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In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer [6], and by Wall and Mattei [11, 13]., The first and third authors were supported by Spanish Ministerio de Ciencia, Innovación y Universidades MTM2015-69135-P, Generalitat de Catalunya 2017SGR-932 projects, and they are with the Barcelona Graduate School of Mathematics (BGSMath), through the project MDM-2014-0445. The second and fourth authors were supported by Spanish Ministerio de Ciencia, Innovación y Universidades MTM2016-76868-C2-1-P

We prove a filtered version of the Homotopy Transfer Theorem which gives an A A -infinity algebra structure on any page of the spectral sequence associated to a filtered dg-algebra. We then develop various applications to the study of the geometry and topology of complex manifolds, using the Hodge filtration, as well as to complex algebraic varieties, using mixed Hodge theory.

Many papers are devoted to study logarithmic sheaves associated to reduced divisors, in particular logarithmic bundles associated to plane curves since forty years in differential and algebraic topology or geometry. An interesting family of these curves are the so-called free ones for which the associated logarithmic sheaf is the direct sum of two line bundles. When the curve is a finite set of distinct lines (i.e. a line arrangement), Terao conjectured thirty years ago that its freeness depends only on its combinatorics. A lot of efforts were done to prove it but at this time it is only proved up to 12 lines. If one wants to find a counter example to this conjecture a new family of curves arises naturally: the nearly free curves introduced by Dimca and Sticlaru. We prove here that the logarithmic bundle associated to a nearly free curve possesses a minimal non zero section that vanishes on one single point $P$, called jumping point, and that characterizes the bundle. Then we give a precise description of the behaviour of $P$. In particular we show, based on detailed examples, that the position of $P$ relatively to its corresponding nearly free arrangement of lines may or may not be a combinatorial invariant, depending on the chosen combinatorics., 24 pages, Comments are welcome. We improve the exposition of the paper with more details

We extend Deligne's weight filtration to the integer cohomology of complex analytic spaces (endowed with an equivalence class of compactifications). In general, the weight filtration that we obtain is not part of a mixed Hodge structure. Our purely geometric proof is based on cubical descent for resolution of singularities and Poincar\'e-Verdier duality. Using similar techniques, we introduce the singularity filtration on the cohomology of compactificable analytic spaces. This is a new and natural analytic invariant which does not depend on the equivalence class of compactifications and is related to the weight filtration., Comment: examples added + minor corrections

Introduced by B. Totaro, the weight filtration on the homology of real algebraic varieties, which is a real analog to P. Deligne's weight filtration for complex algebraic varieties, has been realized via a filtered chain complex by C. McCrory and A. Parusinski, especially through the study of the induced spectral sequence. Among the several pieces of information held by this weight spectral sequence, one can recover the virtual Betti numbers. In this thesis, we show the existence of an equivariant weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action. We realize it by a filtered complex and, via the construction of several spectral sequences, we make significative progress toward the extraction of additive invariants. During our study, we functorially define a weight complex with action and we show an analog of the Smith exact sequence, taking into account the Nash-constructible filtration, that follows from a result on the splitting of Nash manifolds with algebraic involutions. Through the construction of an invariant weight complex in the frame of algebraic involutions, we also recover G. Fichou's equivariant virtual Betti numbers. Finally, applying the relevant functors on T. Limoges' results on the products of real weight filtrations, we give results on the products of equivariant weight filtrations.; Introduite par B. Totaro, la filtration par le poids sur l'homologie des variétés algébriques réelles, analogue réel de la filtration par le poids de P. Deligne sur les variétés algébriques complexes, a été réalisée via un complexe de chaînes filtré par C. McCrory et A. Parusinski, qui en ont enrichi la compréhension, notamment à travers l'étude de la suite spectrale induite. Au milieu des nombreuses informations recelées par cette suite spectrale de poids, on retrouve les nombres de Betti virtuels. Dans cette thèse, on montre l'existence d'une filtration par le poids équivariante sur l'homologie équivariante des variétés algébriques réelles munies d'une action d'un groupe fini. On la réalise par un complexe filtré et, via la construction de plusieurs suites spectrales, on effectue des avancées significatives pour extraire des invariants additifs. Lors de notre étude, on définit fonctoriellement un complexe de poids avec action et on montre qu'un résultat de découpage d'une variété Nash munie d'une involution algébrique entraîne un analogue de la suite exacte de Smith, tenant compte de la filtration Nash-constructible. A travers la construction d'un complexe de poids invariant dans le cadre d'involutions algébriques, on retrouve également les nombres de Betti virtuels équivariants de G. Fichou. Enfin, en appliquant les bons foncteurs aux résultats sur les produits de filtrations par le poids réelles de T. Limoges, on donne des résultats sur les produits de filtrations par le poids équivariantes.

Let X be a smooth complex algebraic variety. Morgan showed that the rational homotopy type of X is a formal consequence of the differential graded algebra defined by the first term E1.X;W / of its weight spectral sequence. In the present work, we generalize this result to arbitrary nilpotent complex algebraic varieties (possibly singular and/or non-compact) and to algebraic morphisms between them. In particular, our results generalize the formality theorem of Deligne, Griffiths, Morgan and Sullivan for morphisms of compact Kahler varieties, filling a gap in Morgan’s theory concerning functoriality over the rationals. As an application, we study the Hopf invariant of certain algebraic morphisms using intersection theory. 32S35, 55P62