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Schieder, Simon

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Schieder

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Schieder, Simon

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    Picard-Lefschetz Oscillators for the Drinfeld-Lafforgue-Vinberg Compactification
    (2015-05-01) Schieder, Simon; Gaitsgory, Dennis; Bezrukavnikov, Roman; Lurie, Jacob
    We study the singularities of the Drinfeld-Lafforgue-Vinberg compactification of the moduli stack of G-bundles on a smooth projective curve for a reductive group G. The study of these compactifications was initiated by V. Drinfeld (for G=GL_2) and continued by L. Lafforgue (for G=GL_n) in their work on the Langlands correspondence for function fields; unlike the work of Drinfeld and Lafforgue, however, we focus on questions about the singularities of these compactifications which arise naturally in the geometric Langlands program. A definition of the compactification for a general reductive group G is also due to Drinfeld (unpublished) and relies on the Vinberg semigroup of G; this case will be dealt with in the forthcoming work [Sch]. In the present work we focus on the case G=SL_2. In this case the compactification can alternatively be viewed as a canonical one-parameter degeneration of the moduli space of SL_2-bundles. We study the singularities of this one-parameter degeneration via the weight-monodromy theory of the associated nearby cycles construction: We give an explicit description of the nearby cycles sheaf together with its monodromy action in terms of certain novel perverse sheaves which we call "Picard-Lefschetz oscillators", and then use this description to determine the intersection cohomology sheaf and other invariants of the singularities. Our proofs rely on the construction of certain local models for the one-parameter degeneration which themselves form one-parameter families of spaces which are factorizable in the sense of Beilinson and Drinfeld. We also include a first application on the level of functions.