Tannaka duality on quotient stacks
The paper under review is concerned with the general problem of constructing dual objects in the category of projective schemes, meaning that a given scheme should be canonically isomorphic to its associated double dual. Prominent examples in algebraic geometry are provided by the dual of an abelian variety, by S. Mukai's duality theory of polarized \(K3\) surfaces, or by certain moduli spaces of stable vector bundles over a smooth projective curve [à la \textit{M. S. Narasimhan} and \textit{S. Ramanan}, Ann. Math. (2) 101, 391--417 (1975; Zbl 0314.14004)]. With these classical examples in mind, the author's underlying philosophy is that certain stacks of vector bundles should be natural categorical candidates for dual objects in algebraic geometry. Indeed, pursuing this strategy, he is able to construct dual objects for quotient stacks of schemes in the following way. Let \({\mathcal X}= [X/G]\) be the quotient stack associated with a scheme \(X\) acted on by an affine group scheme \(G\), and denote by \(\text{VB}({\mathcal X})\) the category of all vector bundles on the stack \({\mathcal X}\). For any affine base scheme \(S\), a \(S\)-valued fiber functor on \(\text{VB}({\mathcal X})\) is defined to be an exact additive tensor functor from \(\text{VB}({\mathcal X})\) to \(\text{VB}(S)\), and these fiber functors form a fibered category denoted by \(\text{FIB}(\text{VB}({\mathcal X}))\). The basic technical framework for this construction is explained in the first sections of the present paper. Then it is shown that there is a natural morphism of fibered categories \(D_{{\mathcal X}}:{\mathcal X}\to\text{FIB} (\text{VB}({\mathcal X}))\), and the author's main theorem establishes the following fact: Assume that the scheme \(X\) is quasi-compact and that there is a line bundle on the quotient stack \({\mathcal X}= [X/G]\) whose underlying line bundle on \(X\) is very ample. Then the associated ``duality functor'' \(D_{{\mathcal X}}:{\mathcal X}\to\text{FIB} ({\mathcal VB}({\mathcal X}))\) is a categorical equivalence. In particular, if the affine group \(G\) is trivial, i.e., if \(G= \text{Spec}(k)\) for a field \(k\), then one is dealing with true schemes, in which case the author's main theorem yields that for a quasi-compact \(k\)-scheme \(X\) admitting an ample line bundle, the duality morphism \(D_X\) provides a duality construction in the ordinary sense. The author's main result may be regarded as a stack-theoretic analogue of the classical Tannaka duality for affine groups in the sense of \textit{P. Deligne} and \textit{J. S. Milne} [in: Hodge cycles, motives, and Shimura varieties, Lect. Notes Math. 900, 101--228 (1982; Zbl 0477.14004)]. On the other hand, the author's result allows to derive a just as interesting and important conclusion, which makes the analogy to the classical duality theorems immediately recognizable. This second main result ensures the existence of another categorical equivalence, the so-called bi-duality morphism, between the quotient stack \({\mathcal X}\) and some tensor category fibered in groupoids. This construction uses the stack of all vector bundles on \({\mathcal X}=[X/G]\) as well as the existing universal bundle for them. In the course of the proofs, the framework of sheaves on fibered tensor categories appears as an essential ingredient. Accordingly, the technical details are thoroughly developed and explained, thereby making this rather advanced work easily accessible even for non-experts.
scientific article; zbMATH DE number 5014046