{
  "id": "1601.04781",
  "version": "v1",
  "published": "2016-01-19T03:08:22.000Z",
  "updated": "2016-01-19T03:08:22.000Z",
  "title": "Degeneration at $E_2$ of Certain Spectral Sequences",
  "authors": [
    "Dan Popovici"
  ],
  "comment": "40 pages",
  "categories": [
    "math.DG",
    "math.AG",
    "math.CV"
  ],
  "abstract": "We propose a Hodge theory for the spaces $E_2^{p,\\,q}$ featuring at the second step either in the Fr\\\"olicher spectral sequence of an arbitrary compact complex manifold $X$ or in the spectral sequence associated with a pair $(N,\\,F)$ 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 $X$ whose kernel in every bidegree $(p,\\,q)$ is isomorphic to $E_2^{p,\\,q}$ 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 $E_2^{p,\\,q}$ to give sufficient conditions for the degeneration at $E_2$ of the spectral sequence considered in each of the two cases in terms of the existence of certain metrics on $X$. For example, in the Fr\\\"olicher case we prove degeneration at $E_2$ if there exists an SKT metric $\\omega$ (i.e. such that $\\partial\\bar\\partial\\omega=0$) whose torsion is small compared to the spectral gap of the elliptic operator $\\Delta' + \\Delta\"$ defined by $\\omega$. In the foliated case, we obtain degeneration at $E_2$ under a hypothesis involving the Laplacians $\\Delta'_N$ and $\\Delta'_F$ associated with the splitting $\\partial = \\partial_N + \\partial_F$ induced by the foliated structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-19T03:08:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "spectral sequence",
      "degeneration",
      "arbitrary compact complex manifold",
      "complementary regular holomorphic foliations",
      "differential operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160104781P"
    }
  }
}