Real vs complex surfaces - MathOverflow

Real vs complex surfaces

2012-05-17T09:42:55Z

Hi! My background is in (complex) algebraic geometry. For the first time I'm considering varieties defined over $\mathbb{R}$ and I have some very basic questions about these. In particular I'm trying to understand MMP for real surfaces.

Let $X$ be a smooth projective surface over $\mathbb{R}$. We can consider its complexification $X_{\mathbb{C}}$, smooth complex projective surface.

Then Cartier divisors on $X$ correspond to Cartier divisors on $X_{\mathbb{C}}$ that are fixed by conjugation. Is intersection theory on $X$ just inherited from intersection theory on $X_{\mathbb{C}}$ via this identification? Is this the right way to look at intersection theory on $X$?

What is the relation between the canonical divisors $K_X$ and $K_{X_{\mathbb{C}}}$? Is it true that $K_{X_{\mathbb{C}}}=\overline{K_{X_{\mathbb{C}}}}$, the divisor defined by conjugated equations and it thus corresponds to the divisor $K_X$ on $X$?

Also, how to define blow-ups of points on $X$? The definition in Hartshorne cap II.7 works but I imagine one can find an easier definition.

Whatever reference about these and related basic facts would be appraciated.

Answer by Jérémy Blanc for Real vs complex surfaces

2012-05-17T11:14:06Z

Hello. For a good reference, I suggest the following text of J. Kollar http://arxiv.org/abs/alg-geom/9712003

For me, the right way to look is to see real divisors as complex divisor invariant by the anti-holomorphic involution. The intersection form is just the same as the one which comes from the complex, as you mention.

For the blow-up of smooth points, you have either a blow-up of a real point or a blow-up of two imaginary conjugate points.

Note that taking the quadric of equation $w^2=x^2+y^2+z^2$ in the projective space, this admits a real Mori fibration onto a point. The real Picard group has dimension 1, whereas the complex one has dimension 2 (over the complex, it is equal to $\mathbb{P}^1\times\mathbb{P}^1$). In particular, there are more Mori fibrations over $\mathbb{R}$ than over $\mathbb{C}$. Another example is given by del Pezzo surfaces of degree $1$ or $2$ with real Picard group of dimension $1$.

Answer by Daniel Loughran for Real vs complex surfaces

2012-05-17T12:46:23Z

I think a lot of your questions apply to arbitrary Galois field extensions, not just the field extension $\mathbb{R} \subset \mathbb{C}$. This might help clarify the problems you are having by putting them into a general context.

Let $E \subset F$ be a finite Galois extension of fields and let $X$ be a smooth variety over $E$. Then, the canonical divisor $K_X$ is always defined over the base field $E$. Moreover, it is well-behaved with respect to smooth base change, so that yes indeed the base change of $K_X$ to $X_F$ is the canonical divisor $K_{X_F}$ of $X_F$.

With regards to divisors, we have natural homomorphisms $\mathrm{Div}(X) \to \mathrm{Div}(X_F)$ and $\mathrm{Pic}(X) \to \mathrm{Pic}(X_F)$ given by base change. Moreover, in the case of surfaces this morphism respects the intersection pairing as your desire, as the intersection number of two divisors is defined geometrically. Also in the first case, it is true that the image consists of those divisors which are invariant under the action of $\mathrm{Gal}(F/E)$.

However, in general this does not hold for Picard groups. By which I mean, there may exist divisor classes which are Galois invariant, but nonetheless there does not exists a divisor in that class defined over $E$.

As an example, consider a conic $X$ defined over $E$ which has no rational points, such that $X_F$ has rational points. Then the natural map $\mathrm{Pic}(X) \to \mathrm{Pic}(X_F)$ corresponds to the inclusion $2\mathbb{Z} \to \mathbb{Z}$, as lowest degree of any divisor is $2$ (given by the anticanonical divisor). However, the action of $\mathrm{Gal}(F/E)$ on $\mathrm{Pic}(X_F)$ is trivial as it preserves the degree of a divisor.

As for blow-ups, one can define blow-ups of closed points in a similar manner to how one defines the blow-up of a rational point. Closed points correspond to Galois invariant collections of rational points on $X_F$, therefore the map given by blowing up each of these rational points is Galois invariant and so descends to a morphism defined over $E$.

A lot of these ideas can be found in Manin's book on cubic forms.