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Given a principally polarized abelian surface $A$ with CM of signature $(1,1)$ by an imaginary quadratic number field $K$, I am interested in studying the Néron-Severi group $\text{NS}(A\times A)$. Can one find explicit classes in this group? In other words, can we give explicit examples of divisors in $A\times A$?

I know the answer will depend on e.g. whether $A$ is simple or not, whether it has quaternionic multiplication or not, but I am interested in all of these cases. However, even a partial answer would be greatly appreciated.

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    $\begingroup$ The Picard scheme of a product $(X,x) \times (Y,y) = (X \times Y, (x,y))$ of smooth proper pointed varieties $(X,x)$, $(Y,y)$ satisfies $\mathbf{Pic}_{X \times Y} \cong \mathbf{Pic}_X \times \mathbf{Pic}_Y \times \operatorname{Hom}(\mathbf{Alb}_X,\mathbf{Pic}_Y^0)$. This gives a recipe for producing classes in $\operatorname{NS}(A \times A)$: most of the interesting ones come from endomorphisms of $A$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Apr 30 at 17:32
  • $\begingroup$ Thank you! Would you be so kind as to give me a reference for such isomorphism? Maybe it's not hard to prove it but I'd still like to double check. $\endgroup$
    – Fra
    Commented May 2 at 18:32
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    $\begingroup$ Hmm, at the time I wrote my dissertation, I did not know a reference for this, so I wrote it up myself (§4.4). Over the complex numbers, there is an easier and pretty well-known argument for $\operatorname{NS}$ using the Lefschetz (1,1)-theorem. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 2 at 20:43
  • $\begingroup$ I see. I guess I wasn't precise enough in my question, but I am indeed interested in the complex part of the story, so if you could give me a reference for the well-known argument I would be extremely grateful. Thank you. $\endgroup$
    – Fra
    Commented May 7 at 11:27

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Unfortunately I do not really know a reference, but the proof is simple enough, so let me reproduce it here. For this answer, I denote by $\operatorname{NS}(X)$ the image of $c_1 \colon \operatorname{Pic}(X) \to H^2(X,\mathbf Q)$. This should maybe be denoted $\operatorname{NS}(X)/\text{torsion}$, as the classical definition uses $H^2(X,\mathbf Z)$.

(I think the lemma below should still hold for the integral version, but I'm currently having some difficulty combining the universal coefficients spectral sequence for cohomology with the Künneth spectral sequence for homology, and I don't know a direct Künneth spectral sequence for cohomology. Rationally, you don't need spectral sequences at all.)

Lemma. Let $X$ and $Y$ be smooth projective complex varieties. Then $$\operatorname{NS}(X \times Y) \cong \operatorname{NS}(X) \times \operatorname{NS}(Y) \times \operatorname{Hom}(\mathbf{Alb}_X,\mathbf{Pic}^0_Y).$$

Proof. By the Lefschetz (1,1)-theorem, we have $\operatorname{NS}(V) = H^2(V,\mathbf Z) \cap H^{1,1}(V)$ for any smooth projective variety $V$. The Künneth theorem gives an isomorphism of $\mathbf Z$-Hodge structures (where everything should be understood modulo torsion) $$H^2(X \times Y) = \big(H^2(X) \otimes H^0(Y)\big) \oplus \big(H^1(X) \otimes H^1(Y)\big) \oplus \big(H^0(X) \otimes H^2(Y)\big).$$ By Poincaré duality, the middle term equals $\operatorname{Hom}(H^{2\dim X -1}(X),H^1(Y))$. Taking integral (1,1)-classes everywhere gives the result: the middle term then picks out the homomorphisms of Hodge structures, which is $\operatorname{Hom}(\mathbf{Alb}_X,\mathbf{Pic}_Y^0)$ by the theory of abelian varieties. $\square$

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