With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory ... more With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory and polarization, Deligne's Mixed Hodge Structure on cohomology of complex algebraic varieties is described.
Let (X, 0) be a germ of an analytic space and f the germ of an analytic function on X. We show th... more Let (X, 0) be a germ of an analytic space and f the germ of an analytic function on X. We show that the polar filtration of the local Milnor fiber, defined by a projection on a complex disc, is diffeomorphic to a valuative filtration of the Milnor fiber called Hironaka's filtration, a result which links invariants associated with the singularities of the projection to those associated with a desingularisation.
Suppose that the critical locus Σ of a complex analytic function f on affine space is, itself, a ... more Suppose that the critical locus Σ of a complex analytic function f on affine space is, itself, a space with an isolated singular point at the origin 0, and that the Milnor number of f restricted to normal slices of Σ -{0} is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of f at 0, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.
Une nouvelle démonstration du théorème de décomposition est donnée, en établissant une relation a... more Une nouvelle démonstration du théorème de décomposition est donnée, en établissant une relation avec une version du théorème de pureté locale de Deligne et Gabber adaptée aux variétés algébriques complexes.
In this paper we give a detailed proof that the Milnor fiber X t of an analytic complex isolated ... more In this paper we give a detailed proof that the Milnor fiber X t of an analytic complex isolated singularity function defined on a reduced n-equidimensional analytic complex space X is a regular neighborhood of a polyhedron P t ⊂ X t of real dimension n -1. Moreover, we describe the degeneration of X t onto the special fiber X 0 , by giving a continuous collapsing map Ψ t : X t → X 0 which sends P t to {0} and which restricts to a homeomorphism X t \P t → X 0 \{0}.
We show that given any germ of complex analytic function f : (X, x) → (C, 0) on a complex analyti... more We show that given any germ of complex analytic function f : (X, x) → (C, 0) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that f ∈ m 2 X,x , where mX,x is the maximal ideal of OX,x. This result generalizes a wellknown theorem of the second named author when X is smooth at x and it also implies the A'Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that X has maximal rectified homotopical depth at x and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.
Abstract: A complex variety has two intrinsic metric space structures in neighborhood of any poin... more Abstract: A complex variety has two intrinsic metric space structures in neighborhood of any point (" inner" and" outer" metric) which are uniquely determined from the complex structure up to bilipschitz change of the metric (changing distances by at most a constant factor). In dimension $1 $ the inner metric (given by minimal arclength within the variety) carries no interesting information, and it is only very recently, starting with a 2008 paper [1] of Birbrair and Fernandes, that it has become clear how rich metric information is in higher ...
<p>This book provides a comprehensive and up-to-date introduction to Hodge theory—one of th... more <p>This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.</p>
In this paper we define the notion of relative rectified homotopical depth for complex analytic m... more In this paper we define the notion of relative rectified homotopical depth for complex analytic morphisms which generalizes the rectified homotopical depth introduced by Grothendieck in [G, Sect. 6, Chap. XIII] (see also Sect. 1]) in the absolute case of complex analytic spaces. As shown in I-H-L3] and conjectured by Grothend~eck (loc.cit.), in theorems analogous to the theorem of Lefschetz on hyperplane sections, the rectified homotopical depth gives an estimate for the level of coincidence of the homotopy of an algebraic variety and one of its hyperplane sections when there are singularities. Similarly we obtain relative Lefschetz type theorems for morphisms where the relative rectified homotopical depth gives also an estimate for coincidence of homotopy. Following the procedure of [H-L3], we show that an adequate stratification of the morphism gives a way to calculate the relative rectified homotopical depth. On the other hand, as corollaries of our main result (see Theorem 2.1.3 below), we obtain a generalization of a conjecture of Deligne ID, Conjecture 1.3] already proven by Goresky and MacPherson in [G-M, II, Sect. 1.1, Sect. 5.1] and theorems of Lefschetz type on a singular space obtained by us in [H-L3] or in a different form by Goresky and MacPherson [G-M, II, Sect. 1.2, Sect. 5.2). 1 Relative rectified homotopieal depth 1.1. First recall the definition of the homotopical depth of X along a locally closed complex analytic subspace Y at a point x ~ Y [G
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1998
We give vanishing theorems for the hypercohomology of complexes of algebraically constructible sh... more We give vanishing theorems for the hypercohomology of complexes of algebraically constructible sheaves on a complex algebraic variety. The proofs of these theorems in this situation are easier because algebraic morphisms are compactifiable.
With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory ... more With a basic knowledge of cohomology theory, the background necessary to understand Hodge theory and polarization, Deligne's Mixed Hodge Structure on cohomology of complex algebraic varieties is described.
Let (X, 0) be a germ of an analytic space and f the germ of an analytic function on X. We show th... more Let (X, 0) be a germ of an analytic space and f the germ of an analytic function on X. We show that the polar filtration of the local Milnor fiber, defined by a projection on a complex disc, is diffeomorphic to a valuative filtration of the Milnor fiber called Hironaka's filtration, a result which links invariants associated with the singularities of the projection to those associated with a desingularisation.
Suppose that the critical locus Σ of a complex analytic function f on affine space is, itself, a ... more Suppose that the critical locus Σ of a complex analytic function f on affine space is, itself, a space with an isolated singular point at the origin 0, and that the Milnor number of f restricted to normal slices of Σ -{0} is constant. Then, the general theory of perverse sheaves puts severe restrictions on the cohomology of the Milnor fiber of f at 0, and even more surprising restrictions on the cohomology of the Milnor fiber of generic hyperplane slices.
Une nouvelle démonstration du théorème de décomposition est donnée, en établissant une relation a... more Une nouvelle démonstration du théorème de décomposition est donnée, en établissant une relation avec une version du théorème de pureté locale de Deligne et Gabber adaptée aux variétés algébriques complexes.
In this paper we give a detailed proof that the Milnor fiber X t of an analytic complex isolated ... more In this paper we give a detailed proof that the Milnor fiber X t of an analytic complex isolated singularity function defined on a reduced n-equidimensional analytic complex space X is a regular neighborhood of a polyhedron P t ⊂ X t of real dimension n -1. Moreover, we describe the degeneration of X t onto the special fiber X 0 , by giving a continuous collapsing map Ψ t : X t → X 0 which sends P t to {0} and which restricts to a homeomorphism X t \P t → X 0 \{0}.
We show that given any germ of complex analytic function f : (X, x) → (C, 0) on a complex analyti... more We show that given any germ of complex analytic function f : (X, x) → (C, 0) on a complex analytic space X, there exists a geometric local monodromy without fixed points, provided that f ∈ m 2 X,x , where mX,x is the maximal ideal of OX,x. This result generalizes a wellknown theorem of the second named author when X is smooth at x and it also implies the A'Campo theorem that the Lefschetz number of the monodromy is equal to zero. Moreover, we give an application to the case that X has maximal rectified homotopical depth at x and show that a family of such functions with isolated critical points and constant total Milnor number has no coalescing of singularities.
Abstract: A complex variety has two intrinsic metric space structures in neighborhood of any poin... more Abstract: A complex variety has two intrinsic metric space structures in neighborhood of any point (&amp;quot; inner&amp;quot; and&amp;quot; outer&amp;quot; metric) which are uniquely determined from the complex structure up to bilipschitz change of the metric (changing distances by at most a constant factor). In dimension $1 $ the inner metric (given by minimal arclength within the variety) carries no interesting information, and it is only very recently, starting with a 2008 paper [1] of Birbrair and Fernandes, that it has become clear how rich metric information is in higher ...
<p>This book provides a comprehensive and up-to-date introduction to Hodge theory—one of th... more <p>This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.</p>
In this paper we define the notion of relative rectified homotopical depth for complex analytic m... more In this paper we define the notion of relative rectified homotopical depth for complex analytic morphisms which generalizes the rectified homotopical depth introduced by Grothendieck in [G, Sect. 6, Chap. XIII] (see also Sect. 1]) in the absolute case of complex analytic spaces. As shown in I-H-L3] and conjectured by Grothend~eck (loc.cit.), in theorems analogous to the theorem of Lefschetz on hyperplane sections, the rectified homotopical depth gives an estimate for the level of coincidence of the homotopy of an algebraic variety and one of its hyperplane sections when there are singularities. Similarly we obtain relative Lefschetz type theorems for morphisms where the relative rectified homotopical depth gives also an estimate for coincidence of homotopy. Following the procedure of [H-L3], we show that an adequate stratification of the morphism gives a way to calculate the relative rectified homotopical depth. On the other hand, as corollaries of our main result (see Theorem 2.1.3 below), we obtain a generalization of a conjecture of Deligne ID, Conjecture 1.3] already proven by Goresky and MacPherson in [G-M, II, Sect. 1.1, Sect. 5.1] and theorems of Lefschetz type on a singular space obtained by us in [H-L3] or in a different form by Goresky and MacPherson [G-M, II, Sect. 1.2, Sect. 5.2). 1 Relative rectified homotopieal depth 1.1. First recall the definition of the homotopical depth of X along a locally closed complex analytic subspace Y at a point x ~ Y [G
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1998
We give vanishing theorems for the hypercohomology of complexes of algebraically constructible sh... more We give vanishing theorems for the hypercohomology of complexes of algebraically constructible sheaves on a complex algebraic variety. The proofs of these theorems in this situation are easier because algebraic morphisms are compactifiable.
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