2015/01/31 by Dominic Joyce · 1 citation
Mathematics · #math.DG
published as Advances in Mathematics 299 (2016), 760-862 · 97 pages, LaTeX. (v3) final version, to appear in Advances in Mathematics
arxiv created 2016/06/07 · arxiv updated 2016/07/27
In conventional Differential Geometry one studies manifolds, locally modelled on \mathbb Rn, manifolds with boundary, locally modelled on [0,∞)×\mathbb Rn-1, and manifolds with corners, locally modelled on [0,∞)k×\mathbb Rn-k. They form categories \bf Man⊂\bf Manb⊂\bf Manc. Manifolds with corners X have boundaries ∂ X, also manifolds with corners, with \mathop\rm dim∂ X=\mathop\rm dim X-1. We introduce a new notion of 'manifolds with generalized corners', or 'manifolds with g-corners', extending manifolds with corners, which form a category \bf Mangc with \bf Man⊂\bf Manb⊂\bf Manc⊂\bf Mangc. Manifolds with g-corners are locally modelled on XP=\mathop\rm Hom\bf Mon(P,[0,∞)) for P a weakly toric monoid, where XP≅[0,∞)k×\mathbb Rn-k for P=\mathbb Nk×\mathbb Zn-k. Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries ∂ X. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in \bf Mangc exist under much weaker conditions than in \bf Manc. This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of J-holomorphic curves can be manifolds or Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than ordinary corners. Our manifolds with g-corners are related to the 'interior binomial varieties' of Kottke and Melrose in arXiv:1107.3320 (see also Kottke arXiv:1509.03874), and to the 'positive log differentiable spaces' of Gillam and Molcho in arXiv:1507.06752.