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Atiyah classes with values in the truncated cotangent complex. (English) Zbl 1282.14019
For a complex analytic vector bundle \(E\) on a complex manifold \(X\), M. F. Atiyah [Trans. Am. Math. Soc. 85, 181–207 (1957; Zbl 0078.16002)] defined an element of \(\text{H}^1(X, {\mathcal H}om(E,E)\otimes \Omega_X)\) which allowed him to prove a criterion for the existence of complex analytic connections. In the algebraic context, if \(X \rightarrow S\) is a separated morphism of schemes and \(\mathcal E\) is a locally free sheaf on \(X\) then the classical Atiyah class of \(\mathcal E\) is the element \(\text{At}_{\text{cl}}(\mathcal E)\) of \(\text{Ext}^1(\mathcal E, \mathcal E\otimes \Omega_{X/S})\) corresponding to the extension\(\, :\) \[ 0 \rightarrow \pi_{2\ast}(\pi_1^\ast \mathcal E\otimes {\mathcal I}_{\Delta_X}/{\mathcal I}^2_{\Delta_X}) \rightarrow \pi_{2\ast}(\pi_1^\ast \mathcal E\otimes {\mathcal O}_{X\times X}/{\mathcal I}^2_{\Delta_X}) \rightarrow \pi_{2\ast}(\pi_1^\ast \mathcal E\otimes {\mathcal O}_{\Delta_X}) \rightarrow 0 \] where \(\pi_i : X\times X \rightarrow X\), \(i = 1, 2\), are the canonical projections (here \(X\times X\) means \(X\times_SX\); we shall omit the index \(S\) in all kinds of notation) and \(\Delta_X\) is the image of the diagonal embedding \(\delta_X : X \rightarrow X\times X\). B. Angéniol and M. Lejeune-Jalabert [Calcul différentiel et classes charactéristiques en Géometrie Algèbrique. Travaux en Cours 38. Paris: Hermann (1989; Zbl 0749.14008)] described \(\text{At}_{\text{cl}}(\mathcal E)\) in terms of Čech cohomology.
On the other hand, L. Illusie [Complexe cotangent et déformations I. Lecture Notes in Mathematics. 239. Berlin-Heidelberg-New York: Springer-Verlag. (1971; Zbl 0224.13014)] defined a general variant of the Atiyah class of a perfect complex \({\mathcal E}^\bullet\) as an element of \(\text{Ext}^1({\mathcal E}^\bullet , {\mathcal E}^\bullet\otimes^{\text{L}}{\widetilde {\mathbb L}}_X)\), where \({\widetilde {\mathbb L}}_X\) is the cotangent complex of \(X/S\). Recently, D. Huybrechts and R. P. Thomas [Math. Ann. 346, No. 3, 545–569 (2010; Zbl 1186.14014)] defined, in an elementary manner, a truncated Atiyah class which is an element of \(\text{Ext}^1({\mathcal E}^\bullet , {\mathcal E}^\bullet\otimes^{\text{L}}{\mathbb L}_X)\), where \({\mathbb L}_X\) is the truncated cotangent complex of \(X/S\). Their construction suffices for the applications of the Atiyah class to the deformation theory of complexes as objects in the derived category.
The paper under review, which is its author’s diploma thesis at the University of Bonn, provides a concrete description of the truncated Atiyah class in terms of Čech resolutions. Before stating the results it contains, we need to recall the construction of Huybrechts and Thomas. Assume that \(S\) is a Noetherian separated scheme and that the \(S\)-scheme \(X\) can be embedded, as a closed subscheme, into a smooth separated and quasi-compact \(S\)-scheme \(U\). Let \(\mathcal I \subset {\mathcal O}_U\) denote the ideal sheaf of \(X\), and let \(p_i : U\times U \rightarrow U\), \(i = 1, 2\), be the canonical projections. The truncated cotangent complex \({\mathbb L}_X\) is the complex \({\mathcal I}/{\mathcal I}^2 \overset{d_{\mathbb L}^{-1}}\longrightarrow \Omega_U\otimes {\mathcal O}_X\), with \(d_{\mathbb L}^{-1}\) induced by the Kähler derivation \(d_U : {\mathcal O}_U \rightarrow \Omega_U\). Of course, \(\text{Coker}\, d_{\mathbb L}^{-1} \simeq \Omega_X\).
Now, an obvious observation asserts that if \(P\), \(Q\) are closed subschemes of a scheme \(R\) such that \(P \subset Q\) then one has an exact sequence\(\, :\) \[ 0 \rightarrow {\mathcal I}_Q\otimes {\mathcal O}_P \rightarrow {\mathcal I}_P\otimes {\mathcal O}_Q \rightarrow {\mathcal I}_{P,\, Q} \rightarrow 0\, , \] where \({\mathcal I}_{P,\, Q} = {\mathcal I}_P/{\mathcal I}_Q \subset {\mathcal O}_Q\). Since \({\mathcal O}_{\Delta_U}\) is flat over \(U\) via \(p_2 : U\times U \rightarrow U\), one sees that, by tensorizing with \(p_2^\ast{\mathcal O}_X\) the exact sequence \( 0 \rightarrow {\mathcal I}_{\Delta_U} \rightarrow {\mathcal O}_{U\times U} \rightarrow {\mathcal O}_{\Delta_U} \rightarrow 0\, , \) one gets an isomorphism \({\mathcal I}_{\Delta_U}\otimes {\mathcal O}_{U\times X} \overset\sim\rightarrow {\mathcal I}_{\Delta_X,\, U\times X}\). Applying the above observation to \(\Delta_X \subset X\times X \subset U\times X\) one gets an exact sequence\(\, :\) \[ p_1^\ast{\mathcal I}\otimes {\mathcal O}_{\Delta_X} \overset\beta\longrightarrow {\mathcal I}_{\Delta_U}\otimes {\mathcal O}_{X\times X} \rightarrow {\mathcal I}_{\Delta_X,\, X\times X} \rightarrow 0 \] (notice that the image of \(p_1^\ast{\mathcal I}\otimes {\mathcal O}_{U\times X} \rightarrow {\mathcal O}_{U\times X}\) is \({\mathcal I}_{X\times X,\, U\times X}\)). If \(f\) is a local section of \(\mathcal I \subset {\mathcal O}_U\) then \(d_U(f)\) is the image into \({\mathcal I}_{\Delta_U}/{\mathcal I}^2_{\Delta_U} \simeq \delta_{U\ast}\Omega_U\) of the local section \(p_2^\ast(f) - p_1^\ast(f)\) of \({\mathcal I}_{\Delta_U}\). Since \(p_2^\ast(f)|_{U\times X} = 0\) it follows, from the above definitions, that \(\beta(p_1^\ast(f)\otimes 1) = (p_1^\ast(f) - p_2^\ast(f))\otimes 1\), hence the diagram \[ \begin{tikzcd}[column sep=large] p_1^\ast{\mathcal I}\otimes {\mathcal O}_{\Delta_X} \ar[r, "\beta"] \ar[d, "\wr" '] & \mathcal I_{\Delta_U}\otimes \mathcal O_{X\times X}\ar[d]\\ \delta_{X\ast}(\mathcal I/\mathcal I^2) \ar[r,"-\delta_{X\ast}d_{\mathbb L}^{-1}" '] & \delta_{X\ast}(\Omega_U\otimes \mathcal O_X) \end{tikzcd} \] is commutative. Moreover, if \(X\) is flat over \(S\), which we shall assume from now on, then the restriction \({\overline p}_1 : U\times X \rightarrow U\) of \(p_1\) is flat, hence \(p_1^\ast{\mathcal I}\otimes {\mathcal O}_{U\times X} \simeq {\overline p}_1^\ast{\mathcal I} \overset\sim\rightarrow {\mathcal I}_{X\times X,\, U\times X}\) and \(\beta\) is a monomorphism.
Denoting by \({\mathcal G}^\bullet\) the left resolution \[ 0 \rightarrow p_1^\ast{\mathcal I}\otimes {\mathcal O}_{\Delta_X} \overset\beta\longrightarrow {\mathcal I}_{\Delta_U}\otimes {\mathcal O}_{X\times X} \rightarrow {\mathcal O}_{X\times X} \rightarrow 0 \] of \({\mathcal O}_{\Delta_X}\), Huybrechts and Thomas define the universal truncated Atiyah class of \(X\) to be the morphism \(\text{At}_X : {\mathcal O}_{\Delta_X} \rightarrow \delta_{X\ast}{\mathbb L}_X[1]\) in the derived category \(\text{D}(X\times X)\) defined by the diagram \({\mathcal O}_{\Delta_X} \overset{\text{qis}}\longleftarrow {\mathcal G}^\bullet \overset\rho\longrightarrow \delta_{X\ast}{\mathbb L}_X[1]\), where \(\rho\) is the morphism of complexes corresponding to the above commutative diagram. If \({\mathcal E}^\bullet\) is a bounded complex of locally free sheaves on \(X\) then the truncated Atiyah class \(\text{At}({\mathcal E}^\bullet)\) is the morphism \(\text{R}\pi_{2\ast}(\pi_1^\ast{\mathcal E}^\bullet \otimes \text{At}_X) : {\mathcal E}^\bullet \rightarrow {\mathcal E}^\bullet \otimes {\mathbb L}_X[1]\) in the derived category \(\text{D}(X)\).
Now, returning to the paper under review, let \((U_i)_{i\in \Gamma}\) be a finite affine open cover of \(U\) and let \((X_i = U_i\cap X)_{i\in \Gamma}\) be the induced affine open cover of \(X\). Assume that \({\mathcal E}^s |_{X_i}\) is trivial, \(\forall \, s\in {\mathbb Z}\), \(\forall \, i\in \Gamma\). If \(\mathcal H\) is a coherent sheaf on \(X\), let \({\check{\mathcal C}}^\bullet(\mathcal H)\) denote the Čech resolution of \(\mathcal H\) corresponding to the above open cover of \(X\). If \({\mathcal H}^\bullet\) is a bounded complex of coherent sheaves on \(X\), let \({\check{\mathcal C}}^\bullet({\mathcal H}^\bullet)\) be the total complex of the double complex \(({\check{\mathcal C}}^i({\mathcal H}^j))_{i,j\in {\mathbb Z}}\). The main result of the paper under review consists in the construction of an explicit morphism of complexes \(\mu : {\mathcal E}^\bullet \rightarrow {\mathcal E}^\bullet \otimes {\check{\mathcal C}}^\bullet({\mathbb L}_X[1])\) which equals \((\text{id}_{\mathcal E}\otimes \text{can})\circ \text{At}({\mathcal E}^\bullet)\) in \(\text{D}(X)\) (\(\text{can} : {\mathbb L}_X[1] \rightarrow {\check{\mathcal C}}^\bullet({\mathbb L}_X[1])\)). If \({\mathcal E}^\bullet\) consists of a single term \(\mathcal E\) (in cohomological degree 0), of rank \(n\), with transition matrices \(M_{ij} \in \text{GL}_n({\mathcal O}_X(X_{ij}))\), then \(\mu\) is determined by the morphism\(\, :\) \[ \mu^0 : \mathcal E \longrightarrow {\mathcal E}\otimes {\check{\mathcal C}}^2({\mathcal I}/{\mathcal I}^2) \oplus {\mathcal E}\otimes {\check{\mathcal C}}^1(\Omega_U\otimes {\mathcal O}_X) \] defined by \(((M_{ik}\cdot ({\widetilde M}_{kj}\cdot {\widetilde M}_{ji} - {\widetilde M}_{ki}))_{i,j,k},\, (M_{ij}\cdot d_U{\widetilde M}_{ji})_{i,j})\), where the element \({\widetilde M}_{ij}\) of \(\text{Mat}_{n\times n}({\mathcal O}_U(U_{ij}))\) is a lifting of \(M_{ij}\). In the general case, the formula of \(\mu\) is more complicated.
Considering the trace map \(\text{tr} : \text{Hom}_{\text{D}}({\mathcal E}^\bullet ,\, {\mathcal E}^\bullet \otimes {\mathbb L}_X[1]) \rightarrow \text{Hom}_{\text{D}}({\mathcal O}_X,\, {\mathbb L}_X[1]) = {\mathbb H}^1(X,\, {\mathbb L}_X)\) one can define the truncated first Chern class \(c_1({\mathcal E}^\bullet)\) to be \(\text{tr}(\text{At}({\mathcal E}^\bullet))\). Using his description of \(\text{At}({\mathcal E}^\bullet)\), the author of the paper under review shows, in a natural manner, that \(c_1(\text{det}\, {\mathcal E}^\bullet) = c_1({\mathcal E}^\bullet)\) and, thus, simplifies an argument from the paper of Huybrechts and Thomas proving the existence of a perfect obstruction theory of stable pairs.
14D15 Formal methods and deformations in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
Full Text: DOI arXiv
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