NDLI: Stability of quadric bundles

Content Provider	Springer Nature Link
Author	Pustetto, Andrea Giudice, Alessio Lo
Copyright Year	2013
Abstract	A decorated vector bundle on a smooth projective curve $$X$$ is a pair $$(E,\varphi )$$ consisting of a vector bundle and a morphism $$\varphi :(E^{\otimes a})^{\oplus b}\rightarrow (\det E)^{\otimes c}\otimes \mathsf {N}$$ , where $$\mathsf {N}\in \text {Pic}(X)$$ . There is a suitable semistability condition for such objects which has to be checked for any weighted filtration of $$E$$ . We prove, at least when $$a=2$$ , that it is enough to consider filtrations of length $$\le $$ 2. In this case decorated bundles are very close to quadric bundles and to check semistability condition one can just consider the former. A similar result for L-twisted Higgs bundles and quadric bundles was already proved (García-Prada et al. in The Hitchin–Kobayashi correspondence, Higgs pairs and surface group representations, 2012; Schmitt in Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich, 2008). Our proof provides an explicit algorithm which requires a destabilizing filtration and ensures a destabilizing subfiltration of length at most two. Quadric bundles can be thought as a generalization of orthogonal bundles. We show that the simplified semistability condition for decorated bundles coincides with the usual semistability condition for orthogonal bundles. Finally we note that our proof can be easily generalized to decorated vector bundles on nodal curves.
Ending Page	242
Page Count	16
Starting Page	227
File Format	PDF
ISSN	00465755
e-ISSN	15729168
Journal	Geometriae Dedicata
Issue Number	1
Volume Number	173
Language	English
Publisher	Springer Netherlands
Publisher Date	2013-12-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Geometry Semistability Decorated vector bundles Quadric bundles Orthogonal bundles Algebraic moduli problems, moduli of vector bundles
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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