Automorphisms of generic complete intersections This question concerns a seemingly folk lore result, which states that automorphism groups of generic complete intersections are trivial, under certain assumptions.
To state the question, let $r \geq 1$ and let $d_1,\ldots,d_r, n \in \mathbb{N}$ be such that


*

*$n \geq 3$. 

*$d_i \geq 2$ for each $i$.

*$(d_1,\ldots,d_r;n) \neq (2;n)$ nor $(2,2;n)$.



Let $X$ be a generic smooth complete intersection of dimension $n$ in projective space $\mathbb{P}^{n+r}$ over $\mathbb{C}$ with 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.
 A: You may try the following:
I believe (I did not check the details) that the monodromy representation on the primitive part of $H^n(X,\mathbb{C})_{prim}$ is irreducible, as in the case of hypersurfaces.
If this is true then for a very general $X$ the Hodge structure $H^n(X,\mathbb{C})_{prim}$ does not have any nontrivial sub-Hodge structures.
Let $\sigma$ be an automorphism of $X$. As you mentioned $\sigma^*$ respects the hyperplane class and hence it induces an automorphism of $H^n(X,\mathbb{C})_{prim}$. The 1-eigenspace of $\sigma^*$ is a sub-Hodge structure and there either all of $H^n(X,\mathbb{C})_{prim}$ or this  eigenspace is the zero-space. In the first case you have that $\sigma^*$ is the identity and you claimed that you then can deduce  that  $\sigma$ is trivial. In the second case you obtain that the quotient $X/\langle \sigma \rangle$ has the same Betti numbers as $\mathbb{P}^n$. I do not see how you can exclude this case, but it seems very weird that this happens when the $d_i$ are large.
A: I think the result you are looking for is worked out in Theorem 1.3 of Chen-Pan-Zhang "Automorphism and Cohomology II: Complete intersections" (+ Remark 1.4), see: https://arxiv.org/pdf/1511.07906.pdf
