On the local monodromy of a variation of Hodge structure (original) (raw)
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Acta Appl Math, 2003
We present a survey of the properties of the monodromy of local systems on quasiprojective varieties which underlie a variation of Hodge structure. In the last section, a less widely known version of a Noether-Lefschetz-type theorem is discussed. : 14D07, 32G20.
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.
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, ...
Variation of mixed Hodge structure. I
Inventiones Mathematicae, 1985
A) Singularities of the period mapping (Schmid [9]). As can be seen on each punctured disc separately, {.~P} extends to a filtration {~P} of ~ on S. While this is not so hard to see by other means in the geometric case over a curve, Schmid's proof shows that the filtration of Y/" that has constant value {.@P(0)} with respect to a standard flame for ~/? is itself a variation of Hodge structure on some deleted neighborhood of 0; moreover, it carries over to the case of more variables, i.e., to the general local situation of the normalized problem. In the case of one variable again, the general variation of Hodge structure is asymptotic (in a specified way) to a special (locally homogeneous) one associated to a representation of SL 2. This gives rise to asymptotic formulas for the Hodge norms: ][vH2=fl (Cv,~). In addition, one obtains a clear picture of the interaction between the filtration {~-P(0)} of ~(0) and the weight filtration M, centered at m, (see our (2.1) and (2.4)) of the logarithm N O of the unipotent Jordan factor of the local monodromy transformation (which acts naturally on ~7 ). This is the so-called limit mixed Hodge structure, and N O acts as a morphism of type (-1, -1).
L 2 and intersection cohomologies for a polarizable variation of Hodge structure
Inventiones Mathematicae, 1987
We consider a polarized variation of Hodge structure (V, Vz, S, F) of weight k, over a complex manifold X 1-16]. Here V denotes a locally constant sheaf of finite dimensional complex vector spaces, V z a sheaf of lattices in V, and F a decreasing filtration of Ox | by locally free sheaves of Ox-modules F p. By assumption, the filtration F induces Hodge structure of weight k on the stalks of V, and satisfies the Riemann bilinear relations, as welt as the transversality relation, relative to the fiat bilinear form S. Typically variations of Hodge structure arise from the cohomology of the fibres in a family of smooth projective varieties.
Journal of the American Mathematical Society, 1995
Our notations are spelled out in 2.1. Note that in the Hodge decomposition, we do not assume p, q ≥ 0. If H is a Hodge structure of even weight 2p, shifting the Hodge filtration, i.e., replacing H by the Tate twist H(p), one obtains a Hodge structure of weight 0. Classes of type (p, p) become of type (0, 0). This allows us to restrict our attention to variations of weight 0 and classes of type (0, 0). For simplicity, we will assume the parameter space S to be non singular.