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HODGE GENERA OF ALGEBRAIC VARIETIES, I
, 2006
"... The aim of this paper is to study the behavior of intersection homology genera and their associated characteristic classes under morphisms of projective algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a projective variety X to such invariants of si ..."
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Cited by 24 (13 self)
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The aim of this paper is to study the behavior of intersection homology genera and their associated characteristic classes under morphisms of projective algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a projective variety X to such invariants of singularities of proper algebraic maps defined on X. Such formulae severely constrain, both topologically and analytically, the singularities of complex maps, even between smooth varieties. Similar results were announced by the first and third author in [10, 22].
Hodge genera and characteristic classes of complex algebraic varieties
"... Abstract. We announce Hodge theoretic formulae of AtiyahMeyer type for genera and characteristic classes of complex algebraic varieties. Our results are formulated in terms of the generalized (motivic) Hirzebruch characteristic classes, and the arguments used in the proofs rely in an essential way ..."
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Cited by 14 (8 self)
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Abstract. We announce Hodge theoretic formulae of AtiyahMeyer type for genera and characteristic classes of complex algebraic varieties. Our results are formulated in terms of the generalized (motivic) Hirzebruch characteristic classes, and the arguments used in the proofs rely in an essential way on Saito’s theory of algebraic mixed Hodge modules. 1.
Hodgetheoretic AtiyahMeyer formulae and the stratified multiplicative property
, 2008
"... In this note we survey Hodgetheoretic formulae of AtiyahMeyer type for genera and characteristic classes of complex algebraic varieties, and derive some new and interesting applications. We also present various extensions to the singular setting of the ChernHirzebruchSerre signature formula. ..."
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Cited by 12 (8 self)
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In this note we survey Hodgetheoretic formulae of AtiyahMeyer type for genera and characteristic classes of complex algebraic varieties, and derive some new and interesting applications. We also present various extensions to the singular setting of the ChernHirzebruchSerre signature formula.
Characteristic classes of mixed Hodge modules
"... Abstract. This paper is an extended version of an expository talk given at the workshop “Topology of stratified spaces ” at MSRI Berkeley in September 2008. It gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge th ..."
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Cited by 6 (4 self)
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Abstract. This paper is an extended version of an expository talk given at the workshop “Topology of stratified spaces ” at MSRI Berkeley in September 2008. It gives an introduction and overview about recent developments on the interaction of the theories of characteristic classes and mixed Hodge theory for singular spaces in the complex algebraic context. It uses M. Saito’s deep theory of mixed Hodge modules as a“black box”, thinking about them as “constructible or perverse sheaves of Hodge structures”, having the same functorial calculus of Grothendieck functors. For the “constant Hodge sheaf”, one gets the “motivic characteristic classes ” of BrasseletSchürmannYokura, whereas the classes of the “intersection homology Hodge sheaf ” were studied by CappellMaximShaneson. The classes associated to “good ” variation of mixed Hodge structures where studied in connection with understanding the monodromy action by CappellLibgoberMaximShaneson and the author. There are two versions of these characteristic classes. The Ktheoretical classes capture information about the graded pieces of the filtered de Rham complex of the filtered Dmodule underlying a mixed Hodge module. Application of a suitable Todd class transformation then gives classes in homology. These classes are functorial for proper
Motivic Milnor classes
, 2009
"... The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern–Schwartz–MacPherson’s class and Fulton–Johnson’s canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a “motivic ” Grothendieck group ..."
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Cited by 6 (2 self)
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The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern–Schwartz–MacPherson’s class and Fulton–Johnson’s canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a “motivic ” Grothendieck group K Prop ℓ.c.i (V/X h − → S) and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch–Milnor class, as a special value of a distinguished element under these natural transformations. We also show a Verdiertype Riemann–Roch formula for our motivic HirzebruchMilnor class. We use Fulton–MacPherson’s bivariant theory and the motivic Hirzebruch class.
The Decomposition Theorem and the topology of algebraic maps
, 2007
"... We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of alg ..."
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Cited by 5 (1 self)
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We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the Decomposition Theorem, indicate some important applications and examples.
Equivariant genera of complex algebraic varieties
 Int. Math. Res. Notices
"... Abstract. For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. In this note we obtain AtiyahMeyer type formulae for equivariant Hodgetheoretic genera of complex algebraic varieties. Equivariant Hirzebruch genera χy(X; g) of a quasiprojective variety X act ..."
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Cited by 4 (3 self)
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Abstract. For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. In this note we obtain AtiyahMeyer type formulae for equivariant Hodgetheoretic genera of complex algebraic varieties. Equivariant Hirzebruch genera χy(X; g) of a quasiprojective variety X acted upon by a finite group of algebraic automorphisms are defined by combining the group action with the information encoded by the Hodge filtration of the mixed Hodge structure in cohomology. While for a projective algebraic manifold χy(X; g) can by computed by the AtiyahSinger holomorphic Lefschetz theorem, we derive a AtiyahMeyer type formula for χy(X; g) in the case when X is not necessarily smooth or compact, but just fibers equivariantly (in the complex topology) over a compact algebraic manifold. These results apply to computing Hodgetheoretic invariants of orbit spaces. We also obtain some results comparing equivariant Hodgetheoretic genera of the range and domain of an equivariant algebraic map in terms of its singularities. 1.