κ R is invariant under birational diffeomorphisms, every fake real plane S birationally diffeomorphic to A 2 R satisfies κ R (S) = −∞ (Corollary 2.15), so every of the examples which we construct in Theorem 2 is a fake real plane not birationally diffeomorphic to A 2 R . In contrast with the cases κ = 1, 2, the only smooth algebraic models of R 2 of Kodaira dimension −∞ and 0 with Q-acyclic complexifications known so far are respectively the affine plane A 2 R and a real model Y (3, 3, 3) of one of Fujita's exceptional surfaces [7] which was constructed in [5]. This motivates the following question: Question 0.1. Are the surfaces A 2 R and the fake real plane Y (3, 3, 3) of real Kodaira dimension 0 given in §2.B.b the unique algebraic models of R 2 with trivial reduced rational homology groups of real Kodaira dimension −∞ and 0, up to birational diffeomorphism ?
The article is organised as follows: Section 1 contains some preliminaries. In Section 2, we define the real Kodaira dimension of a smooth real surface and establish its basic properties. We also give some examples of fake real planes of real Kodaira dimension 0 (the surface Y (3, 3, 3) in §2.B.b) and 2 (the Ramanujam surface in §2.B.b). Then, in Sections 3 and 4, we provide families of pairwise not birational diffeomorphic fake real planes of Kodaira dimension 1 and 2 respectively, which achieve the proof of Theorem 2 hence of Theorem 1. The last subsection ( §4.C) describes pairs of fake real planes having the same complexifications but such that one has real Kodaira dimension 2 and the other is birationally diffeomorphic to A 2 R . We thank the referee for his careful reading and his helpful comments to improve the exposition of this text.

Preliminaries
A k-variety is a geometrically integral scheme X of finite type over a base field k. A morphism of kvarieties is a morphism of k-schemes. In the sequel, k will be equal to either R or C, and we will say that X is a real, respectively complex, algebraic variety. A complex algebraic variety X will be said to be defined over R if there exists a real algebraic variety X 0 and an isomorphism of complex algebraic varieties between X and the complexification X 0,

1.A. Real algebraic varieties and morphisms between them
For a real algebraic variety X, we denote by X(R) and X(C) the sets of R-rational and C-rational points of X respectively. These are endowed in a natural way with the Euclidean topology, locally induced by the usual Euclidean topologies on A n R (R) R 2n and A n C (C) C n respectively. When X is smooth, X(R) and X(C) can be further equipped with natural structures of C ∞ -manifolds. Every morphism f : X → X of real algebraic varieties induces a continuous map X(R) → X (R) for the Euclidean topologies, and an isomorphism of real algebraic varieties f : X −→ X induces a homeomorphism X(R) −→ X (R), which is a diffeomorphism when X and X are both smooth. In the context of the study of real algebraic models of a C ∞ -manifold, it is natural to consider a broader class of isomorphisms, induced by appropriate rational maps. Recall that the domain of definition of a rational map ϕ : X Y between two k-schemes X and Y is the largest open subset dom ϕ on which ϕ is represented by a morphism. We say that ϕ is regular at a closed point x if x ∈ dom ϕ . A rational map ϕ : X Y is called birational if it admits a rational inverse ψ : Y X.
Definition 1.1. Let ϕ : X X be a rational map between real algebraic varieties such that X(R) and X (R) are not empty.
(1) We say that ϕ is R-regular, or that ϕ induces a morphism X(R) → X (R) (that we will again write ϕ), if the rational map ϕ is regular at every R-rational point of X. Equivalently, the real locus X(R) of X is contained in the domain of definition of ϕ.
(2) We say that ϕ is R-biregular, or that ϕ is an isomorphism X(R) −→ X (R), if it is birational and ϕ and its inverse are R-regular.
(3) A birational diffeomorphism is an R-biregular rational map ϕ : X X between smooth real algebraic varieties (or equivalently an isomorphism X(R) −→ X (R), where X and X are smooth).
We can then consider the category most often used in real algebraic geometry (for instance in [1,13,2]) whose objects are the non-empty real loci X(R) of real algebraic varieties and whose morphisms correspond to R-regular rational maps X(R) → X (R). Note that the class of morphisms considered is in general much larger than the class of usual regular maps. For instance, if X is a projective real algebraic surface, the group Aut(X) of biregular automorphisms of X is often quite small: its neutral component is an algebraic group and has thus finite dimension. In contrast, the group of birational diffeomorphisms Aut(X(R)) can be very large. If X is smooth and rational, then Aut(X(R)) acts infinitely transitively on X(R) [13,Theorem 1.4]. A similar behaviour can also happen if X is not rational but only geometrically rational [2, Theorem 2].

1.B. Pairs and (logarithmic) Kodaira dimension
Recall that a Smooth Normal Crossing (SNC) divisor B on a smooth surface S defined over k is a curve B on S whose base extension B k to the algebraic closure k of k has smooth irreducible components and ordinary double points only as singularities. Equivalently, for every closed point p ∈ B k ⊆ S k , the local equations of the irreducible components of B k passing through p form a part of a regular sequence in the maximal ideal m S k ,p of the local ring O S k ,p of S k at p. The (logarithmic) Kodaira dimension κ(S) of S is then defined as the Iitaka dimension κ(V , ω V (log B)) [14], where is independent of the choice of a smooth SNC completion (V , B) of S [15], and it coincides with the usual notion of Kodaira dimension in the case where S is already complete. Furthermore, it is invariant under arbitrary extensions of the base field k, as a consequence of the flat base change theorem [11, Proposition III.9.3]. In particular a smooth real surface S and its complexification S C = S × Spec(R) Spec(C) have the same Kodaira dimension.

The real Kodaira dimension of open real surfaces 2.A. A variant of logarithmic Kodaira dimension
For a smooth real surface S, the Kodaira dimension κ(S) is in general not a birational invariant, unless S is complete: for instance, the affine plane A 2 R and the product of the punctured affine line A 1 R \ {0} with itself are birational to each other but have Kodaira dimensions −∞ and 0 respectively. We now introduce a variant of Kodaira dimension more adapted to the study of equivalence classes of open real surfaces up to birational diffeomorphisms.
If furthermore B R has no imaginary loop, then we define the real Kodaira dimension of S = V \ B to be By definition, given a smooth SNC pair (V , B) defined over R, the curve B R contains imaginary loops if and only if it has some pairs of non-real singular points q and q. The following lemma provides a simple procedure to eliminate imaginary loops.

Lemma 2.5. Let (V , B) be a smooth SNC pair defined over R and let
be its exceptional locus. Then the following hold: is a smooth SNC pair defined over R for whichB R has no imaginary loops and such that τ induces an isomorphismV \B → V \ B.
Proof. (1): As B is SNC, the morphism τ only blows-up ordinary double points of B, soB is again SNC. Every irreducible curve onV C contracted by τ is not defined over R and does not intersect its conjugate, so does not contain any real point. This implies thatB R is the strict transform of B R . Every singular Crational point of B R which was not real has been blown-up, and every singular C-rational point ofB R is an R-rational point. Hence,B R has no imaginary loop. The fact that τ induces an isomorphismV \B → V \B follows from the fact thatB = τ * (B) red and that all points blown-up by τ and all exceptional divisors of E are contained in B andB respectively.
(2): Since the points blown-up by τ are ordinary double points of B R hence of B, we haveB = τ * B − E whereasB R = τ * B R − 2E because E does not contain any real point. Denoting by KV and K V the canonical divisors onV and V respectively, we have the ramification formula KV = τ * K V + E for τ. This yields the two equalities The first equality gives κ(V , The following example shows that the inequality of Lemma 2.5 (2) can be strict.
is the sum of the exceptional divisors over q and q respectively, and whereL,C are the strict transforms of L and C. We The aim of this section is to show that the definition of κ(S(R)) (or κ R (S)) only depends on the birationnal diffeomorphism class of S(R), or equivalently of the real surface S, up to birational diffeomorphism.
The following notion is natural to compare two possible pairs, up to birational diffeomorphism.
Example 2.8. Let (V , B) and (V , B ) be two smooth SNC pairs defined over R, and let τ : V → V be a birational morphism, defined over R. In each of the following cases, τ yields a birational map of pairs  Proof. By definition, ϕ : V V is a birational map defined over R, inducing an isomorphism between and is thus of the type of Example 2.8 (1). Otherwise, we can take a minimal resolution of the indeterminacies of ϕ given by where W is a smooth projective real surface and τ and τ are birational morphisms defined over R.
Recall that since ϕ and ϕ −1 are defined over R, the union of their base-points, including infinitely near ones, is defined over R, hence consists of either real points or pair of conjugate non-real points. The minimality assumption implies in particular that τ and τ are the blow-ups of the base-points of ϕ and ϕ −1 respectively. This gives back the classical decomposition of τ and τ into simple blow-ups (one real or a pair of conjugate non-real points) and thus the real Zariski strong factorisation of ϕ, as explained for instance in [22, Chapter II, Proposition 6.4].
We proceed by induction on the number of such points, the case where there is no base-point being 2.8 (1).
If q ∈ V (R) is a base-point of ϕ, then q belongs to B(R), since ϕ induces an isomorphism between (V \ B)(R) and (V \ B )(R). We can write τ as τ = τ q •τ, where τ q :V → V is the blow-up of q and τ : W →V is a birational morphism defined over R. WritingB = (τ q ) −1 (B) red , the birational map τ q yields a birational maps of pairs (V ,B) (V , B) of type 2.8 (2). Moreover, if B R has no imaginary loop, the same holds forB R = ((τ q ) −1 (B R )) red . As ϕ • τ q : (V ,B) (V , B ) is again a birational maps of pairs, whose minimal resolution has less base-points, we conclude by induction.
The same argument works with a point q ∈ V (R) which is a base-point of ϕ −1 . We can thus assume that no point of V (R) or V (R) is a base-point of ϕ or ϕ −1 .
If τ is not an isomorphism, there is a pair of conjugate non-real points q, q ∈ V (C), both base-points of ϕ, blown-up by τ. As before, we write τ as τ = τ q •τ, where τ q :V → V is the blow-up of q and q, which is thus defined over R. Thenτ : W →V is a birational morphism defined over R. The strict transformB of B onV is then again an SNC-divisor, defined over R, and τ q induces a birational map of pairs (V ,B) (V , B) of type 2.8 (3). Moreover, if B R has no imaginary loop, the same holds forB R , which is the strict transform of B R . As before, the result follows by induction. The same works when τ is not a regular morphism.
In case (1), ϕ is an isomorphism V −→ V that sends B(R) isomorphically onto B (R). It thus maps the Zariski closure of B(R) isomorphically onto that of B (R). This implies that ϕ(B R ) = B R (see Remark 2.2). This achieves the proof in this case.
We then do the two cases (2)-(3), and denote, in both cases, by E ⊆ V the divisor contracted by ϕ.
In case (2), ϕ is the blow-up of a point q ∈ B (R), B = ϕ −1 (B ) red and E = ϕ −1 (q). In case (3), ϕ is the blow-up of a pair of conjugate non-real points q, q ∈ V (C), B is the strict transform of B and E = ϕ −1 (q) + ϕ −1 (q). As B R is an SNC-divisor with no imaginary loop, the points q, q cannot be singular points of B R .
We find respectively where ∈ {−1, 0, 1} and δ = m + n ≥ m − |n| ≥ 0 as m ≥ |n| by hypothesis. As a consequence, the natural inclusion is a bijection. Indeed, for each integer r ≥ 0, an effective divisor D equivalent to ϕ * (mK V + nB R ) + rE is equal tõ D + rE for some effective divisorD equivalent to ϕ * (mK V + nB R ). This is clear for r = 0, and for r > 0 we just compute E · D = rE 2 ≤ −r < 0 and obtain that D − E is effective, which yields the result by induction.
The case where m = n shows that κ R (V , B ) = κ R (V , B).
As a consequence of Lemma 2.11, we obtain:

is thus a well-defined invariant of S(R).
The following result summarises immediate consequences of the definition and Lemma 2.11.

Proposition 2.13. The real Kodaira dimension κ R (S) of a smooth real surface S enjoys the following properties:
(1) κ R (S) = κ R (S ) for every smooth surface S defined over R birationally diffeomorphic to S. Example 2.14. The inequality κ R (S) ≤ κ(S) is strict in general: for instance, let B ⊆ P 2 R be a general arrangement consisting of 0 ≤ r ≤ 2 real lines and a collection of p ≥ 0 pairs of non-real complex conjugate lines. Then for S = P 2 R \ B, we have κ R (S) = −∞ independently of r and p while The equality κ R (S) = −∞ follows from the fact that S is birationally diffeomorphic to the complement S of r ≤ 2 lines in P 2 R , which satisfies κ R (S ) = κ(S) = −∞. On the other hand, since B is an SNC divisor, As a consequence of Proposition 2.13 (1), we obtain: Proof. Follows from Proposition 2.13 (1), and the fact that

2.B.a. An algebraic model of real Kodaira dimension 0: the exceptional fake plane Y (3, 3, 3)
Let us recall from [5, §5.1.1] the following construction of a fake plane S of Kodaira dimension 0 whose complexification S C is Q-acyclic 1 , with H 1 (S C ; Z) Z 9 . Let D be the union of four general real lines i P 1 R , i = 0, 1, 2, 3 in P 2 R and let τ : V → P 2 R be the real projective surface obtained by first blowingup the real points p ij = i ∩ j with exceptional divisors E ij , i, j = 1, 2, 3, i j and then blowing-up the real points 1 ∩ E 12 , 2 ∩ E 23 and 3 ∩ E 13 with respective exceptional divisors E 1 , E 2 and E 3 . We let 13 . The dual graphs of D, of its total transform τ −1 (D) in V and of B are depicted in Figure 1.

2.B.b. An algebraic model of real Kodaira dimension 2: the real Ramanujam surface
The real Ramanujam surface S is a real model of the complex Ramanujam surface [21] which is constructed as follows: let D ⊆ P 2 R = Proj(R[x, y, z]) be the union of the cuspidal cubic C = {x 2 z + y 3 = 0} with its osculating conic Q at an R-rational point q ∈ C(R) distinct from the singular point [0 : 0 : 1] of C and its flex [0 : 0 : 1]. Up to change of coordinates, one can for instance choose q = [1 : 1 : −1], which implies that the equation of Q is So Q is a smooth R-rational conic intersecting C at q with multiplicity 5 and transversally at a second R-rational point p. We let β : F 1 → P 2 R be the blow-up of p with exceptional divisor E P 1 R and we let S be the complement in F 1 of the proper transformD of D. The total transform B ofD in a minimal log-resolution τ : (V , B) → (F 1 ,D) of the pair (F 1 ,D) is a tree of R-rational curves depicted in Figure 2. The surface S is a fake real plane of Kodaira dimension 2 with contractible complexification S C : the contractibility of S C was first established by Ramanujam [21], the fact that κ(S) = 2 follows for instance from the classification of contractible complex surfaces of Kodaira dimension ≤ 1 established in [8] (see also [16]), and the fact that S(R) R 2 was proven in [5,Example 3.8].
Since the smooth SNC completion (V , B) of S has the property that B R = B is a tree, the equality κ R (S) = κ(S) = 2 holds by virtue of Proposition 2.13 (1), and so, S is not birationally diffeomorphic to A 2 R .
Remark 2.16. The same argument as above also applies to the three examples of fake real planes S of loggeneral type with contractible complexification S C constructed in [6, §5.1] from arrangements of real lines and irreducible singular R-rational quartics in P 2 R : all these surfaces have the property to admit a smooth SNC completion (V , B) defined over R for which B R = B is a tree, so that their real Kodaira dimension κ R coincides with their usual Kodaira dimension. All of them are therefore non birationally diffeomorphic to A 2 R .

Families of algebraic models of Kodaira dimension 1
Fake real planes S of Kodaira dimension 1 whose complexifications S C are Z-acyclic manifolds, that is topological manifolds with trivial reduced homology groupsH i (S C ; Z), have been classified up to isomorphism in [5] (see also [8] and [3] for the complex case). One obtains the following: Proof. By virtue of [5,Theorem 3.2], every such surface admits a completion into a smooth projective surface V defined over R obtained from P 2 R by blowing-up specific sequences of real points, and whose boundary B = V \ S consists of a tree of P 1 R 's. In particular, such a smooth pair (V , B) satisfies B R = B, and we deduce from Proposition 2.13 (1) that κ R (S) = κ(S) = 1. The result then follows from Corollary 2.15.
In the rest of this section, we build on a blow-up construction of certain fake real planes of Kodaira dimension 1 with contractible complexifications [5,Example 3.5] to derive the existence of infinitely many pairwise non birationally diffeomorphic such surfaces. The main ingredient is the uniqueness of the logcanonical fibration, given by Lemma 2.11. Let 1 < a < b be a pair of coprime integers and consider the rational pencil  S(a, b) = X(a, b) \ (C a,b ∪ L z ) where we identified a curve in P 2 R with its proper transform in X(a, b).
The dual graph of the total transform of C a,b ∪L x ∪L y ∪L z in the minimal resolution α : Figure 3. The boundary B(a, b) R is a P 1 -fibration having the last exceptional divisors C 0 and C 1 of α over the points q 0 and q ∞ as disjoint sections.  V (a, b), B(a, b)) and (V , B ) = (V (a , b ), B(a , b )) be the smooth pairs obtained by taking the minimal resolutions of the pencils Ψ • β and Ψ • β respectively. The structures of B and B imply that ϕ extends to a birational diffeomorphism of pairs Φ : (V , B) (V , B ). Indeed, otherwise either Φ or its inverse, say Φ, would contract an irreducible component of B onto a real point of B . But B does not contain any irreducible curve whose proper transform by a birational morphism W → V defined over R whose center is supported on B is (−1)-curve which can be contracted while keeping the property that the total transform of B is an SNC divisor.
By virtue of [17, Lemma 4.5.3 p. 237], the positive part of the Zariski decomposition of where denotes a general real fiber of the P 1 -fibration f : V → P 1 R . Since 1 < a < b, it follows that f coincides with the log-canonical fibration f |m(K V +B)| : V → P 1 R for every integer m ≥ 1. The same holds for the log-canonical fibration f = f |m(K V +B )| : V → P 1 R on V . Since B = B R and similarly for B , it follows from Lemma 2.11 that for every m ≥ 1, Φ induces an isomorphism between H 0 (V , m(K V + B)) and H 0 (V , m(K V + B )). Consequently, there exists an automorphism γ of P 1 R defined over R such that The curves E, L x and L y have multiplicities 1, a and b as irreducible components of the scheme theoretic fibers of f : V → P 1 R over the points [1 : 1], [1 : 0] and [0 : 1] respectively. Similarly, the curves E , L x and L y have multiplicities 1, a and b as irreducible components of the scheme theoretic fibers of f : V → P 1 R over these points. Since 1 < a < b and 1 < a < b , and Φ(S(R)) ⊆ S (R), it follows that Φ * (E) = E , Φ * (L x ) = L x and Φ * (L y ) = L y . Thus γ = id, from which we conclude in turn that a = a and b = b .

Families of algebraic models of Kodaira dimensions 2
Here we construct infinite families of pairwise non birationally diffeomorphic fake real planes of real Kodaira dimension 2 with contractible complexifications. We also give examples of Z-acyclic complex surfaces of log-general type with two real forms: one of them has negative logarithmic Kodaira real dimension and is in fact birationally diffeomorphic to A 2 R whereas the other one has real Kodaira dimension 2, hence is not birationally diffeomorphic to A 2 R .

4.A. A criterion for isomorphism
is a morphism, which restricts to an isomorphism between S and its image.
are irreducible components of ∆. So ϕ = ψ • h restricts to an isomorphism between S and its image.
By combining Lemma 2.11 and Lemma 4.1, we obtain the following: . We thus get a commutative diagram where the right-hand side isomorphism is induced by θ. This shows that f is an isomorphism.

4.B. Miyanishi-Sugie surfaces: a countable family of pairwise non birationally diffeometric fake real planes of log-general type
We consider the following real counterpart of a family of smooth complex topologically contractible surfaces of log-general type constructed by Miyanishi-Sugie [18]. For each integer s ≥ 1, we will construct a surface S s , defined over R, which corresponds to the surface X (1) s+1,1 of [18], i.e. to the construction of [18] with n = s + 1, m = 1 and r = s. We recall the construction here for self-containedness.
We define C s and L s to be the irreducible curves in P 2 R = Proj(R[x, y, z]) given by the zero loci of the polynomials respectively. Note that the polynomials above correspond to −y n−1 z +x 2 (x n−2 − n−2 i=2 (i −1) n i x n−i−2 y i ) and (n 2 − 2n)x + (n − 1)y − z, with n = s + 1, and thus the equations of C s and L s are the same as those given in Let τ : V 0,s → P 2 R be the birational morphism defined over R obtained by first blowing-up q and then blowing-up s + 1 times the intersection point of the proper transform of C s with that of the previous exceptional divisor produced. Let E 1 , . . . , E s+2 P 1 R be the corresponding successive exceptional divisors.
where we identified each curve with its proper transform in V 0,s and let S s = V 0,s \ B 0,s . The dual graph of B 0,s is given in the following figure, where the double arrow corresponds to a multiple intersection at a point (corresponding here to the point p).
The smooth surface S s is defined over R. After blowing-up the singular point of C s on V 0,s , the total transform of B 0,s is given by the following dual graph, where F is the exceptional curve contracted on the singular point of C s .
The total transform B s of B 0,s is thus a tree of R-rational curves whose dual graph is depicted on Figure 4 (which is the same as the graph of X (1) n,m given in [18, Theorem 1, Page 339], with m = 1 and n = s + 1). Proof. The fact that the complexification of S s is contractible and of log-general type is proven in [18]: the log-general type is the purpose of the construction and the contrability is given in [18,Theorem 2].
Since B s = B s,R by construction, we have κ R (S s ) = κ(S s ) = 2, and S n is a fake real plane by virtue of [5,Proposition 2.4]. By Proposition 4.2, every birational diffeomorphism between S s and S s is an isomorphism. But the description of the dual graphs of the boundaries B s in Figure 4 implies that every isomorphism between S s and S s extends to an isomorphism of pairs between (V s , B s ) and (V s , B s ) and that two such pairs are non isomorphic for different s and s .

4.C. Fake real planes of general type with nontrivial real forms
To complete this section, we reconsider a family of fake real planes of general type with two real forms intoduced in [5, §3.2.2]. We start with the projective duals Γ 1 and Γ 2 of real nodal cubic curves C 1 , C 2 ⊆ P 2 R , such that the two branches at the singular point of C 1 , C 2 are real, respectively non-real.
Note that C 1 , C 2 are not equivalent under Aut(P 2 R ) = PGL 2 (R), and that every real nodal cubic curve of P 2 R is projectively equivalent to either C 1 or C 2 . The latter can be checked by looking at the parametrisations P 1 R → C 1 , C 2 , given by polynomials of degree 3 having the same value at two points, which are either real or pairs of non-real complex conjugates. Explicitely, one can choose, for instance, the equations of C 1 and C 2 to be x 2 z − y 2 z + xy 2 = 0 and x 2 z + y 2 z − xy 2 = 0. 2 With these coordinates, we find that C 1 and C 2 are exchanged by the non-real complex projective transformation The curves Γ 1 and Γ 2 are thus rational quartics with three cusps: an ordinary real cusp p 0 corresponding to the common R-rational flex of C 1 and C 2 , and either a pair of non-real conjugate cusps q and q for Γ 1 or an additional pair or real ordinary cusps q 1 and q 2 for Γ 2 . So Γ 1 and Γ 2 are not isomorphic over R, but their respective complexifications are both projectively equivalent over C. In fact, after change of coordinates, the curves Γ 1 and Γ 2 can be given by the equations (x 2 + y 2 ) 2 + z(2x 3 + 2xy 2 − y 2 z) = 0 and x 2 y 2 + x 2 z 2 + y 2 z 2 + 2xyz(x + y − z) = 0, For i = 1, 2, the tangent line L i = T p 0 (Γ i ) to Γ i at p 0 (given respectively by y = 0 and x = y and satisfying θ(L 1 ) = L 2 ) intersects Γ i transversally in a unique other real point p i different from p 0 (being given by p 1 = [1 : 0 : −1/2] and p 2 = [1 : 1 : −1/4] = θ(p 1 )). Let (a, b) be a pair of positive integers such that 4b − a = ±1 and let τ i : V i → P 2 R be the real birational morphism obtained by first blowing-up p i with exceptional divisor E 1 P 1 R and then blowing-up a sequence of real points on the successive total transforms of E 1 in such a way that the following two conditions are satisfied: a) the inverse image of p i is a chain of curves isomorphic to P 1 R containing a unique (−1)-curve A and b) the coefficients of A in the total transform of Γ i and L i = T p 0 (Γ i ) are equal to a and b respectively. We denote the corresponding exceptional divisors by E 1 , . . . , E r−1 , E r = A and we let B i = Γ i ∪ T p 0 (Γ i ) ∪ r−1 j=1 E j , i = 1, 2. b) κ R (S 2 ) = 2, in particular S 2 is not birationally diffeomorphic to A 2 R . c) κ R (S 1 ) = −∞, and S 1 is actually birationally diffeomorphic to A 2 R . Proof. The complex surfaces S 1,C and S 2,C are isomorphic, by lifting the projective transformation θ. The fact that S 1 and S 2 are Z-acyclic fake real planes of log-general type is established in [5, Proposition 3.10]. Since by construction B 2 is a tree of R-rational curves, we have κ R (S 2 ) = κ(S 2 ) = 2 by Proposition 2.13 (2), and so S 2 is not birationally diffeomorphic to A 2 R . The fact that S 1 is birationally diffeomorphic to A 2 R is proven in [6,Proposition 21].