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**Chern numbers for singular varieties and elliptic homology.**
*(English)*
Zbl 1050.14500

From the paper: 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.

The author uses 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 M. Goresky and R. MacPherson as a next natural problem after the definition of intersection homology. The author finds 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 presents two main results. First (theorem 4.1), a rational linear combination of Chern numbers, viewed as an invariant of compact complex manifolds, is unchanged under “classical flops” if and only if it is a linear combination of the coefficients of the complex elliptic genus studied by I. M. Krichever [Math. Notes 47, No. 2, 132–142 (1990); translation from Mat. Zametki 47, No. 2, 34–45 (1990; Zbl 0702.57006)] and G. Höhn. This elliptic genus can be viewed as a power series associated to any compact complex manifold, the coefficients of the series being certain fixed linear combinations of the Chern numbers of the manifold. A more precise form of this result determines a geometrically meaningful version of complex elliptic homology over the ring \(\mathbb{Z}[1/2]\) (theorem 6.1, remark 1).

Second, we can ask when a given rational linear combination of Chern numbers, viewed as an invariant of smooth compact varieties, can be extended to an invariant of singular varieties, subject to a natural condition (compatibility with “IH-small resolutions”). This compatibility condition implies that the given linear combination of Chern numbers is invariant under classical flops; so every such linear combination of Chern numbers is a linear combination of the coefficients of the elliptic genus. We conjecture that, conversely, the elliptic genus can be defined for arbitrary singular varieties (compatibly with IH-small resolutions). The second main result of this paper is that at least a certain weaker invariant, which is called the twisted \(\chi_y\) genus, can be defined for arbitrary singular varieties (compatibly with IH-small resolutions): see theorems 8.1 and 8.2

The author uses 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 M. Goresky and R. MacPherson as a next natural problem after the definition of intersection homology. The author finds 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 presents two main results. First (theorem 4.1), a rational linear combination of Chern numbers, viewed as an invariant of compact complex manifolds, is unchanged under “classical flops” if and only if it is a linear combination of the coefficients of the complex elliptic genus studied by I. M. Krichever [Math. Notes 47, No. 2, 132–142 (1990); translation from Mat. Zametki 47, No. 2, 34–45 (1990; Zbl 0702.57006)] and G. Höhn. This elliptic genus can be viewed as a power series associated to any compact complex manifold, the coefficients of the series being certain fixed linear combinations of the Chern numbers of the manifold. A more precise form of this result determines a geometrically meaningful version of complex elliptic homology over the ring \(\mathbb{Z}[1/2]\) (theorem 6.1, remark 1).

Second, we can ask when a given rational linear combination of Chern numbers, viewed as an invariant of smooth compact varieties, can be extended to an invariant of singular varieties, subject to a natural condition (compatibility with “IH-small resolutions”). This compatibility condition implies that the given linear combination of Chern numbers is invariant under classical flops; so every such linear combination of Chern numbers is a linear combination of the coefficients of the elliptic genus. We conjecture that, conversely, the elliptic genus can be defined for arbitrary singular varieties (compatibly with IH-small resolutions). The second main result of this paper is that at least a certain weaker invariant, which is called the twisted \(\chi_y\) genus, can be defined for arbitrary singular varieties (compatibly with IH-small resolutions): see theorems 8.1 and 8.2

### MSC:

14C17 | Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry |

14B05 | Singularities in algebraic geometry |

55N34 | Elliptic cohomology |

14E30 | Minimal model program (Mori theory, extremal rays) |

57R20 | Characteristic classes and numbers in differential topology |

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

58J26 | Elliptic genera |

11F50 | Jacobi forms |

32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |