Toric vector bundles: GAGA and Hodge theory
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Mathematische Zeitschrift
Toric vector bundles: GAGA and Hodge theory Jonas Stelzig1 Received: 3 June 2019 / Accepted: 1 September 2020 © The Author(s) 2020
Abstract We prove a GAGA-style result for toric vector bundles with smooth base and give an algebraic construction of the Frölicher approximating vector bundle that has recently been introduced by Dan Popovici using analytic techniques. Both results make use of the Rees-bundle construction.
1 Introduction A toric variety over a field k is an algebraic variety X over k with a Gnm -action that has a dense open orbit on which the group acts simply transitively. A vector bundle on such X is called toric if it is equipped with a Gnm -action s.t. the projection is an equivariant map. Toric varieties and vector bundles are an important source of examples in algebraic geometry. Just as normal toric varieties can be studied by combinatorial data, toric vector bundles (and also more general classes of equivariant sheaves) on a given normal toric variety X have been classified in terms of linear-algebra-data (roughly as vector spaces and filtrations with certain compatibility conditions), c.f. [10,11,13,14,18]. If k = C, for every toric variety X over C one also has a natural notion of holomorphic toric vector bundles over X an , the latter meaning (the set of complex points of) X seen as a complex analytic space. One obtains an analytification functor: toric vector bundles on X −→ holomorphic toric vector bundles on X an The first main result of this article is that for smooth toric varieties, this functor is an equivalence of categories: Theorem A For a smooth toric variety X over C, analytification induces an equivalence of categories between algebraic toric vector bundles on X and holomorphic toric vector bundles on X an . The same is true for toric vector bundles with equivariant connections. Of course, there is the known GAGA-principle by Serre [22], asserting an equivalence of the categories of coherent sheaves on a complex projective variety and its analytification.
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Jonas Stelzig [email protected] LMU, Munich, Germany
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J. Stelzig
The above theorem is not a formal consequence of this. For instance, X is not assumed to be projective and even for projective X , the equivariant structure is sheaf-theoretically described by an isomorphism of sheaves over Gnm × X , which is not projective. We do not need the full classification of toric vector bundles. In fact, it is enough for our purposes to consider algebraic toric vector bundles on affine spaces and we include a brief but largely self-contained treatment of these in Sect. 2. A key notion is the Rees-bundle construction, which associates to (suitable) multifiltered vector spaces (V , F1 , . . . , Fn ) a toric vector bundle ξAn (V , F1 , . . . , Fn ) on An . Toric vector bundles have also been studied in connection with Hodge theory, see e.g. [5–7,12,16,17,23,24], in part apparently independent of the classification. A basic idea is that, since a Hodge structure is a multifiltered vector space,
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