Abstract
We study the basic properties of Higgs sheaves over compact Kähler manifolds and establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we prove some properties of semistable Higgs sheaves.
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Notes
Notice that \(\phi \) is a morphism of sheaves, then \(\phi (T)\) is contained in the torsion of \(E\otimes \Omega ^{1}_{X}\), which is exactly \(T\otimes \Omega ^{1}_{X}\) because \(\Omega ^{1}_{X}\) is locally free.
If it is not, we can destabilize \(\mathfrak H \) with a Higgs subsheaf \(\mathfrak{H }^{\prime }\). If it is semistable we stop, if it is not, then we repeat this procedure. Clearly this finishes after a finite number of steps, since in the extreme case we get a Higgs sheaf of rank one, which is in particular stable by Proposition 2.3, part (i).
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Acknowledgments
The author would like to thank his thesis advisor, Prof. U. Bruzzo, for his constant support and encouragement and also The International School for Advanced Studies (SISSA) at Trieste, Italy. The main part of this work has been done when he was a Ph.D. student there. The author wants to thank also the Centro de Investigación en Matemáticas (CIMAT A.C.) at Guanajuato, Mexico for the hospitality. Finally, the author would like to thank O. Iena, G. Dossena and C. Marin for some comments. Some results presented in this article were part of the author’s Ph.D. thesis in the Mathematical-Physics sector at SISSA.
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Cardona, S.A.H. Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles II: Higgs sheaves and admissible structures. Ann Glob Anal Geom 44, 455–469 (2013). https://doi.org/10.1007/s10455-013-9376-y
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DOI: https://doi.org/10.1007/s10455-013-9376-y