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Jacobi-Bernoulli cohomology and deformations of schemes and maps. (English) Zbl 1279.14013

The study of deformation theory ordinarily includes the study of a differential graded Lie algebra (dgla) \(\mathfrak g\). That is, one studies the canonically defined deformation ring \(R(\mathfrak g)\). In this article, the deformation theory is mainly the study of complex algebraic structures, and so the deformation ring \(R(\mathfrak g)\) is constructed with basis in the Jacobi complex associated to \(\mathfrak g\). This deformation theory is not broad enough to include embedded deformations, deformations of a manifold \(X\) embedded in a fixed ambient space \(Y\). In earlier work, the author has introduced the notion of a Lie atom and an associated Jacobi-Bernoulli complex to handle this deformation theory.
The present article establishes a notion of (dg) semi-simplicial Lie algebra (SELA) as a convenient setting for deformation theory. This is a generalization of the Lie atom, and it handles deformations of arbitrary singular schemes over \(\mathbb C\).
Let \(A\) be a totally ordered index-set. A simplex in \(A\) is a finite nonempty subset \(S\subset A\), a biplex is a pair \(\{S_1\subset S_2\}\) of simplices with \(|S_1|+1=|S_2|\), and further for triplex etc. Also, there is assigned a sign rule \(\epsilon(S_1,S_2)\). A semi-simplicial Lie algebra (SELA) \(\mathfrak g.\) on \(A\) is an assignment \(\mathfrak g_S\) for each simplex \(S\) on \(A\), and for each biplex a restriction morphism compatible with triplexes and sign rules. The \(\mathfrak g_S\) can be assembled into a complex \(K^.(\mathfrak g.)=\bigoplus_{|S|=i+1}\mathfrak g_S\) and with differential constructed by the restriction maps.
To compute the deformation theory of a SELA, a complex called the Jacobi-Bernoulli complex is introduced. This transforms a gluing condition from a nonabelian cocycle condition to an ordinary additive cocycle condition via the multilinearity of the groups making up the complex. The transformation is given by using the exponential operator, and letting \(\Psi=\exp\psi\), the Baker-Campbell-Hausdorff (BCH) formula gives a formal expression \[ \exp X\exp Y\exp Z=\sum W_{i,j,k}(X,Y,Z) \] with \(W_{i,j,k}(X,Y,Z)\) the homogeneous ad-polynomial of tridegree \(i,j,k\) called the BCH-polynomial. Then the Jacobi-Bernoulli complex \(J(\mathfrak g.)\) for the SELA \(\mathfrak g.\) is designed to encompass the various BCH-polynomials \(w_{i,j,k}\), and the dual of the cohomology of \(J(\mathfrak g.)\) yields the deformation ring associated to the SELA \(\mathfrak g.\) Another main results is that the deformation theory of an algebraic scheme over \(\mathbb C\) can be expressed in terms of a SELA, constructed by an affine covering. Associated to an embedding \(X\rightarrow P\) where \(P\) is either affine or projective, a dgla is associated which is called the tangent dgla, and denoted \(\mathcal T_X(P)\). This \(\mathcal T_X(P)\) is the mapping cone of a constructed map \(T_P\otimes A_X\rightarrow N_{X/P}\) where \(N_{X/P}\) is the normal atom to \(X\) in \(P\). The author show that \(\mathcal T_X(P)\) admits a dgla structure, a dgla action on the coordinate ring of \(X\), \(A_X\), and an \(A_X\) module structure. Then an essential result is that up to what is called weak equivalence, the dgla \(\mathcal T_X(P)\) depends only on the scheme \(X\) and not on the embedding in \(P\). This kind of independence gives the possibility of defining the global SELA \(\mathcal T_X\), behaving polite on an affine covering. Given this, global deformations of \(X\) amount to collections of deformations of each \(X_\alpha\) given by Kodaira-Spencer theory, and suitable gluing data. The necessary compatibilities are expressed as a cocycle condition in the Jacobi-Bernoulli complex \(J(\mathcal T_X.)\).
The main results in this article, is the construction of the proposed complexes and SELAs, and the proofs of these behaving well on arbitrary schemes. Also, the functorial properties of the constructions and their contribution to deformation theory is important and interesting. It illustrates general techniques in deformation theory, and finally, explicit examples are given proving that the theoretical results work in practice.

MSC:

14D15 Formal methods and deformations in algebraic geometry
58H15 Deformations of general structures on manifolds
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