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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 $B$?

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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.

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

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