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Degeneration at of certain spectral sequences.

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Τίτλος: Degeneration at of certain spectral sequences.
Συγγραφείς: Popovici, Dan1
Πηγή: International Journal of Mathematics. Dec2016, Vol. 27 Issue 14, p1. 31p.
Θεματικοί όροι: *Spectral sequences (Mathematics), *Algebraic topology, *Algebraic equations, *Algebraic varieties, *Hodge theory
Περίληψη: We propose a Hodge theory for the spaces featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold or in the spectral sequence associated with a pair of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on whose kernel in every bidegree is isomorphic to in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for to give sufficient conditions for the degeneration at of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on . For example, in the Frölicher case, we prove degeneration at if there exists an SKT metric (i.e. such that ) whose torsion is small compared to the spectral gap of the elliptic operator defined by . In the foliated case, we obtain degeneration at under a hypothesis involving the Laplacians and associated with the splitting induced by the foliated structure. [ABSTRACT FROM AUTHOR]
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  Data: Degeneration at of certain spectral sequences.
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  Data: <searchLink fieldCode="JN" term="%22International+Journal+of+Mathematics%22">International Journal of Mathematics</searchLink>. Dec2016, Vol. 27 Issue 14, p1. 31p.
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  Data: We propose a Hodge theory for the spaces featuring at the second step either in the Frölicher spectral sequence of an arbitrary compact complex manifold or in the spectral sequence associated with a pair of complementary regular holomorphic foliations on such a manifold. The main idea is to introduce a Laplace-type operator associated with a given Hermitian metric on whose kernel in every bidegree is isomorphic to in either of the two situations discussed. The surprising aspect is that this operator is not a differential operator since it involves a harmonic projection, although it depends on certain differential operators. We then use this Hodge isomorphism for to give sufficient conditions for the degeneration at of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on . For example, in the Frölicher case, we prove degeneration at if there exists an SKT metric (i.e. such that ) whose torsion is small compared to the spectral gap of the elliptic operator defined by . In the foliated case, we obtain degeneration at under a hypothesis involving the Laplacians and associated with the splitting induced by the foliated structure. [ABSTRACT FROM AUTHOR]
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    Identifiers:
      – Type: doi
        Value: 10.1142/S0129167X16501111
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      – Code: eng
        Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 31
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    Subjects:
      – SubjectFull: Spectral sequences (Mathematics)
        Type: general
      – SubjectFull: Algebraic topology
        Type: general
      – SubjectFull: Algebraic equations
        Type: general
      – SubjectFull: Algebraic varieties
        Type: general
      – SubjectFull: Hodge theory
        Type: general
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      – TitleFull: Degeneration at of certain spectral sequences.
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              Text: Dec2016
              Type: published
              Y: 2016
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              Value: 27
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              Value: 14
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            – TitleFull: International Journal of Mathematics
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