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.

  • 1
    $\begingroup$ 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 $E$. $\endgroup$
    – naf
    May 30, 2017 at 8:56
  • $\begingroup$ 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. $\endgroup$
    – user347489
    May 30, 2017 at 9:27


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