This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under suitable assumptions.

To state the question, let $r \geq 1$ and let $d_1,\ldots,d_r, n \in \mathbb{N}$ be such that

1. $n \geq 3$.
2. $d_i \geq 2$ for each $i$.
3. $(d_1,\ldots,d_r;n) \neq (2;n)$ nor $(2,2;n)$.

Let $X$ be a generic smooth complete intersection of dimension $n$ in projective space $\mathbb{P}^{n+r}$ over $\mathbb{C}$ of equations of degrees $d_1,\ldots,d_r$. Is $\mathrm{Aut}(X)$ trivial?

Remarks:

• The conditions (1), (2), (3) imply that $\mathrm{Aut}(X)$ is finite and preserves the hyperplane class, for each such smooth complete intersection $X$ (not necessarily generic).
• The result is not true without condition (3), as here the generic such complete intersection has a non-trivial automorphism group (hopefully I did not miss any other ''bad'' cases).
• The answer is well-known to be yes when $r=1$, i.e. for generic hypersurfaces.

I need a version of this result in a paper I am currently writing. We are able to use this to show that the action of $\mathrm{Aut}(X)$ on $H^n(X,\mathbb{C})$ is faithful for any such smooth complete intersection (not necessarily generic). I would be most interested if anyone has any ideas on alternative approaches to this problem as well.

We are able to handle many special cases, such as when the degrees $d_i$ are all different or $X$ is of general type. A critical case is for example when all the degrees are the same and $X$ is Fano of high codimension.