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Register a new userOn the monodromy of elliptic surfaces
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Authors
Genival da Silva Jr
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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.
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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 the Picard numbers of elliptic surfaces with complex multiplication.
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 Jacobian fibration. We use ideas from a descent paper of Poonen and Schaefer to remove this assumption on the rational 2-torsion.
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 $mathrm{CH}_0(S)$, unless the geometric genus and the irregularity satisfy $p_g(S)=q(S)in{1,2}$. In the exceptional case, the image of the homomorphism $mathrm{Aut}_s(S)rightarrow mathrm{Aut}(mathrm{CH}_0(S)_{mathrm{alb}})$ is either trivial or possibly isomorphic to $mathbb{Z}/3mathbb{Z}$. Our main arguments actually take care of the group $mathrm{Aut}_f(S)$ of fibration preserving automorphisms of elliptic surfaces $fcolon Srightarrow B$. We prove that, if $sigmainmathrm{Aut}_f(S)$ induces the trivial action on $H^{i,0}(S)$ for $i>0$, then it induces trivial action on $mathrm{CH}_0(S)_mathrm{alb}$. Also, if $S$ is additionally a K3 surface, then $mathrm{Aut}_f(S)cap mathrm{Aut}_s(S)$ acts trivially on $mathrm{CH}_0(S)_{mathrm{alb}}$.
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. We describe conditions under which a slope stable torsion-free sheaf is taken by a Fourier-Mukai transform to a $Z^l$-stable object, and describe a modification upon which a $Z^l$-semistable object is taken by the inverse Fourier-Mukai transform to a slope semistable torsion-free sheaf. We also study wall-crossing for Bridgeland stability, and show that 1-dimensional twisted Gieseker semistable sheaves are taken by a Fourier-Mukai transform to Bridgeland semistable objects.
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 regulator for $K_1$ of a very general $K3$ surface. We also construct an explicit family of $K_1$ cycles on $H oplus E_8 oplus E_8$-polarized $K3$ surfaces, and show they are indecomposable by a direct evaluation of the real regulator. Critical use is made of natural elliptic fibrations, hypersurface normal forms, and an explicit parametrization by modular functions.
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