### Questions About the Néron–Severi Group

#### Posted by John Baez

A friend of mine with good intuitions sometimes says things without proof, and sometimes I want to know why — or even whether — these things are true.

Here are some examples from algebraic geometry.

First some background, just to see if I understand the basics, and maybe make this post interesting to other people who are just starting to learn algebraic geometry.

If you have a smooth complex projective variety $X$, the set of isomorphism classes of holomorphic line bundles on $X$ is called the **Picard group** of $X$. This has a topology — to indicated this it’s sometimes called the **Picard scheme** — where each connected component consists of different holomorphic line bundles that are isomorphic as *topological* line bundles. So, the identity component consists of different ways to give the trivial topological line bundle a holomorphic structure.

If you mod out the Picard scheme by its identity component you get the Néron–Severi group of $X$, or $NS(X)$ for short. Elements of this group —- in other words, the group of connected components of the Picard scheme — are isomorphism classes of topological line bundles over $X$ that actually admit a holomorphic structure. (Not all of them do, in general.) And this group is isomorphic to $\mathbb{Z}^n$ for some number $n$ called the **Picard number** of $X$.

(As a Star Trek fan all this terminology never ceases to amuse me. I just wish that Néron had been called something like Dukat.)

Now, my friend and I have been looking at the case where $X$ is a 2-dimensional abelian variety, also called an **abelian surface**. Then he claims that $NS(X)$ can be seen as a discrete subgroup of the Doubeault cohomology group $H^{1,1}(X)$. How generally is this true? I just need it for an abelian surface right now.

For $X$ an abelian surface, $H^{1,1}(X) \cong \mathbb{C}^4$. And he claims $NS(X) \cong \mathbb{Z}^4$. Is that right?

If so, we can form the real vector subspace $V \subset H^{1,1}(X)$ spanned by $NS(X)$, and $NS(X)$ will be a lattice in this.

Next, he claims something that I’m interpreting like this: the intersection pairing on $H^{1,1}(X)$ gives $V$ a symmetric bilinear form $B$ of signature $+++-$. Is that right?

If so, we can think of $NS(X)$ as a lattice in Minkowski spacetime.

Next, he claims that the ample line bundles — that is, those line bundles $L$ such that sufficiently high tensor powers $L^{\otimes p}$ have enough holomorphic sections to separate points — give precisely the elements of the Néron–Severi group that lie in the **future cone** of the Minkowski spacetime $V$. Here I’m borrowing some more terminology from physics: I mean that they lie in one of the two components of the set

$\{L \in NS(X) : B(L,L) > 0 \}$

Is that right?

I also have some independent evidence to support another guess of my own: the ‘principal polarizations’ of $X$ correspond to elements $L$ of the Néron–Severi group that not only lie in the future cone but have

$B(L,L) = 2$

Is that right?

## Questions About the NéronSeveri Group

Those claims are correct. I know that you are looking for arguments that make sense for beginners, not just keywords. Nonetheless, there are some keywords. The claim that the Neron-Severi group is a discrete subgroup of $H^{1,1}$ follows from the long exact sequence of cohomology associated to the short exact sequence called the “exponential sequence”. The claim about the signature of $B$ on $V$ is called the “Hodge index theorem”. The claim about the positive cone is part of the Nakai-Moishezon Criterion (probably it can also be proved in other ways).