The n-Category Café

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?