Uniqueness of the complex structure on Kähler manifolds of certain homotopy types

Libgober, Anatoly; Wood, John W.

doi:10.4310/jdg/1214445041

Cited by 70 publications

(85 citation statements)

References 26 publications

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“…There is a different way to describe this calculation of the image of the χ y genus on SU -manifolds, rationally: the only linear relations satisfied by the χ y genus of an SU -manifold are those coming from Serre duality together with the one other relation found by Libgober and Wood [22]. From Serre duality, the χ y genus of an SU -manifold of complex dimension n is a polynomial χ(y) of degree n such that χ(1/y) = (−1/y) n χ(y).…”

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confidence: 99%

Chern Numbers for Singular Varieties and Elliptic Homology

Totaro¹

2000

The Annals of Mathematics

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A fundamental goal of algebraic geometry is to do for singular varieties whatever we can do for smooth ones. Intersection homology, for example, directly produces groups associated to any variety which have almost all the properties of the usual homology groups of a smooth variety. Minimal model theory suggests the possibility of working more indirectly by relating any singular variety to a variety which is smooth or nearly so.Here we use ideas from minimal model theory to define some characteristic numbers for singular varieties, generalizing the Chern numbers of a smooth variety. This was suggested by Goresky and MacPherson as a next natural problem after the definition of intersection homology [11]. We find that only a subspace of the Chern numbers can be defined for singular varieties. A convenient way to describe this subspace is to say that a smooth variety has a fundamental class in complex bordism, whereas a singular variety can at most have a fundamental class in a weaker homology theory, elliptic homology. We use this idea to give an algebro-geometric definition of elliptic homology: "complex bordism modulo flops equals elliptic homology."This paper was inspired by some questions asked by Jack Morava. The descriptions of elliptic homology given by Gerald Höhn [13] were also an important influence. Thanks to Dave Bayer, Mike Stillman, John Stembridge, Sheldon Katz, and Stein Stromme for their computer algebra programs Macaulay, SF, and Schubert, which helped in guessing the right answer. Contents 1. Statements 2. Weakly complex manifolds 3. The complex elliptic genus 4. Complex bordism modulo flops 5. Complex bordism modulo flops, continued 6. SU-bordism modulo flops 7. Saito's homology classes χ n−k i 8. The Chern numbers c k 1 χ n−k i 9. The twisted χ y genus

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confidence: 99%

Chern Numbers for Singular Varieties and Elliptic Homology

Totaro¹

2000

The Annals of Mathematics

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“…It was discovered by Libgober and Wood that the product of the Chern classes c 1 (X)c n−1 (X) depends only on the Hodge numbers of X [11]. This result has been used by Eguchi, Jinzenji and Xiong in their approach to the quantum cohomology of X via a representation of the Virasoro algebra with the central charge c n (X) [8,9].…”

Section: Introductionmentioning

confidence: 99%

Stringy Hodge numbers and Virasoro algebra

Batyrev

2000

Mathematical Research Letters

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Abstract. In this paper we define for singular varieties X a rational number c 1,n−1 st (X) which is a stringy version of the product of Chern numbers c 1 and c n−1 We show that the number c 1,n−1 st (X) can be expressed via stringy Hodge numbers of singular X in the same way as c 1 c n−1 expresses via usual Hodge numbers for smooth manifolds. Our result provides some evidences for the existence of quantum cohomology theory of singular varieties X based on representation of the Virasoro algebra whose central charge is the rational number e st (X) which equals the stringy Euler number of X.

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“…This is a well-known open problem (see e.g. [10]), and it is known that if this is true when n = 3 then there is no complex manifold diffeomorphic to S 6 (another famous open problem, see e.g. [9]): Proposition 3.1.…”

Section: Closing Remarksmentioning

confidence: 99%

“…As a partial generalization of Theorems 1.1 and 1.2, Libgober-Wood [10] proved that a compact Kähler manifold of complex dimension n ≤ 6 which is homotopy equivalent to CP n must be biholomorphic to it. A natural question is whether the Kähler hypothesis is really necessary in Theorem 1.1, and so one can ask whether a compact complex manifold diffeomorphic to CP n must be biholomorphic to it.…”

Section: Closing Remarksmentioning

confidence: 99%