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Let $X$ be a compact Kähler manifold, $x \in X$ be a base point and $ρ : π_{1} (X, x) \to G L_{N} (C)$ be the monodromy representation of a $𝒞$ -VHS. Building on Goldman–Millson’s classical work, we construct a mixed Hodge structure on the complete local ring of the representation variety at $ρ$ and a variation of mixed Hodge structures whose monodromy is the universal deformation of $ρ$ .

Variations of moduli of parabolic bundles.

Hans U. Boden, Yi Hu (1995)

Mathematische Annalen

Variations on a theme of groups splitting by a quadratic extension and Grothendieck-Serre conjecture for group schemes $F_{4}$ with trivial $g_{3}$ invariant.

Chernousov, V. (2010)

Documenta Mathematica

Variations on a theme of rationality of cycles

Nikita Karpenko (2013)

Open Mathematics

We prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s...

Variations on a theme of Runge: effective determination of integral points on certain varieties

Aaron Levin (2008)

Journal de Théorie des Nombres de Bordeaux

We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge’s theorem valid for higher-dimensional varieties, generalizing a uniform version of Runge’s theorem due to Bombieri. We then take up the study of how Runge’s method may be expanded by taking advantage of certain coverings. We prove both a result for arbitrary curves and a more explicit result for superelliptic curves. As an application of our...