Kuranishi spaces as a 2-category

Preprint English OPEN
Joyce, Dominic;
(2015)
  • Subject: Mathematics - Symplectic Geometry | Mathematics - Differential Geometry
    arxiv: Mathematics::Algebraic Geometry | Mathematics::Differential Geometry | Mathematics::Complex Variables | Mathematics::Symplectic Geometry

This is a survey of the author's paper arXiv:1409.6908 and in-progress book. 'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1503.07631), as the geometric structure on moduli spaces of $J$-holomorphic cur... View more
  • References (41)
    41 references, page 1 of 5

    atlas on X, as in [42, Def. 2.3.1], if it also satisfies:

    • Let V = (V0 ⊃ V1 ⊃ · · · ) be an sc-Banach space and U ⊆ V0 be open.

    (O, V ) and a homeomorphism ψ : O → Im ψ with an open set Im ψ ⊂ Z. • M-polyfold charts (O, V , ψ), (O˜, V˜, ψ˜) on Z are compatible if ψ˜−1 ◦ ψ ◦ r :

    U → V˜ and ψ−1 ◦ ψ˜ ◦ r˜ : U˜ → V are sc-smooth, where U ⊂ V0, U˜ ⊂ V˜0 are

    and Im r = ψ−1(Im ψ˜) ⊆ O, Im r˜ = ψ˜−1(Im ψ) ⊆ O˜. • An M-polyfold is roughly a metrizable topological space Z with a maximal

    atlas of pairwise compatible M-polyfold charts. That is, composition of 1-morphisms is associative up to canonical 2-isomorph-

    ism, as for weak 2-categories in §A.1.

    If g : X → Z and h : Y → Z are 1-morphisms in Kur with g a w-

    then g, h are strongly transverse. Thus Theorem 5.7 implies:

    Now let f : Y1 → Y2 be a smooth map of manifolds. Define the pushforward

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