*Simon Scott*

- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780198568360
- eISBN:
- 9780191594748
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568360.003.0006
- Subject:
- Mathematics, Analysis

The final chapter focuses on applications of trace and determinant structures to geometric families of pseudodifferential operators. This is closely entwined with developments in QFT and string ...
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The final chapter focuses on applications of trace and determinant structures to geometric families of pseudodifferential operators. This is closely entwined with developments in QFT and string theory. The first section reviews the construction of families of pseudodifferential operators parameterized by a fibration. A generalized trace on complex powers of geometric families is used to construct zeta trace forms. Specifically,a natural logarithm map from geometric families to the de Rham algebra on the parameter manifold is constructed whose superconnection character form is a canonical representative for the Chern class. In cohomology this is a canonical form representative for the index bundle. The final part of the chapter gives a of the determinant line bundle endowed with its zeta function metric,. The chapter closes with a detailed computation of the zeta metric for the case of a family of Cauchy Riemann operators on a surface.Less

The final chapter focuses on applications of trace and determinant structures to geometric families of pseudodifferential operators. This is closely entwined with developments in QFT and string theory. The first section reviews the construction of families of pseudodifferential operators parameterized by a fibration. A generalized trace on complex powers of geometric families is used to construct zeta trace forms. Specifically,a natural logarithm map from geometric families to the de Rham algebra on the parameter manifold is constructed whose superconnection character form is a canonical representative for the Chern class. In cohomology this is a canonical form representative for the index bundle. The final part of the chapter gives a of the determinant line bundle endowed with its zeta function metric,. The chapter closes with a detailed computation of the zeta metric for the case of a family of Cauchy Riemann operators on a surface.

*Charles P. Boyer and Krzysztof Galicki*

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0004
- Subject:
- Mathematics, Geometry / Topology

This chapter reviews some basic facts about Kähler manifolds with special emphasis on projective algebraic varieties. All standard material is covered: complex structures, curvature properties, Hodge ...
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This chapter reviews some basic facts about Kähler manifolds with special emphasis on projective algebraic varieties. All standard material is covered: complex structures, curvature properties, Hodge theory, Chern classes, positivity and Fano varieties, line bundles and divisors. Of particular interest is Yau's famous proof of the Calabi conjecture which ends this chapter.Less

This chapter reviews some basic facts about Kähler manifolds with special emphasis on projective algebraic varieties. All standard material is covered: complex structures, curvature properties, Hodge theory, Chern classes, positivity and Fano varieties, line bundles and divisors. Of particular interest is Yau's famous proof of the Calabi conjecture which ends this chapter.

*Paula Tretkoff*

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number ...
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This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.Less

This chapter deals with topological invariants and differential geometry. It first considers a topological space *X* for which singular homology and cohomology are defined, along with the Euler number *e*(*X*). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space *X*. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold *X*.