Deformation of \(\ell \)-adic sheaves with undeformed local monodromy (Q1930123)

Let \(k\) be an algebraically closed field of characteristic \(p\). Let \(X\) be a smooth connected projective curve over \(k\), let \(S\) be a finite closed subset of \(X\), and let \(\ell\) be a prime number distinct from \(p\). For any \(s\in S\), let \(\eta_s\) be the generic point of the strict Henselization of \(X\) at \(s\). A lisse \(\bar{\mathbb Q}_\ell\)-sheaf \(\mathcal F\) of \(X\setminus S\) is called \textit{physically rigid} if for any lisse \(\bar{\mathbb Q}_\ell\)-sheaf \(\mathcal G\) on \(X\setminus S\) with the property \(\mathcal F|_{\eta_s}\cong\mathcal G|_{\eta_s}\) for any closed point \(s\in S\), we have \(\mathcal F\cong\mathcal G\). The lisse \(\bar{\mathbb Q}_\ell\)-sheaf \(\mathcal F\) on \(X\setminus S\) corresponds to a Galois representation \[ \rho:\text{Gal}\left(\overline{K(X)}/K(X)\right)\rightarrow\text{GL}(n,\bar{\mathbb Q}_\ell) \] of the function field \(K(X)\) unramified everywhere on \(X\setminus S\). \(\mathcal F\) is physically rigid if and only if, for any Galois representation \(\rho^\prime\) of \(\text{Gal}(\overline{K(X)}/K(X))\) such that \(\rho\) and \(\rho^\prime\) induce isomorphic Galois representations of the local field obtained by taking completion of \(K(X)\) at any place of \(K(X)\), we have \(\rho\cong\rho^\prime\). That is, a physically rigid sheaf \(\mathcal F\) is completely determined by all the Galois representations of local fields defined by \(\mathcal F\). Assume \(X=\mathbb P^1_k\). If \(X\) has genus \(g\geq 1\), then there exists a lisse \(\bar{\mathbb Q}_{\ell}\)-sheaf \(\mathcal L\) of \(\text{rk} 1\) on \(X\) s.t. \(\mathcal L{^\otimes n}\) is nontrivial for all \(n\). For any lisse \(\bar{\mathbb Q}_{\ell}\)-sheaf \(\mathcal F\) on \(X\setminus S\), the lisse sheaf \(\mathcal G=\mathcal F\otimes_k\mathcal L\) is not isomorphic to \(\mathcal F\) as they have non-isomorphic determinant, but \(\mathcal F|_{\eta_s}\cong\mathcal G|_{\eta_s}\) for all \(s\in X\). Hence \(\mathcal F\) is not rigid. A lisse \(\bar{\mathbb Q}_{\ell}\)-sheaf \(\mathcal F\) on \(X\setminus S\) is called \textit{cohomologically rigid} if we have \(H^1(X,j_\ast\mathcal E nd(\mathcal F))=0\;,\) where \(j:X\setminus S\hookrightarrow X\) is the canonical open immersion. Katz shows that for an irreducible lisse sheaf, cohomological rigidity implies physical rigidity. It is conjectured that the converse is true. Bloch and Esnault studied deformations of locally free \(\mathcal O_{X\setminus S}\)-modules provided with connections while keeping local (formal) data undeformed. They prove that physically rigidity and cohomological rigidity are equivalent for locally free \(\mathcal O_{X\setminus S}\)-modules provided with connections. In this article, the author studies the deformation of lisse \(\ell\)-adic sheaves while keeping the local monodromy undeformed. That is, let \(F\) be any one of the following fields: a finite extension of the finite field \(\mathbb F_\ell\) with \(\ell\) elements, an algebraic closure \(\bar{\mathbb F}_\ell\) of \(\mathbb F_\ell\), a finite extension of the \(\ell\)-adic number field \(\mathbb Q_\ell\), or an algebraic closure \(\bar{\mathbb Q}_\ell\) of \(\mathbb Q_\ell\). Let \(\mathcal F\) be a lisse \(F\)-sheaf on \(X\setminus S\). The author studies the deformation of \(\mathcal F\) so that \(\mathcal F|_{\eta_s}\), \(s\in S\), is undeformed. Let \(\eta\) be the generic point of \(X\). An \(F\)-representation of \(\pi_1(X\setminus S,\bar\eta)\) of rank \(r\) is a homomorphism \(\rho:\pi_1(X\setminus S,\bar\eta)\rightarrow\text{GL}(F^r)\) such that i) If \(F\) is a finite extension of \(F_\ell\) or \(\mathbb Q_\ell\), \(\rho\) is supposed to be continuous, where the topology on \(\text{GL}(F^r)\) is the discrete topology if \(F\) is a finite field, and is induced by the \(\ell\)-adic topology if \(F\) is a finite extension of \(\mathbb Q_\ell\). ii) If \(F\) is an algebraic closure of \(\mathbb F_\ell\), respectively \(\mathbb Q_\ell\), there should exist a finite extension \(E\) of \(\mathbb F_\ell\), respectively \(\mathbb Q_\ell\), such that \(\rho\) factors through a continuous homomorphism \(\pi_1(X\setminus S,\bar\eta)\rightarrow\mathrm{GL}(E^r)\). Let \(V=\mathcal F_{\bar\eta}\). Then the lisse \(\mathcal F\)-sheaf \(\mathcal F\) on \(X\setminus S\) defines an \(F\)-representation \(\rho_0:\pi_1(X\setminus S,\bar\eta)\rightarrow\text{GL}(V)\). Fix an embedding \(\text{Gal}(\bar\eta_s/\eta_s)\hookrightarrow\pi_1(X\setminus S,\bar\eta)\) for each \(s\in S\). Then the problem of this article can be interpreted as the deformation of the representation \(\rho_0\) so that \(\rho_0|_{\text{Gal}(\bar\eta_s/\eta_s)}\;,s\in S\) remain undeformed. The treatment is somewhat similar to Mazur's theory of deformation of Galois representations. \(\mathcal C\) denotes the category of Artinian local \(F\)-algebras with residue field \(F\). Morphisms in \(\mathcal C\) are local \(F\)-algebra homomorphisms, inducing the identity on the residue fields. For an object \(A\in\mathcal C\), \(\mathfrak m_A\) denotes the maximal ideal, and a homomorphism \(\rho:\pi_1(X\setminus S,\bar\eta)\rightarrow\text{GL}(A^r)\) is called a representation if by regarding \(A^r\) as a finite dimensional \(F\)-vector space, \(\rho\) is an \(F\)-representation of \(\pi_1(X\setminus S,\bar\eta)\). Two representations \(\rho_1,\rho_2:\pi_1(X\setminus S,\bar\eta)\rightarrow\text{GL}(A^r)\) are equivalent with respect to the subgroup \(G\subseteq\pi_1(X\setminus S,\bar\eta)\), \(\rho_1|G\cong\rho_2|G\) if there exits \(P\in\text{GL}(A^r)\) such that \(P^{-1}\rho_1(g)P=\rho_2(g)\) for all \(g\in G\). They are called \textit{strictly equivalent} if if the above condition holds for some \(P\) with the property \(P\equiv I\mod\mathfrak m_A\). Fix an \(F\)-representation \(\rho_0:\pi_1(X\setminus S,\bar\eta)\rightarrow\text{GL}(F^r)\). For any \(A\in\mathcal C\), \(R(A)\) is defined as the set of strict equivalence classes of representations \(\rho:\pi_1(X\setminus S,\bar\eta)\rightarrow\text{GL}(A^r)\) such that \(\rho\equiv\rho_0\mod\mathfrak m_A\) and \(\rho|_{\mathrm{Gal}(\bar\eta_s/\eta_s)}\cong\rho_0|_{\text{Gal}(\bar\eta_s/\eta_s)}\) for all \(s\in S\). Each element in \(R(A)\) is called a \textit{deformation} of \(\rho_0\) with \(\rho_0|_{\text{Gal}(\bar\eta_s/\eta_s)}\) being undeformed. The main result in this article can the be summarized in the following: Assume all elements in the set \(\mathrm{End}_{F|_{\pi_1(X\setminus S,\bar\eta)}}(F^r)\) are scalar multiplications. (i) The functor \(R:\mathcal C\rightarrow\text{(sets)}\) is pro-representable (ii) Let \(F[\epsilon]\) be the ring of dual numbers over \(F\). The tangent space \(R(F[\epsilon])\) of the functor \(R\) is isomorphic to \(H^1(X,j_\ast\mathcal E nd(\mathcal F))\), where \(j:X\setminus S\rightarrow X\) is the open immersion and \(\mathcal F\) is the lisse \(F\)-sheaf on \(X\) corresponding to the representation \(\rho_0\). (iii) The prorepresenting ring \(R_{\text{univ}}\) is isomorphic to \(k[[t_1,\dots,t_m]]\), \(m=\dim_FH^1(X,j_\ast\mathrm{End}(\mathcal F)).\) (iv) If there exist endomorphisms above that are not scalar multiplications, the functor above has a prorepresenting hull, as defined by M. Schlessinger. The article is very well written, and it takes a complicated problem down to ``ordinary'' deformation theory in nice and readable way. The obstruction theory is included, and there is a lot to gain also in invariant theory by reading the article.