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The Hodge ring of Kähler manifolds

Part of: Surfaces and higher-dimensional varieties Complex manifolds Differential topology Birational geometry Cycles and subschemes

Published online by Cambridge University Press:  28 February 2013

D. Kotschick  and
S. Schreieder
D. Kotschick
Affiliation:
Mathematisches Institut, LMU München, Theresienstr. 39, 80333 München, Germany email [email protected]
S. Schreieder
Affiliation:
Mathematisches Institut, LMU München, Theresienstr. 39, 80333 München, Germany email [email protected] Trinity College, Cambridge, CB2 1TQ, UK email [email protected]
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Abstract

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We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kähler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kähler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruch’s.

Keywords

Hodge numbersChern numbersHirzebruch problem

MSC classification

Secondary: 32Q15: Kähler manifolds 14C30: Transcendental methods, Hodge theory , Hodge conjecture 14E99: None of the above, but in this section 14J80: Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) 57R77: Complex cobordism (${rm U}$- and ${rm SU}$-cobordism)
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 4 , April 2013 , pp. 637 - 657
Copyright
© The Author(s) 2013

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