# Calabi-Yau threefold with an automorphism of infinite order

I am looking for a (hopefully simple) example of a Calabi-Yau threefold (projective, simply connected, with trivial canonical bundle) admitting an automorphism of infinite order.

A Schoen manifold $$X$$ is a generic complete intersection in $$\mathbb{P}^2 \times \mathbb{P}^2 \times \mathbb{P}^1$$ of two hyper-surfaces of degree (3,0,1) and (0,3,1) respectively. Alternately, you can describe this Calabi-Yau threefold as the fiber product of two generic rational elliptic surfaces $$X = S\times _{\mathbb{P}^1} S'$$ where $$S$$ and $$S'$$ are hyper-surfaces in $$\mathbb{P}^2\times \mathbb{P}^1$$ of degree (3,1) and projection to $$\mathbb{P}^1$$ induces the elliptic fibration. The surface $$S$$ has an infinite automorphism given by fiberwise addition by a non-zero section $$\sigma: \mathbb{P}^1 \to S$$ and this induces an infinite automorphism on $$X$$.

• More generally, there are lots of CY3s which admit elliptic fibrations and I would expect there to be lots of examples where the Mordell-Weil group of sections is infinite. – Jim Bryan Feb 16 at 22:24
• Certainly. My problem was to get an explicit construction — I guess the one you give is reasonably explicit. – abx Feb 17 at 7:58
• For sure there are many examples of CY3s with elliptic fibration of positive MW rank. But typically these translations only give birational automorphisms, not biregular ones. – Bort Feb 17 at 10:24
• (For clarity, my comment was in reference to Jim Bryan's comment, not to his answer.) – Bort Feb 17 at 10:32
• @abx I think that with a little bit of work, one can get extremely explicit with a Schoen manifold (writing down the equations of $X$ and then writing the automorphism in terms of the variables). I haven't done it, but I would start by writing down the group law for a pencil of cubics. I don't know what you are trying to do, but I would be surprised if having explicit equations is any more useful than the more conception description in terms of translation by a section in the Abelian fibration. – Jim Bryan Feb 18 at 16:33

Another nice example is by Oguiso and Truong, "Explicit Examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy".

Briefly, take $$E$$ to be an elliptic curve with an order $$3$$ automorphism $$\tau$$ and form the abelian variety $$A = E \times E \times E$$. The quotient of $$A$$ by the diagonal action of $$\tau$$ is mildly singular, but it has only isolated singularities and it's easy to check/standard that there's a crepant resolution $$X$$ that's a CY3.

Then it's easy to get automorphisms of infinite order: $$SL(3,\mathbb Z)$$ acts on $$X$$ in an obvious way. In fact, [OT] show that some of these automorphisms are "primitive", meaning they don't preserve any nontrivial fibration (unlike in the elliptically fibered examples). This is done using the theory of dynamical degrees.

(Note that this obviously gives honest automorphisms, not just birational maps; there's nothing subtle to check on singular fibers, unlike what you usually run into looking at elliptically fibered examples, as Bort alludes in the comments above.)

• Very useful, thanks! – abx Feb 22 at 17:29