Potentials and Chern forms for Weil-Petersson and Takhtajan-Zograf metrics on moduli spaces

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• 作者：Jinsung Park<sup>asup><sup> ; sup><sup>jinsung@kias.re.kr" class="auth_mail" title="E-mail the corresponding authorsup> ; Leon A. Takhtajan<sup>bsup><sup> ; sup><sup>csup><sup> ; sup><sup>leontak@math.sunysb.edu" class="auth_mail" title="E-mail the corresponding authorsup> ; Lee-Peng Teo<sup>dsup><sup> ; sup><sup>lpteo@xmu.edu.my" class="auth_mail" title="E-mail the corresponding authorsup>
• 关键词：primary ; 14H60 ; 32G15 ; secondary ; 53C80
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：856-894
• 全文大小：631 K

文摘

For the TZ metric on the moduli space <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si1.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=0ef4c0900e91c930d6acfbe98ebcc9b6" title="Click to view the MathML source">M<sub>0,nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>script">M0,nsub>span>span>span> of n  -pointed rational curves, we construct a Kähler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si2.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=082ac70425503cfffe4f36b606c622a5" title="Click to view the MathML source">S<sub>g,nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Sg,nsub>span>span>span> as holomorphic fibration <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si3.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=cef116e0fcbc7c05f764baad31d315e4" title="Click to view the MathML source">S<sub>g,nsub>→S<sub>gsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Sg,nsub>stretchy="false">→sub>Sgsub>span>span>span> over the Schottky space <span id="mmlsi4" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si4.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=d5a11d01b9b68b1dcc8800a492859b11" title="Click to view the MathML source">S<sub>gsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Sgsub>span>span>span> of compact Riemann surfaces of genus g, where the fibers are configuration spaces of n   points. For the tautological line bundles <span id="mmlsi21" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si21.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=3d5f378089a3be2f660ec640afc7b993" title="Click to view the MathML source">L<sub>isub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>script">Lisub>span>span>span> over <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si2.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=082ac70425503cfffe4f36b606c622a5" title="Click to view the MathML source">S<sub>g,nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Sg,nsub>span>span>span>, we define Hermitian metrics <span id="mmlsi343" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si343.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=7512e6237c51f5e31429f4e8181c4cca" title="Click to view the MathML source">h<sub>isub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>hisub>span>span>span> in terms of Fourier coefficients of a covering map J of the Schottky domain. We define the regularized classical Liouville action S   and show that <span id="mmlsi39" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si39.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=e25a109b45c6a842087c5a55780dd6e3" title="Click to view the MathML source">exp⁡{S/π}span><span class="mathContainer hidden"><span class="mathCode">scroll">exp⁡stretchy="false">{Sstretchy="false">/πstretchy="false">}span>span>span> is a Hermitian metric in the line bundle <span id="mmlsi9" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si9.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=7bc83484836b060081b10fc5db7c461a">ss="imgLazyJSB inlineImage" height="16" width="93" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816301670-si9.gif">script>style="vertical-align:bottom" width="93" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0001870816301670-si9.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">script">L=subsup>&otimes;i=1nsubsup>sub>script">Lisub>span>span>span> over <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si2.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=082ac70425503cfffe4f36b606c622a5" title="Click to view the MathML source">S<sub>g,nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Sg,nsub>span>span>span>. We explicitly compute the Chern forms of these Hermitian line bundles We prove that a smooth real-valued function <span id="mmlsi11" class="mathmlsrc">source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si11.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=b8b60a4d1ac037fb01da529eaba12c24">ss="imgLazyJSB inlineImage" height="18" width="186" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816301670-si11.gif">script>style="vertical-align:bottom" width="186" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0001870816301670-si11.gif">script><span class="mathContainer hidden"><span class="mathCode">scroll">&minus;script">S=&minus;S+πsubsup>&sum;i=1nsubsup>log⁡sub>hisub>span>span>span> on <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si2.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=082ac70425503cfffe4f36b606c622a5" title="Click to view the MathML source">S<sub>g,nsub>span><span class="mathContainer hidden"><span class="mathCode">scroll">sub>Sg,nsub>span>span>span>, a potential for this special difference of WP and TZ metrics, coincides with the renormalized hyperbolic volume of a corresponding Schottky 3-manifold. We extend these results to the quasi-Fuchsian groups of type <span id="mmlsi12" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si12.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=46bf961eaec02c7956e2adaee582d856" title="Click to view the MathML source">(g,n)span><span class="mathContainer hidden"><span class="mathCode">scroll">stretchy="false">(g,nstretchy="false">)span>span>span>.