Small contraction for Hyperkähler Varieties

Question

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 $B$?

Answer 1

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.

Answer 2

Another way to see it is the following. Let $f\colon X\to Y$ be any birational contraction from a projective hyperkähler manifold $X$ onto a normal projective variety $Y$. The exceptional locus of $f$ is equal to the augmented base locus $\mathbf{B}_+(f^*(A))$, where $A$ is any ample Cartier divisor on $Y$. Since $f^*(A)$ is big, and $K_X=0$, using a result of S. Takayama (Theorem 1.2, On uniruledness os stable base loci), any irreducible component of $\mathbf{B}_+(f^*(A))$ is uniruled. Even more simply, one can use Theorem 1 in this paper by Y. Kawamata. Note that the statement is true also if $X$ is any smooth (complex) projective variety with numerically trivial canonical divisor, and can be further generalised if $X$ has mild singularities.

Another nice property of exceptional loci of birational contractions of a projective hyperkähler manifold $X$ is that they have dimension at least $\mathrm{dim}(X)/2$. This follows, for example, from the fact that their irreducible components are algebraically coisotropic, as mentioned in the original post.