A generalization of manifolds with corners

DOI10.1016/J.AIM.2016.06.004zbMATH Open1359.58001arXiv1501.00401OpenAlexW2964325357MaRDI QIDQ2629117

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Publication date: 20 July 2016

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Abstract: In conventional Differential Geometry one studies manifolds, locally modelled on ${mathbbR}^n$, manifolds with boundary, locally modelled on $[0,infty)imes{mathbbR}^{n-1}$, and manifolds with corners, locally modelled on $[0,infty{)}^{k}imes{mathbbR}^{n-k}$. They form categories $$. Manifolds with corners $X$ have boundaries $partialX$, also manifolds with corners, with $mathopmdimpartialX=mathopmdimX-1$. We introduce a new notion of 'manifolds with generalized corners', or 'manifolds with g-corners', extending manifolds with corners, which form a category $$ with $$. Manifolds with g-corners are locally modelled on $$ for $P$ a weakly toric monoid, where $X_{P}cong[0,infty{)}^{k}imes{mathbbR}^{n-k}$ for $P={mathbbN}^{k}imes{mathbbZ}^{n-k}$. Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries $partialX$. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in $$ exist under much weaker conditions than in $$. 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.

Full work available at URL: https://arxiv.org/abs/1501.00401

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