Automorphisms of surfaces of general type with \(q\geqslant 2\) acting trivially in cohomology (Q2860717)

We say that a compact complex manifold \(X\) is \textit{cohomologically rigidified} if its automorphism group \({\text{ Aut}}(X)\) acts faithfully on the cohomology ring \(H^*(X, \mathbb{Z})\), \textit{rationally cohomologically rigidified} if \({\text{ Aut}}(X)\) acts faithfully on \(H^*(X, \mathbb{Q})\), and \textit{rigidified} if \({\text{ Aut}}(X) \cap {\text{ Diff}}^0(X)=\{{\text{ id}}_X\}\). There are the following implications:NEWLINENEWLINE{rationally \, cohomologically \, rigidified \, } \(\Rightarrow\) {cohomologically \, rigidified\, } \(\Rightarrow\) rigidifiedNEWLINENEWLINEA question posed by Catanese, inspired by the case of curves, is whether every surface of general type is rigidified. Up to now, it is only known, in contrast with the curve case, that not all surfaces of general type are cohomologically rigidified.NEWLINENEWLINEIn the paper under review the three authors prove that surfaces of general type with irregularity \(q(S) \geq 3\) are rationally cohomologically rigidified (hence rigidified) and so are minimal surfaces \(S\) with irregularity \(q(S)=2\) unless \(K^2_S=8\chi(\mathcal{O}_S)\).NEWLINENEWLINEA particular class of surfaces with \(K^2_S=8\chi(\mathcal{O}_S)\) is the one of surfaces isogenous to a product of curve. Let us recall that a surface S is said to be \textit{isogenous to a product} if \(S\) is an unramified quotient of a product of two curves of genera greater than or equal to two, i.e., \(S \cong (C_1 \times C_2)/G\) where \(G\) is a finite group acting freely on the product. These surfaces are divided into two types. The \textit{mixed} type in which the group \(G\) exchanges the two factors. And the \textit{unmixed} type in which the group \(G\) acts diagonally.NEWLINENEWLINEThe authors prove that all surfaces isogenous to a product of mixed type with \(q(S) \geq 2\) are rationally cohomologically rigidified. Moreover, they classify all the surfaces isogenous of unmixed type with \(q(S)=2\) that are not rationally cohomologically rigidified. In particular these surfaces have group \(G\) isomorphic either to \(\mathbb{Z}/2m\mathbb{Z} \times \mathbb{Z}/2mn\mathbb{Z}\) or to \(\mathbb{Z}/2\mathbb{Z} \times\mathbb{Z}/2m\mathbb{Z} \times \mathbb{Z}/2mn\mathbb{Z}\).NEWLINENEWLINEAll the questions about rigidity are closely related to the local moduli problem for \(S\). This is to study the local map \(\phi\colon\text{Def}(S) \rightarrow \mathcal{T}(M)_{[S]}\), from the Kuranishi space to the germ of the Teichmüller space at \([S]\). The authors deduce from their results that, if \(S\) is a minimal surface of general type with \(q(S) \geq 3\) and \(K_S\) ample, then \(\phi\) is a local homomorphism.NEWLINENEWLINEFinally, they generalize some of their results in higher dimensions. More precisely, they prove that if \(X\) is a smooth projective variety of general type, of maximal Albanese dimension and with canonical sheaf with positive generic vanishing index then \(X\) is cohomologically rigidified.