# Geometry of super-rigidid elliptic curve

Consider a Calabi-Yau threefold $X$ with an elliptic curve $E$ living inside it. Furthermore, assume $E$ is super-rigid, i.e., for any stable map from a nonsingular curve $\mu:C\to E$ we have $H^0(C,\mu^*(N_{E|X}))=0$. In particular it's rigid, i.e., $H^0(E,N_{E|X})=0$. I'm trying to understand the geometric implications of super-rigidity.

Then (see 1.1 here) rigidity implies that there is a short exact sequence: $$0\to L\to N\to L^\vee\to 0,$$ where $L$ is a non-trivial degree 0 line bundle (I think I can show this).

On the other hand, in the paper linked above it's claimed that super-rigidity is equivalent to $L$ being a nontorsion element of $Pic^0 E$, which I don't know how to prove. Any help with it is greatly appreciated.

• If $L$ is non torsion, then its pullback to any curve mapping dominantly to $E$ is also non torsion (the pullback map has finite kernel). On the other hand if $L$ is torsion then it becomes trivial on some curve (which can be taken to be elliptic) mapping dominantly to $E$.
– naf
May 30, 2017 at 8:56
• Dear @ulrich, thanks for your reply! Your comment solves my question and made me realize its elementary nature. If you feel like posting it as an answer I'll gladly accept it. May 30, 2017 at 9:27