Andreas Höring’s research while affiliated with Université Côte d'Azur and other places

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Publications (2)


Symmetric tensors on the intersection of two quadrics and Lagrangian fibration
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November 2024

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19 Reads

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2 Citations

Arnaud Beauville

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Andreas Höring

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Claire Voisin

Let X be an n -dimensional (smooth) intersection of two quadrics, and let TX{T^{\rm{*}}}X be its cotangent bundle. We show that the algebra of symmetric tensors on X is a polynomial algebra in n variables. The corresponding map Φ:TXCn{\rm{\Phi }}:{T^{\rm{*}}}X \to {\mathbb{C}^n} is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is a Zariski open subset of an abelian variety, which is a quotient of a hyperelliptic Jacobian by a 2 -torsion subgroup. In dimension 3 , Φ{\rm{\Phi }} is the Hitchin fibration of the moduli space of rank 2 bundles with fixed determinant on a curve of genus 2 .

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Symmetric tensors on the intersection of two quadrics and Lagrangian fibration

April 2023

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33 Reads

Let X be a n-dimensional (smooth) intersection of two quadrics, and let T*X be its cotangent bundle. We show that the algebra of symmetric tensors on X is a polynomial algebra in n variables. The corresponding map F: T*X -- > C^n is a Lagrangian fibration, which admits an explicit geometric description; its general fiber is a Zariski open subset of an abelian variety, quotient of a hyperelliptic Jacobian by a 2-torsion subgroup. For n = 3 F is the Hitchin fibration of the moduli space of rank 2 bundles with fixed determinant on a curve of genus 2.

Citations (1)


... Theorem 1.4 offers a potential approach to address Conjecture 1.1 when the total dual VMRT is a hypersurface in the projectivized tangent bundle. For precise definitions, we direct the reader to [ Recall that when X is a smooth hypersurface of degree d, the joint paper of Höring, Liu and the first author shows that the tangent bundle T X is pseudo-effective if and only if d ≤ 2 (see [16,Theorem 1.4]); when X is a smooth complete intersection of two quadrics, [4] verifies that the tangent bundle T X is Q-effective but not big; when X is a smooth Fano complete intersection of dimension at least 3 and of Fano index 1 or 2, [15, Theorem 1.1] implies that T X is not big. We refer the reader to Theorem 4.5 written by Liu, which shows that a general smooth finite cover over a general complete intersection of two quadrics cannot have a pseudo-effective tangent bundle. ...

Reference:

Bigness of tangent bundles and dynamical rigidity of Fano manifolds of Picard number 1 (with an appendix by Jie Liu)
Symmetric tensors on the intersection of two quadrics and Lagrangian fibration