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There have been several constructions of family of varieties with exceptional monodromy group. In most cases, these constructions give Hodge structures with high weight(Hodge numbers spread out). N. Katz was the first to obtain Hodge structures with low weight(Hodge numbers equal to (2,3,2)) and geometric monodromy group G2. In this article I will give an alternative description of Katzs construction and give an extension of his result.
We recast elliptic surfaces over the projective line in terms of the non-commutative tori with real multiplication. The correspondence is used to study the Picard numbers, the ranks and the minimal models of such surfaces. As an example, we calculate
Transcendental Brauer elements are notoriously difficult to compute. Work of Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion transcendental classes on surfaces that have a genus 1 fibration with rational 2-torsion in the Jaco
Let $S$ be a complex smooth projective surface of Kodaira dimension one. We show that the group $mathrm{Aut}_s(S)$ of symplectic automorphisms acts trivially on the Albanese kernel $mathrm{CH}_0(S)_{mathrm{alb}}$ of the $0$-th Chow group $mathrm{CH}_
On a Weierstra{ss} elliptic surface $X$, we define a `limit of Bridgeland stability conditions, denoted as $Z^l$-stability, by moving the polarisation towards the fiber direction in the ample cone while keeping the volume of the polarisation fixed. W
Using Gauss-Manin derivatives of normal functions, we arrive at some remarkable results on the non-triviality of the transcendental regulator for $K_m$ of a very general projective algebraic manifold. Our strongest results are for the transcendental