Aeppli–Bott-Chern cohomology and Deligne cohomology from a viewpoint of Harvey–Lawson’s spark complex

详细信息    查看全文

• 作者：Jyh-Haur Teh
• 关键词：Differential cohomology ; Differential character ; Aeppli cohomology ; Bott ; Chern cohomology ; Deligne cohomology ; Spark complex ; Refined Chern class
• 刊名：Annals of Global Analysis and Geometry
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：50
• 期：2
• 页码：165-186
• 全文大小：531 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Mathematical and Computational Physics
  Group Theory and Generalizations
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9060
• 卷排序：50

文摘

By comparing Deligne complex and Aeppli–Bott-Chern complex, we construct a differential cohomology \(\widehat{H}^*(X, *, *)\) that plays the role of Harvey–Lawson spark group \(\widehat{H}^*(X, *)\), and a cohomology \(H^*_{\mathrm{ABC}}(X; \mathbb Z(*, *))\) that plays the role of Deligne cohomology \(H^*_{\mathcal {D}}(X; \mathbb Z(*))\) for every complex manifold X. They fit in the short exact sequence $$\begin{aligned} 0\rightarrow H^{k+1}_{\mathrm{ABC}}(X; \mathbb Z(p, q)) \rightarrow \widehat{H}^k(X, p, q) \overset{\delta _1}{\rightarrow } Z^{k+1}_I(X, p, q) \rightarrow 0 \end{aligned}$$and \(\widehat{H}^{\bullet }(X, \bullet , \bullet )\) possess ring structure and refined Chern classes, acted by the complex conjugation, and if some primitive cohomology groups of X vanish, there is a Lefschetz isomorphism. Furthermore, the ring structure of \(H^{\bullet }_{\mathrm{ABC}}(X; \mathbb Z(\bullet , \bullet ))\) inherited from \(\widehat{H}^{\bullet }(X, \bullet , \bullet )\) is compatible with the one of the analytic Deligne cohomology \(H^{\bullet }(X; \mathbb Z(\bullet ))\). We compute \(\widehat{H}^*(X, *, *)\) for X the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.KeywordsDifferential cohomologyDifferential characterAeppli cohomologyBott-Chern cohomologyDeligne cohomologySpark complexRefined Chern classMathematics Subject Classification32C3053C5655N20References1.Angella, D.: The cohomologies of the Iwasawa manifold and of its small deformations. J. Geom. Anal. 23, 1355–1378 (2013)MathSciNetCrossRefMATHGoogle Scholar2.B\(\ddot{a}\)r, C., Becker, C.: Differential characters, lecture notes in mathematics, vol. 2112. Springer, New York (2014)3.Bunke, U.: Differential cohomology (2012). arXiv:1208.3961 4.Bunke, U., Kreck, M., Schick, T.: A geometric description of differential cohomology. Ann. Math. Blaise Pascal 17(1), 1–16 (2010)MathSciNetCrossRefMATHGoogle Scholar5.Bunke, U., Nikolaus, T., Völkl, M.: Differential cohomology theories as sheaves of spectra (2014). arXiv:1311.3188 6.Bunke, U., Schick, T.: Differential K-theory: a survey. Global Differ. Geom., Springer Proc. Math., vol. 17, pp. 303–357. Springer, Heidelberg (2012)7.Cenkl, B., Porter, R.: Cohomology of Nilmanifolds, algebraic topology-rational homotopy. In: Felix, Y. (ed.) Proceedings of a Conference held in Louvain-la-Neuve, LNM1318, pp. 73–86. Springer, New York (1988)8.Cheeger, J., Simons, J.: Differential character and geometric invariants, Geometry and Topology, LNM1167. Springer, New York (1985)MATHGoogle Scholar9.Freed, D.S.: Dirac charge quantization and generalized differential cohomology. Surv. Differ. Geom. VII, 129–194 (2000)10.Hao, N.: Ph.D. Thesis, D-Bar spark sheory and Deligne cohomology, Stony Brook University (2007)11.Harvey, R., Lawson, B., Zweck, J.: The de Rham-Federer theory of differential characters and character duality. Am. J. Math. 125, 791–847 (2003)MathSciNetCrossRefMATHGoogle Scholar12.Harvey, R., Lawson, B.: D-bar sparks. Proc. LMS 3(97), 1–30 (2008)MathSciNetMATHGoogle Scholar13.Harvey, R., Lawson, B.: From sparks to grundles-differential characters. Comm. Anal. Geom. 1(14), 25–58 (2006)MathSciNetCrossRefMATHGoogle Scholar14.Hopkins, M.J., Singer, I.M.: Quadratic functions in geometry, topology, and M-theory. JDG 70(3), 329–452 (2005)MathSciNetMATHGoogle Scholar15.Harvey, F.R., Zweck, J.: Divisors and Euler sparks of atomic sections. Indiana Univ. Math. J. 50, 243–298 (2001)MathSciNetCrossRefMATHGoogle Scholar16.Lambe, L.A., Priddy, S.B.: Cohomology of nilmanifolds and torsion-free, nilpotent groups. Trans. AMS 273(1), 39–55 (1982)MathSciNetCrossRefMATHGoogle Scholar17.Nakamura, I.: Complex parallelisable manifolds and their small deformations. JDG 10(1), 85–112 (1975)MathSciNetMATHGoogle Scholar18.Nomizu, K.: On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. Math. 59(3), 531–538 (1954)MathSciNetCrossRefMATHGoogle Scholar19.Schweitzer, M.: Autour de la cohomologie de Bott-Chern (2007). arXiv:0709.3528 20.Zweck, J.: Stiefel-Whitney sparks. Houst. J. Math. 27(2), 325–351 (2001)MathSciNetMATHGoogle ScholarCopyright information© Springer Science+Business Media Dordrecht 2016Authors and AffiliationsJyh-Haur Teh1Email author1.National Tsing Hua UniversityHsinchuTaiwan About this article CrossMark Print ISSN 0232-704X Online ISSN 1572-9060 Publisher Name Springer Netherlands About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips