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: 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.)
A: 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 $X$.
