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 mmlsi1" class="mathmlsrc">mulatext 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_0,nmathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll"><msub><mrow><mi mathvariant="script">Mmi>mrow><mrow><mn>0mn><mo>,mo><mi>nmi>mrow>msub>math> of m>n  m>-pointed rational curves, we construct a K&auml;hler potential in terms of the Fourier coefficients of the Klein's Hauptmodul. We define the space mmlsi2" class="mathmlsrc">mulatext 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_g,nmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">Smi>mrow><mrow><mi>gmi><mo>,mo><mi>nmi>mrow>msub>math> as holomorphic fibration mmlsi3" class="mathmlsrc">mulatext 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_g,n→S_gmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">Smi>mrow><mrow><mi>gmi><mo>,mo><mi>nmi>mrow>msub><mo stretchy="false">→mo><msub><mrow><mi mathvariant="fraktur">Smi>mrow><mrow><mi>gmi>mrow>msub>math> over the Schottky space mmlsi4" class="mathmlsrc">mulatext 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_gmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">Smi>mrow><mrow><mi>gmi>mrow>msub>math> of compact Riemann surfaces of genus m>gm>, where the fibers are configuration spaces of m>n  m> points. For the tautological line bundles mmlsi21" class="mathmlsrc">mulatext 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_imathContainer hidden">mathCode"><math altimg="si21.gif" overflow="scroll"><msub><mrow><mi mathvariant="script">Lmi>mrow><mrow><mi>imi>mrow>msub>math> over mmlsi2" class="mathmlsrc">mulatext 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_g,nmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">Smi>mrow><mrow><mi>gmi><mo>,mo><mi>nmi>mrow>msub>math>, we define Hermitian metrics mmlsi343" class="mathmlsrc">mulatext 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_imathContainer hidden">mathCode"><math altimg="si343.gif" overflow="scroll"><msub><mrow><mi>hmi>mrow><mrow><mi>imi>mrow>msub>math> in terms of Fourier coefficients of a covering map m>Jm> of the Schottky domain. We define the regularized classical Liouville action m>S  m> and show that mmlsi39" class="mathmlsrc">mulatext 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}mathContainer hidden">mathCode"><math altimg="si39.gif" overflow="scroll"><mi mathvariant="normal">expmi><mo>⁡mo><mo stretchy="false">{mo><mi>Smi><mo stretchy="false">/mo><mi>πmi><mo stretchy="false">}mo>math> is a Hermitian metric in the line bundle mmlsi9" class="mathmlsrc">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">mg class="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">mathContainer hidden">mathCode"><math altimg="si9.gif" overflow="scroll"><mi mathvariant="script">Lmi><mo>=mo><msubsup><mrow><mo>&otimes;mo>mrow><mrow><mi>imi><mo>=mo><mn>1mn>mrow><mrow><mi>nmi>mrow>msubsup><msub><mrow><mi mathvariant="script">Lmi>mrow><mrow><mi>imi>mrow>msub>math> over mmlsi2" class="mathmlsrc">mulatext 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_g,nmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">Smi>mrow><mrow><mi>gmi><mo>,mo><mi>nmi>mrow>msub>math>. We explicitly compute the Chern forms of these Hermitian line bundles We prove that a smooth real-valued function mmlsi11" class="mathmlsrc">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">mg class="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">mathContainer hidden">mathCode"><math altimg="si11.gif" overflow="scroll"><mo>&minus;mo><mi mathvariant="script">Smi><mo>=mo><mo>&minus;mo><mi>Smi><mo>+mo><mi>πmi><msubsup><mrow><mo>&sum;mo>mrow><mrow><mi>imi><mo>=mo><mn>1mn>mrow><mrow><mi>nmi>mrow>msubsup><mi mathvariant="normal">logmi><mo>⁡mo><msub><mrow><mi>hmi>mrow><mrow><mi>imi>mrow>msub>math> on mmlsi2" class="mathmlsrc">mulatext 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_g,nmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><msub><mrow><mi mathvariant="fraktur">Smi>mrow><mrow><mi>gmi><mo>,mo><mi>nmi>mrow>msub>math>, 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 mmlsi12" class="mathmlsrc">mulatext 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)mathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll"><mo stretchy="false">(mo><mi>gmi><mo>,mo><mi>nmi><mo stretchy="false">)mo>math>.