Acta Applicandae Mathematicae 75: 183–194, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. 183 Monodromy of Variations of Hodge Structure (original) (raw)
A result about Picard-Lefschetz monodromy
2006
Let fff and ggg be reduced homogeneous polynomials in separate sets of variables. We establish a simple formula that relates the eigenspace decomposition of the monodromy operator on the Milnor fiber cohomology of fgfgfg to that of fff and ggg separately. We use a relation between local systems and Milnor fiber cohomology that has been established by D. Cohen and A. Suciu.
Polinomios de Hodge de variedades de caracteres
2016
La tesis Polinomios de Hodge de variedades de caracteres se dedica al estudio de una clase de invariantes algebraicos llamados polinomios de Hodge o Epolinomios, que pueden asociarse a cualquier variedad quasiproyectiva X. Son de naturaleza cohomologica y contienen informacion topologica, geometrica y aritmetica. La tesis aborda el analisis de estos polinomios para un tipo particular de variedades algebraicas llamadas variedades de caracteres, que ocupan un lugar destacado en muchas ramas de la matematica: aparecen en dinamica, teoria de representaciones, geometria algebraica y diferencial. Dado un grupo finitamente presentado F y un grupo algebraico reductivo G, las variedades de caracteres se definen como el espacio de moduli de las representaciones de F en G, donde se cocienta por la accion de G por conjugacion y donde se utiliza la Teoria de Invariantes Geometricos en la construccion del moduli. Un caso de particular relevancia es cuando F es el grupo fundamental de una curva li...
Ju n 20 06 HODGE GENERA OF ALGEBRAIC VARIETIES
2006
The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic 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 [11, 28].
Monodromy of a family of hypersurfaces
2008
Let YYY be an (m+1)(m+1)(m+1)-dimensional irreducible smooth complex projective variety embedded in a projective space. Let ZZZ be a closed subscheme of YYY, and delta\deltadelta be a positive integer such that mathcalIZ,Y(delta)\mathcal I_{Z,Y}(\delta)mathcalIZ,Y(delta) is generated by global sections. Fix an integer dgeqdelta+1d\geq \delta +1dgeqdelta+1, and assume the general divisor Xin∣H0(Y,icZ,Y(d))∣X \in |H^0(Y,\ic_{Z,Y}(d))|Xin∣H0(Y,icZ,Y(d))∣ is smooth. Denote by Hm(X;mathbbQ)perpZtextvanH^m(X;\mathbb Q)_{\perp Z}^{\text{van}}Hm(X;mathbbQ)perpZtextvan the quotient of Hm(X;mathbbQ)H^m(X;\mathbb Q)Hm(X;mathbbQ) by the cohomology of YYY and also by the cycle classes of the irreducible components of dimension mmm of ZZZ. In the present paper we prove that the monodromy representation on Hm(X;mathbbQ)perpZtextvanH^m(X;\mathbb Q)_{\perp Z}^{\text{van}}Hm(X;mathbbQ)perpZtextvan for the family of smooth divisors Xin∣H0(Y,icZ,Y(d))∣X \in |H^0(Y,\ic_{Z,Y}(d))|Xin∣H0(Y,icZ,Y(d))∣ is irreducible.
Hodge genera of algebraic varieties, I
2007
The aim of this paper is to study the behavior of Hodge-theoretic (intersection homology) genera and their associated characteristic classes under proper morphisms of complex algebraic varieties. We obtain formulae that relate (parametrized families of) global invariants of a complex algebraic 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 CS1, S.
Singularities of variations of mixed Hodge structure
Asian Journal of Mathematics, 2003
We give a condition for a variation of mixed Hodge structure on a curve to be admissible. It involves the asymptotic behavior of a grading of the weight filtration, supplementing exactly the description of the graded variation and its monodromy given by Schmid's Orbit Theorems. In many salient cases the condition is equivalent to admissibility.
Karl Weierstraß (1815–1897), 2015
We discuss the history of the monodromy theorem, starting from Weierstraß, and the concept of monodromy group. From this viewpoint we compare then the Weierstraß, the Legendre and other normal forms for elliptic curves, explaining their geometric meaning and distinguishing them by their stabilizer in PSL(2, Z) and their monodromy. Then we focus on the birth of the concept of the Jacobian variety, and the geometrization of the theory of Abelian functions and integrals. We end illustrating the methods of complex analysis in the simplest issue, the difference equation f (z) = g(z + 1)g(z) on C.
2012
ρ: Γ → GLN(C) be a finite dimensional semisimple representation. We assume ρ to be the monodromy of a given polarized C-VHS (Vρ, F • , G • , S) whose weight is zero. If ρ is not irreducible then several distinct polarizations could be chosen, we fix one once for all. In the introduction, we fix an isomorphism Vρ,x → C n. Then, the Zariski closure of its monodromy group is a reductive subgroup G ⊂ GLN. Let R(Γ, GLN) be the variety of its representations in GLN [LuMa85]. R(Γ, GLN) may be viewed as an affine scheme over Z but we will only consider it as an affine scheme over C. The group GLN acts algebraically on R(Γ, GLN) by conjugation and we denote by Ωρ the orbit of ρ. It is a closed smooth algebraic subvariety and we will consider it as a subscheme of R(Γ, GLN) endowing it with its reduced induced structure. Denote by R(Γ, GLN)ρ the formal local scheme which is the germ at [ρ] of R(Γ, GLN). Similarly, denote by ˆ Ωρ the germ of Ωρ at [ρ]. ˆ Ωρ is a closed formal subscheme of R(Γ, ...
Hodge theory of maps: Lectures 1-3
2010
These three lectures summarize classical results of Hodge theory concerning algebraic maps, and presumably contain much more material than I'll be able to cover. Lectures 4 and 5, to be delivered by M. A. de Cataldo, will discuss more recent results. I will not try to trace the history of the subject nor attribute the results discussed. Coherently with this policy, the bibliography only contains textbooks and a survey, and no original paper. Furthermore, quite often the results will not be presented in their maximal generality; in particular I'll alway stick to projective maps, even though some results discussed hold more generally. Contents 1 Introduction. 2 2 The smooth case: E 2 -degeneration 4 3 Mixed Hodge structures. 8 3.1 Mixed Hodge structures on the cohomology of algebraic varieties 8 3.2 The global invariant cycle theorem .