# Ampleness of the normal bundle to the Albanese image

Let $$X$$ be a projective surface of general type over $$\mathbb{C}$$, and assume that $$\Omega_X$$ is globally generated. Then the Albanese map $$a \colon X \to \operatorname{Alb}(X)$$ is a local embedding namely, a finite, étale cover with image $$a(X) \subset \operatorname{Alb}(X)$$. It is also well known that $$a(X)$$ generate $$\operatorname{Alb}(X)$$.

Question. Is the normal bundle of $$a(X)$$ an ample vector bundle?

By a result of Hartshorne (see 1), in order to have a positive answer, it suffices to check that every curve in $$a(X)$$ generates $$\operatorname{Alb}(X)$$, see also MO187149 On a proposition in Hartshorne's paper "Ample vector bundles on curves". In particular, this shows that the answer to the previous question is "yes" when $$\operatorname{Alb}(X)$$ is simple. But what can happen in general?

References.

1. Hartshorne, R., Ample vector bundles on curves, Nagoya Math. J. 43, 73-89 (1971). ZBL0218.14018.

• I think that fails when $X$ is a product of two hyperbolic curves. Sep 22 at 10:09
• Thanks for the comment. Could you please give more details? Sep 22 at 22:36
• If $X$ equals a product $C\times D$, then the Albanese morphism for $X$ is the product of the Albanese morphisms of the factors, $\text{alb}_C\times\text{alb}_D:C\times D \to \text{Alb}(C)\times \text{Alb}(D)$. Thus, the pullback of the tangent bundle by the Albanese morphism is the direct sum, $\text{pr}_C^*\text{alb}_C^*T_{\text{Alb}(C)}\oplus \text{pr}_C^*\text{alb}_D^*T_{\text{Alb}(D)}$. Therefore the normal bundle is also a direct sum $\text{pr}_C^* N_{\text{Alb}(C)/C}\oplus \text{pr}_D^*N_{\text{Alb}(D)/D}$. These are not ample on curves in fibers of $\text{pr}_C,\text{pr}_D$. Sep 23 at 0:26
• "The Albanese map is a finite, étale cover onto its image": not necessarily. You only know that it is unramified, the image could very well be singular.
– abx
Sep 23 at 4:54
• @abx: ok, thanks. In fact, I only know that the differential of the Albanese map is everywhere of maximal rank. Do you know any example of surface such that $\Omega_X$ is globally generated and $a(X)$ is singular? Sep 23 at 5:22