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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeNov 2nd 2012

    brief paragraph at Dolbeault cohomology

    • CommentRowNumber2.
    • CommentAuthorColin Tan
    • CommentTimeJun 28th 2016

    Recall the philosophy which interprets cohomology as the homset in a (oo,1)-topos. Has such an interpretation been found for Dolbeault cohomology?

    Namely, is there an (oo,1)-topos \mathcal{E} where the compact Kaehler manifolds form a subcategory, such that, for each p, q, there exists a classifying object D p,qD_{p, q} such that the Dolbeault cohomology H (p,q)(X)H^{(p,q)}(X) is naturally isomorphic to the homset of the mapping space Hom (X,D p,q)Hom_{\mathcal{E}}(X, D_{p, q})?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2016
    • (edited Jun 28th 2016)

    By the Dolbeault theorem (see there) Dolbeault cohomology is equivalently the abelian sheaf cohomology H q(X;Ω X p)H^q(X;\Omega_X^p), of the abelian sheaf Ω X p\Omega_X^p which is the Dolbeault complex of holomorphic p-forms.

    And all abelian sheaf cohomology theories are given by hom-spaces in an \infty-topos.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2016

    For some discussion along these lines see also page 2 of Differential cohomology is Cohesive homotopy theory (schreiber).

    • CommentRowNumber5.
    • CommentAuthorColin Tan
    • CommentTimeJun 28th 2016
    Thanks Urs!
    • CommentRowNumber6.
    • CommentAuthorColin Tan
    • CommentTimeJul 1st 2016

    As I understand it, the abelian sheaf cohomology H q(X;Ω X p)H^q(X;\Omega^p_X) is interpreted as Π 0Hom(X,B qΩ X p)\Pi_0 Hom(X, \mathbf{B}^q \Omega^p_X), where the Hom is taken in the oo-topos Sh (,1)(X)Sh_{(\infty,1)}(X) of simplicial sheaves over XX. Thus, as XX varies, the oo-topos Sh (,1)(X)Sh_{(\infty,1)}(X) in consideration also varies.

    This unlike ordinary cohomology of topological spaces where there is a single object B qA\mathbf{B}^q A in a single oo-topos Grpd\infty Grpd whereby π oHom(X,B qA)H q(X;A)\pi_o Hom(X, \mathbf{B}^q A) \simeq H^q(X ; A) gives the ordinary cohomology for all topological spaces XX.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2016
    • (edited Jul 1st 2016)

    No, you are to consider the \infty-topos over the site of all complex manifolds, see at complex analytic ∞-groupoid

    HAnalyticGrpdSh (CplxMfd). \mathbf{H} \coloneqq \mathbb{C}Analytic \infty Grpd \coloneqq Sh_\infty(CplxMfd) \,.

    Then given any particular complex manifold XX, it represents an object in that \infty-topos, and its Dolbeault cohomology is

    p,q(X)H q(X,Ω p)π 0H(X,B qΩ p). \mathcal{H}^{p,q}(X) \simeq H^q(X, \Omega^p) \simeq \pi_0 \mathbf{H}(X, \mathbf{B}^q \Omega^p) \,.
    • CommentRowNumber8.
    • CommentAuthorColin Tan
    • CommentTimeJul 2nd 2016

    Here B qΩ p\mathbf{B}^q\Omega^p is not a single object in Sh (CplxMfd)Sh_\infty (CplxMfd), but really B qΩ X p\mathbf{B}^q\Omega_X^p (which depends on XX) right?

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 2nd 2016

    No, it’s a single object. Ω pSh (CpxMfd)\Omega^p \in Sh_\infty(CpxMfd) is the sheaf on the site of all complex manifolds which assigns to any one its additive group of holomophic pp-forms.

    The Yoneda lemma says that then

    H(X,Ω p)Ω X p(X) \mathbf{H}(X, \Omega ^p) \simeq \Omega^p_X(X)

    and the claim about Dolbeault cohomology follows similarly.

    • CommentRowNumber10.
    • CommentAuthorColin Tan
    • CommentTimeJul 2nd 2016
    Ah! Thanks for your clarifications, Urs.
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