I have the following basic question. Everything is over $\mathbb{C}$.

Let $X$ be a hyperkähler (irreducible holomorphic symplectic) variety and we consider a small contraction $f\colon X \rightarrow Y$. By this I mean that f is birational and surjective (e.g. induced by a big and semiample line bundle) and the exceptional locus $B$, i.e. the locus where $f$ is not an isomorphism, has codimension $\ge$ 2. Is $B$ uniruled?

If B is an irreducible divisor, this is true and follows from Section 1 of Huybrechts - Compact hyper-Kähler manifolds.

However, I could not find a statement in the case that $B$ is not a divisor. The only statement I was able to find is that each irreducible component of $B$ is algebraically coisotropic (Theorem A in https://arxiv.org/abs/math/0111089).

Is more known about the structure of $B$?


Let $f:X\to Y$ be a birational contraction where $X$ is hyperkähler, then $K_X\sim 0$ and $K_Y=f_*K_X\sim 0$, and hence $K_X=f^*K_Y$. In particular, this means that $Y$ has canonical singularities. Hence by a result of Hacon and Mckernan [On Shokurov's rational connectedness conjecture, Duke Math. J. Volume 138, Number 1 (2007), 119-136.], every fiber of $f$ is rationally chain connected, in particular, the exceptional locus is covered by rational curves.

| cite | improve this answer | |
  • $\begingroup$ Thank you very much. This was exactely the type of answer I was looking for! $\endgroup$ – Nico Berger Sep 9 '18 at 7:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.