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

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• 作者：Jinsung Park^a ; ^{jinsung@kias.re.kr" class="auth_mail" title="E-mail the corresponding author} ; Leon A. Takhtajan^b ; ^c ; ^{leontak@math.sunysb.edu" class="auth_mail" title="E-mail the corresponding author} ; Lee-Peng Teo^d ; ^{lpteo@xmu.edu.my" class="auth_mail" title="E-mail the corresponding author}
• 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 d="mmlsi1" class="mathmlsrc">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_0,ndden">de">${\mathcal{M}}_{0,n}$ 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 d="mmlsi2" class="mathmlsrc">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_g,ndden">de">${\mathfrak{S}}_{g,n}$ as holomorphic fibration d="mmlsi3" class="mathmlsrc">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_g,n→S_gdden">de">${\mathfrak{S}}_{g,n}\to {\mathfrak{S}}_g$ over the Schottky space d="mmlsi4" class="mathmlsrc">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_gdden">de">${\mathfrak{S}}_g$ of compact Riemann surfaces of genus g, where the fibers are configuration spaces of n   points. For the tautological line bundles d="mmlsi21" class="mathmlsrc">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_idden">de">${\mathcal{L}}_i$ over d="mmlsi2" class="mathmlsrc">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_g,ndden">de">${\mathfrak{S}}_{g,n}$, we define Hermitian metrics d="mmlsi343" class="mathmlsrc">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_idden">de">$h_i$ 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 d="mmlsi39" class="mathmlsrc">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}dden">de">$\exp \{S/\pi \}$ is a Hermitian metric in the line bundle d="mmlsi9" class="mathmlsrc">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">dth="93" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816301670-si9.gif">dden">de">$\mathcal{L}={\otimes }_{i=1}^n{\mathcal{L}}_i$ over d="mmlsi2" class="mathmlsrc">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_g,ndden">de">${\mathfrak{S}}_{g,n}$. We explicitly compute the Chern forms of these Hermitian line bundles<div class="formula" id="fm0010"><div class="mathml">d="mmlsi10" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0001870816301670&_mathId=si10.gif&_user=111111111&_pii=S0001870816301670&_rdoc=1&_issn=00018708&md5=e42faca9f6d08f181f1445a29fe46d35">dth="360" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816301670-si10.gif">dden">de">d/blank.gif">div>div> We prove that a smooth real-valued function d="mmlsi11" class="mathmlsrc">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">dth="186" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0001870816301670-si11.gif">dden">de">$-\mathcal{S}=-S+\pi {\sum }_{i=1}^n\log h_i$ on d="mmlsi2" class="mathmlsrc">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_g,ndden">de">${\mathfrak{S}}_{g,n}$, 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 d="mmlsi12" class="mathmlsrc">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)dden">de">$(g,n)$.