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 ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si1.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=48b7f747f9a2e99967f5759293d55552" title="Click to view the MathML source">Rⁿer hidden">e">

erflow="scroll"> {e-struck">R}^{n}

, manifolds with boundary, locally modelled on ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si2.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=e1a2886f18ca70d772b3884ebf987941" title="Click to view the MathML source">[0,∞)×Rⁿ⁻¹er hidden">e">

erflow="scroll"> etchy="false">[0, \infty etchy="false">) &times; {e-struck">R}^{n - 1}

, and manifolds with corners, locally modelled on ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si1119.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=7bf1f31d71d523f8b1a65dfedeee46da" title="Click to view the MathML source">[0,∞)^k×R^n−ker hidden">e">

erflow="scroll"> {etchy="false">[0, \infty etchy="false">)}^{k} &times; {e-struck">R}^{n - k}

. They form categories ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si4.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=a6c9b326340ae538730fc11cc678aeb2" title="Click to view the MathML source">Man⊂Man^b⊂Man^cer hidden">e">

erflow="scroll"> Man \subset M a n^{b} \subset M a n^{c}

. Manifolds with corners Xem> have boundaries ∂X em>, also manifolds with corners, with e="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si5.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=66c07a2bceb763e26419d104d45a97bd">eImage" height="12" width="138" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816307186-si5.gif">

eeeeeeeeeeeee

er hidden">e">

erflow="scroll"> \dim \partial X = \dim e width="0.2

em">e>X−1.

We introduce a new notion of manifolds with generalized cornersem>, or manifolds with g-corners em>, extending manifolds with corners, which form a category ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si2149.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=3ef4c425574712ebb5a266cbedcc60c2" title="Click to view the MathML source">Man^gcer hidden">e"> $erflow="scroll"> M a n^{gc}$ with ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si7.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=dc8150a164e8057696946b683ec1b206" title="Click to view the MathML source">Man⊂Man^b⊂Man^c⊂Man^gcer hidden">e"> $erflow="scroll"> Man \subset M a n^{b} \subset M a n^{c} \subset M a n^{gc}$ . Manifolds with g-corners are locally modelled on e="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si14.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=4c286a9b161e8a0b1c4d99e4a0fc54ef">eImage" height="20" width="184" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816307186-si14.gif">er hidden">e"> $erflow="scroll"> X_{P} = {Hom}_{Mon} e width="0.2$ em">e>etchy="true" maxsize="2.4ex" minsize="2.4ex">(P,etchy="false">[0,∞etchy="false">)etchy="true" maxsize="2.4ex" minsize="2.4ex">) for P em> a weakly toric monoid, where ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si1541.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=3880d02da7e06d007fb27348f0e2e523" title="Click to view the MathML source">X_P≅[0,∞)^k×R^n−ker hidden">e"> $erflow="scroll"> X_{P} ≅ {etchy="false">[0, \infty etchy="false">)}^{k} × {e-struck">R}^{n - k}$ for ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si10.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=1655bc898723a7f071f9d3a0f2188ae8" title="Click to view the MathML source">P=N^k×Z^n−ker hidden">e"> $erflow="scroll"> P = {e-struck">N}^{k} × {e-struck">Z}^{n - k}$ .

Most differential geometry of manifolds with corners extends nicely to manifolds with g-corners, including well-behaved boundaries ∂X em>. In some ways manifolds with g-corners have better properties than manifolds with corners; in particular, transverse fibre products in ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si2149.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=3ef4c425574712ebb5a266cbedcc60c2" title="Click to view the MathML source">Man^gcer hidden">e"> $erflow="scroll"> M a n^{gc}$ exist under much weaker conditions than in ext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816307186&_mathId=si11.gif&_user=111111111&_pii=S0001870816307186&_rdoc=1&_issn=00018708&md5=604b2f3437079412a6637307600a113a" title="Click to view the MathML source">Man^cer hidden">e"> $erflow="scroll"> M a n^{c}$ .

This paper was motivated by future applications in symplectic geometry, in which some moduli spaces of Jem>-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].

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