A variation formula for the determinant line bundle. Compact subspaces of moduli spaces of stable bundles over class VII surfaces
| Autor: | Andrei Teleman |
|---|---|
| Přispěvatelé: | Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), ANR-10-BLAN-0118,MNGNK,Méthodes nouvelles en géométrie non-kählerienne(2010) |
| Jazyk: | angličtina |
| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
010102 general mathematics Mathematical analysis Holomorphic function Grothendieck–Riemann–Roch theorem [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] 01 natural sciences Linear subspace Unitary state instanton moduli space Moduli space determinant line bundle Mathematics::Algebraic Geometry Line bundle [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] 0103 physical sciences MSC 2L10 32Q57 58J52 32G13 Mathematics::Differential Geometry 010307 mathematical physics 0101 mathematics Mathematics::Symplectic Geometry Mathematics class VII surfaces |
| Zdroj: | Geometry, Analysis and Probability: In Honor of Jean-Michel Bismut Geometry, Analysis and Probability: In Honor of Jean-Michel Bismut, pp.217-243, 2017, Progress in Mathematics, ⟨10.1007/978-3-319-49638-2_10⟩ Progress in Mathematics ISBN: 9783319496368 |
| DOI: | 10.1007/978-3-319-49638-2_10⟩ |
| Popis: | Proceedings of the Conference in honor of J. M. Bismut, Progress in Mathematics, Birkhäuser; International audience; This article deals with two topics: the first, which has a general character, is a variation formula for the the determinant line bundle in non-K\"ahlerian geometry. This formula, which is a consequence of the non-K\"ahlerian version of the Grothendieck-Riemann Roch theorem proved recently by Bismut, gives the variation of the determinant line bundle corresponding to a perturbation of a Fourier-Mukai kernel on a product B×X by a unitary flat line bundle on the fiber X. When this fiber is a complex surface and is a holomorphic 2-bundle, the result can be interpreted as a Donaldson invariant. The second topic concerns a geometric application of our variation formula, namely we will study compact complex subspaces of the moduli spaces of stable bundles considered in our program for proving existence of curves on minimal class VII surfaces. Such a moduli space comes with a distinguished point a=[\mathcal{A}] corresponding to the canonical extension of X. The compact subspaces $Y\subset {\cal M}^{\mathrm{st}}$ containing this distinguished point play an important role in our program. We will prove a non-existence result: there exists no compact complex subspace of positive dimension $Y\subset {\cal M}^{\mathrm{st}}$ containing a with an open neighborhood $a\in Y_a\subset Y$ such that $Y_a\setminus\{a\}$ consists only of non-filtrable bundles. In other words, within any compact complex subspace of positive dimension $Y\subset {\cal M}^{\mathrm{st}}$ containing a, the point a can be approached by filtrable bundles. Specializing to the case $b_2=2$ we obtain a new way to complete the proof of a theorem in a previous article: any minimal class VII surface with $b_2=2$ has a cycle of curves. Applications to class VII surfaces with higher $b_2$ will be be discussed in a forthcoming article. |
| Databáze: | OpenAIRE |
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