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Formal loops. II: A local Riemann-Roch theorem for determinantal gerbes. (English) Zbl 1129.14022
The development of computational techniques in the last decade has inade possible to attack some classical problems of algebraic geometry from a computational viewpoint. In this survey, we briefly describe some open problems of computational algebraic geometry which can be approached from such viewpoint. Some of the problems we discuss are the decomposition of Jacobians of genus two curves, automorphisrns groups of algebraic curves and the corresponding loci in the moduli space of algebraic curves \({\mathcal M}_g\), inclusions among such loci, decomposition of Jacobians of algebraic curves with automorphisms, invariants of binary forms and the hyperelliptic moduli, theta functions of curves with automorphisms, etc. We decompose Jacobians of genus 3 curves with automorphisms and determine the inclusions among the loci for algebraic curves with automorphisms of genus 3 and 4.
The goal of this article is to relate three subjects of interest: (A) the theory of sheaves of chiral differential operators (CDO) on a complex manifold, (B) the theory of the group \(\text{GL}(\infty)\) developetd by Sato and others, and (C) the refinement of the Grothendieck-Riemann-Roch theorem found by Deligne.
This relation proceeds via the ind-scheme \(\mathcal{L}(X)\) of formal loops which is the algebro-geometric analogs of the Fréchet manifolds \(LX.\) Sato’s theory of \(\text{GL}(\infty)\) can be developed in two versions. The formal version works with a locally linearly compact topological vector space \(V\) such as the space \(\mathbb{C}((t))\) with the \(t\)-adic topology. The group \(\text{GL}(\infty)\) is then interpreted as the group of continuous automorphisms of \(V\). The Hilbert version starts with a Hilbert space \(H\) equipped with a polarization \(H=H_+\oplus H_-.\) The group \(\text{GL}(\infty)\) is interpreted as the group of bounded linear automorphisms preserving the polarization. In both cases one has a Grassmann-type variety \(\mathcal{G}\) and a determinantal line bundle \(\Delta\) on \(\mathcal{G}\times\mathcal{G}\) making \(\mathcal{G}\) into a set of objects of a \(\mathbb{C}^\times\)-gerbe acted upon by \(\text{GL}(\infty)\).
A nonlinear version of the theory should involve infinite-dimensional manifolds with a \(\text{GL}(\infty)\)-structure in the tangent bundle. The authors have developed a formalization of this idea in the algebro-geometric setting. The corresponding objects are locally compact smooth ind-schemes, and their tangent spaces possess a \(\text{GL}(\infty)\)-structure in the formal version. For such an ind-scheme, there is the relative Sato Grassmannian \(\mathcal{G}\to Y\) and a determinantal line bundle \(\Delta\) on \(\mathcal{G}\times_Y\mathcal{G}\) giving an \(\mathcal{O}_Y^\times\)-gerbe \(\mathcal{D}et_Y\). The determinantal anomaly of \(Y\) is the class of \(\mathcal{D}et_Y\) in \(H^2(Y,\mathcal{O}_Y^\times)\) classifying \(\mathcal{O}_Y^\times\)-gerbes. The gerbe \(Y=\mathcal{L}X\) turns out to give sheaves of CDO: The anomaly in constructing CDO is precisely the determinantal anomaly for this loop space.
The main result of the article has the following consequence: The class \([\mathcal{D}et_{\mathcal{L}X}]\in H^2(\mathcal{L}X,\mathcal{O}^\times)\) is equal to the image of the characteristic class (0.1.1) under the transgression map
\[ \tau:H^2(X,K_2(\mathcal{O}_X))\to H^2(\mathcal{L}X,\mathcal{O}^\times). \] This identification of \([{\mathcal D}et_{{\mathcal L}X}]\) can be seen as a particular case of a Riemann-Roch-type result for determinantal gerbes, and this Riemann-Roch theorem for gerbes is the main result of the article.
The main result of the article, theorem 5.3.1, can be seen as a statement comparing two central extensions of the loop group \(\text{GL}_N((t))\). The authors result identifies the determinantal central extension of \(\text{GL}_N((t))\) with the extension coming from \(\text{ch}_2\). Finally, the relation of the Riemann-Roch theorem is similar to the relation of the self-duality of the Jacobian of a curve to the Cartier self-duality of the ind-group scheme \(\text{GL}_1((t))\) established by C. Contou-Carrère. The Contou-Carrère symbol plays an important role in the authors’ approach.
The authors give the necessary definition of \(\mathcal F\)-gerbes on a scheme \(S\), up to equivalence these are identified with \(H^2({\mathcal S},{\mathcal F})\). Also Ind-schemes are defined and studied. The determinantal gerbe of a locally free \({\mathcal O}_S((t))\)-module is given, this includes the study of the twisted affine Grassmannian and the \({\mathcal O}^\times\)-groupoid structure, the last leading to the \({\mathcal O}^\times\)-gerbe \({\mathcal D}et(\mathcal E)\). Restriction of scalars, the evaluation map and the transgression map leads to Chern classes and the local Riemann-Roch theorem. The article ends with an application to the anomaly of the loop space and to chiral differential operators.
This article contains a lot of information, and is nice to read. lt gives a lot of relations between differential and algebraic geometry.

MSC:
14C40 Riemann-Roch theorems
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
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