(cont'd) Also taking a general point on the product of an elliptic curve with itself and blowing it up will have trivial automorphism while the automorphism group of the Hodge structure (fixing Chern classes) will be equal to the automorphism group of the product which is infinite.

@VA: Isn't the situation more complicated already for K3-surfaces? In that case the arithmetic group you are talking about is commensurable with the semi-direct product of the normal subgroup generated by reflections in $-2$-curves and the actual automorphism group of the K3-surface and there are examples when the normal subgroup has finite index. It is true that which elements of $H^\ast(X,\mathbb Z)$ are $-2$-curves is determined by the Hodge structure so that the automorphism group is determined by Hodge structure data.

Dolgachev: Infinite Coxeter groups and automorphisms of algebraic surfaces I just found by searching in SciMath and haven't looked at it. I think the crucial case is probably K3-surfaces and abelian varieties which can be handled by periods. An interesting further case would be to look at K3-surfaces and consider a stabiliser of a point. That will be the automorphism group of the blow up in that point and could possibly be non-f.g. I don't think that can be handled by period theory however.

I do not see that what you say is true. Note that in the case of finite length categories two modules in different blocks have no common simple for their Jordan-Hölder factors so clearly there are no non-zero morphisms between them.

Not that this is an answer to your question (which is an interesting one) but you can hide the nasty details of the reduction to the Brouwer fixed point theorem by using the Michael selection theorem which gives a continuous map $f\colon S \to S$ with $f(x) \in \phi(x)$ for all $x$ to which you then can directly apply the Brouwer theorem.

I don't think it is fair to say that there is a partial proof in the SGA 3 resissue. Exactly at the point where one is to deal with the implication (iv) => (v) they switch to the old version which is over a field.

See addition to my answer.

Does anyone understand the proof of the claim (EGA IV 17.1.6) that formal smoothness is local (in the Zariski topology)? The authors refer to 16.5.17 but to me it seems that what they really are referring to is 16.5.18 and there they have explicitly made the assumption that $\Omega^1_{X/Y}$ should be locally finitely presented an assumption which is not part of 17.1.6. Until I understand that proof I am not sure I believe the statement that formal smoothness is local which should be weaker than what you ask.

@norondion: I have added elaboration in answer.

@Robin: Right I saw potential problems only because for some strange reason I had confused what was subgroup of what.

You are right, what I wrote is true but invalidates the following arguments. Corrected.