# Infinitesimal deformations of fake projective planes (or ball quotients)

This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.

By Mostow Rigidity Theorem, the geometry of a hyperbolic manifold $X$ of real dimension at least $3$ is determined by its fundamental group, and hence it is unique. As a corollary, it follows that a complex ball quotient $X$ of (complex) dimension at least $2$ (for instance, a fake projective plane) has no deformations in the large (i.e. is rigid), in other words the base $\textrm{Def}(X)$ of the Kuranishi family of $X$ consists of a single point.

Question. Is it also true that $X$ is infinitesimally rigid, i.e. that $H^1(X, \, T_X)=0$?

If $H^1(X, \, T_X) \neq 0$ then by Mostow's theorem it would follow that $X$ has obstructed first order deformations. I guess this is not the case, but I was unable to find a precise reference or statement in the wealth of results concerning deformations (or rigidity) of hyperbolic manifolds.

• I do not know the answer, but I have a trivial observation. By GAGA, and using that the universal cover is Stein, $H^1(X,T_X)$ equals the group cohomology $H^1(\pi_1(X);H^0(\widetilde{X},T_{\widetilde{X}}))$. I have a vague recollection that Koll'ar studies this in "Shafarevich Maps and Automorphic Forms" . . . – Jason Starr Jul 8 '15 at 12:35
• This follows from a well known theorem of Calabi and Vesentini which predates Mostow rigidity (Theorem 2 of "On compact, locally symmetric Kähler manifolds" Ann. of Math. (2) 71 1960 472–507.) – ulrich Jul 8 '15 at 13:01
• Ah, thanks! In fact, the paper you quoted says that $H^1(X, T_X)=0$ as soon as the universal cover of $X$ is a Cartan domain of dimension at least $2$, and this in particular implies an affirmative answer to my question. – Francesco Polizzi Jul 8 '15 at 14:15
• @ulrich: may I suggest that you post your comment as an answer? I would be glad to accept it. – Francesco Polizzi Jul 8 '15 at 14:25

Yes, this is true. It follows from a more general theorem of Calabi and Vesentini on the vanishing of $H^i(X, T_X)$ (in a suitable range) for $X$ a compact quotient of an irreducible Hermitian symmetric space of non-compact type (which predates Mostow rigidity).