mjhjmtu

Let $U$ be an affine open subscheme of an abelian variety $A$ over $mathbbC$. Is $A-U$ an ample divisor?

If $dim A =1$ this is true.
If $dim A = 2$, the complement is a divisor $D_1+ldots + D_n$. If all of these are elliptic curves, then the complement is not affine (as it will contain the translate of one of these elliptic curves). So, wlog $D_1$ is not an elliptic curve. But then $D_1$ is ample (by Nakai-Moishezon). And this implies that $D_1 +ldots +D_n$ is also ample.

I couldn't figure it out for abelian threefolds.

Welcome new contributor. Yes, that is true. Let $k$ be any field, let $A$ be an Abelian variety over $k$, and let $Usubset A$ be a dense open affine. Denote by $Dsubset A$ the complementary divisor with its induced reduced structure. Denote the invertible sheaf of this divisor by $mathcalL:=mathcalO_A(D).$ Denote by $s$ the global section of $mathcalL$ whose zero locus equals $D.$

Ampleness can be proved after faithfully flat base change over $textSpec k$, thus assume that $k$ is algebraically closed. By Lemma II.5.14 of Hartshorne, for every $gin mathcalO_A(U)$, there exists an integer $n_0$ and a section $widetildegin mathcalL^otimes n_0(A)$ such that $widetildeg|_U$ equals $gcdot s^n_0|_U$. Notice that for every integer $rgeq 0$, also $s^rwidetildeg|_U$ equals $gcdot s^n_0+r|_U$. Since the $k$-algebra $mathcalO_A(U)$ is finitely generated, there exists an integer $n_0>0$ such that the image of $(s|_U)^-ncdot mathcalL^otimes n(A) to mathcalO_A(U)$ generates $mathcalO_A(U)$ as a $k$-algebra for every $ngeq n_0$. In particular, the base locus $B_n$ of the complete linear system of $mathcalL^otimes n$ is disjoint from $U$, and the induced morphism to projective space, $$phi_n:Asetminus B_n to mathbbP^d_n_k,$$ restricts as a locally closed immersion on $U$.

The set $A(k)_texttor$ of torsion $k$-points $a$ of $A$ is dense. Thus, denoting by $$tau_a:Ato A$$ the morphism of translation by $a$, for every $k$-point $p$ of $A$, the set $ ain A(k)_texttor $ is dense in $A$. In particular, not all of these points can lie in the proper, Zariski closed subset $D$. Thus, there exists an integer $m>0$ and an $m$-torsion point $a$ such that $tau_a^*D$ does not contain $p$. In other words, the point $p$ is contained in the open affine subset $tau_a^-1(U)$.

Since $a$ is $m$-torsion, $tau_a^*mathcalL^otimes n$ is isomorphic to $mathcalL^otimes n$ for every positive integer $n$ that is divisible by $m$. For such $n$, also the base locus $B_n$ is disjoint from $tau_a^-1(U)$, and $phi_n$ restricts to a locally closed immersion on $tau_a^-1(U)$.

The set of translates $tau_a^-1(U)$ for $ain A(k)_texttor$ is an open affine covering of $A$. Since $A$ is quasi-compact, there exist finitely many torsion translates of $U$ that cover $A$. For $m>1$ equal to the least common multiple of the orders of those finitely many torsion points, for every positive integer $n$ that is divisible by $m$, the base locus $B_n$ is disjoint from each of these open translates in this open covering. Thus, the base locus $B_n$ is empty. Moreover, the $k$-morphism $phi_n$ restricts as a locally closed immersion on each of these open translates.

If $phi_n$ had a positive dimensional fiber, that fiber would intersect one of these open translates in a positive dimensional subvariety, and that would contradict that $phi_n$ is an immersion on that open translate. Thus, every fiber of $phi_n$ is finite. Since $phi_n$ is a morphism between proper $k$-schemes that has finite fibers, the morphism $phi_n$ is a finite morphism. Since the pullback of an ample invertible sheaf by a finite morphism is ample, the invertible sheaf $mathcalL^otimes n cong phi_n^*mathcalO(1)$ is ample on $A$. Finally, since $mathcalL^otimes n$ is ample, also $mathcalL$ is ample.