A generalization of manifolds with corners

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• 作者：Dominic Joyce ^{joyce@maths.ox.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Differential geometry ; Manifold with corners ; Boundary ; Monoid ; Category ; Transverse fibre product ; Kuranishi space
• 刊名：Advances in Mathematics
• 出版年：2016
• 出版时间：20 August 2016
• 年：2016
• 卷：299
• 期：Complete
• 页码：760-862
• 全文大小：1623 K

文摘

In conventional Differential Geometry one studies manifolds, locally modelled on Rⁿ${\mathbb{R}}^n$, manifolds with boundary, locally modelled on [0,∞)×Rⁿ⁻¹$[0,\infty )\times {\mathbb{R}}^{n-1}$, and manifolds with corners, locally modelled on [0,∞)^k×R^n−k${[0,\infty )}^k\times {\mathbb{R}}^{n-k}$. They form categories Man⊂Man^b⊂Man^c$\mathbf{Man}\subset \mathbf{M}\mathbf{a}{\mathbf{n}}^{\mathbf{b}}\subset \mathbf{M}\mathbf{a}{\mathbf{n}}^{\mathbf{c}}$. Manifolds with corners X have boundaries ∂X  , also manifolds with corners, with .

We introduce a new notion of manifolds with generalized corners, or manifolds with g-corners  , extending manifolds with corners, which form a category Man^gc$\mathbf{M}\mathbf{a}{\mathbf{n}}^{\mathbf{gc}}$ with Man⊂Man^b⊂Man^c⊂Man^gc$\mathbf{Man}\subset \mathbf{M}\mathbf{a}{\mathbf{n}}^{\mathbf{b}}\subset \mathbf{M}\mathbf{a}{\mathbf{n}}^{\mathbf{c}}\subset \mathbf{M}\mathbf{a}{\mathbf{n}}^{\mathbf{gc}}$. Manifolds with g-corners are locally modelled on for P   a weakly toric monoid, where X_P≅[0,∞)^k×R^n−k$X_P\cong {[0,\infty )}^k\times {\mathbb{R}}^{n-k}$ for P=N^k×Z^n−k$P={\mathbb{N}}^k\times {\mathbb{Z}}^{n-k}$.

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 Man^gc$\mathbf{M}\mathbf{a}{\mathbf{n}}^{\mathbf{gc}}$ exist under much weaker conditions than in Man^c$\mathbf{M}\mathbf{a}{\mathbf{n}}^{\mathbf{c}}$.

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 rather than ordinary corners.

Our manifolds with g-corners are related to the ‘interior binomial varieties’ of Kottke and Melrose [20], and the ‘positive log differentiable spaces’ of Gillam and Molcho [6].