# Extend rational map $f: V \dashrightarrow X$ to Abelian variety

I try to understand the proof of an extension theorem proved in Moonen's and van der Geer's draft Abelian varieties(in online accessible notes Theorem 1.18 on page 14):

(1.18) Theorem.

Let $$X$$ be an abelian variety over a field $$k$$. If $$V$$ is a smooth $$k$$-variety then any rational map $$f: V \dashrightarrow X$$ extends to a morphism $$V \to X$$.

Proof:

We may assume that $$k = \bar{k}$$, for if a morphism $$V_{\overline{k}}= V \times_k \overline{k} \to X_{\overline{k}}$$ is defined over $$k$$ on some dense open subset of $$V_{\overline{k}}$$, then it is defined over $$k$$. Let $$U \subset V$$ be the maximal open subset on which $$f$$ is defined. Our goal is to show that $$U = V$$. [...]

Questions: the assumption $$k = \bar{k}$$ I not understand. if we succeed to prove that $$V_{\overline{k}} \to X_{\overline{k}}$$ extends to a morphism, why $$f$$ is also extendable? Afterwards, if we proof $$V_{\overline{k}} \to X_{\overline{k}}$$ extends to a morphism, why if it is defined over $$k$$ on some dense open subset of $$V_{\overline{k}}$$ (=preimage of a dense open $$U \subset V$$ under projection $$V_{\overline{k}} \to V$$), then it is defined over $$k$$?

• Suppose we know what we want over $\overline{k}$. Then we can descend this to a finite Galois extension $L/k$, i.e., $U_L\to X_L$ extends to a morphism $V_L\to X_L$. Now, we consider the action of $G= Gal(L/k)$ on the morphism $V_L\to X_L$. Note that the restriction morphism $U_L\to X_L$ is fixed. Since $U_L\subset V_L$ is dense open, this implies that $V_L\to X_L$ is in fact fixed by $G$, so it descends to a morphism $V\to X$. For another proof of this extension property: see Lemma 3.5 of arxiv.org/abs/1807.03665 and use the fact that abelian varieties have no rational curves. – Ariyan Javanpeykar Oct 10 at 17:39
• @AriyanJavanpeykar: This is a nice idea, thank you a lot. a little remark: lastly I asked a similar question in a slightly other context about factorization problem reduced to algebraically closed ground field: math.stackexchange.com/questions/3385478/…. although I got a good answer I'm keenly curious if this reduction to $k=\bar{k}$ can also be solved with a descent argument similar to that you used above? My first suspicion was with Stein factorization – katalaveino Oct 11 at 12:51
• I can assume that fibers are geometrically connected and could "somehow go down" using properness (especially closedness of the map, but from there I could not proceed ahead. – katalaveino Oct 11 at 12:51