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Christian Geske

Publications and source records attributed to Christian Geske.

5 recordsLinked to original sources
arXiv2022

Mixed Hodge Structures on Alexander Modules

Motivated by the limit mixed Hodge structure on the Milnor fiber of a hypersurface singularity germ, we construct a natural mixed Hodge structure on the torsion part of the Alexander modules of a smooth connected complex algebraic variety. More precisely, let $U$ be a smooth connected complex algebraic variety and let $f\colon U\to \mathbb{C}^*$ be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of $\mathbb{C}^*$ by $f$ gives rise to an infinite cyclic cover $U^f$ of $U$. The action of the deck group $\mathbb{Z}$ on $U^f$ induces a $\mathbb{Q}[t^{\pm 1}]$-module structure on $H_*(U^f;\mathbb{Q})$. We show that the torsion parts $A_*(U^f;\mathbb{Q})$ of the Alexander modules $H_*(U^f;\mathbb{Q})$ carry canonical $\mathbb{Q}$-mixed Hodge structures. We also prove that the covering map $U^f \to U$ induces a mixed Hodge structure morphism on the torsion parts of the Alexander modules. As applications, we investigate the semisimplicity of $A_*(U^f;\mathbb{Q})$, as well as possible weights of the constructed mixed Hodge structures. Finally, in the case when $f\colon U\to \mathbb{C}^*$ is proper, we prove the semisimplicity and purity of $A_*(U^f;\mathbb{Q})$, and we compare our mixed Hodge structure on $A_*(U^f;\mathbb{Q})$ with the limit mixed Hodge structure on the generic fiber of $f$.

math.AG↗
arXiv2020

Signatures of topological branched covers

Let $X^4$ and $Y^4$ be smooth manifolds and $f: X\to Y$ a branched cover with branching set $B$. Classically, if $B$ is smoothly embedded in $Y$, the signature $σ(X)$ can be computed from data about $Y$, $B$ and the local degrees of $f$. When $f$ is an irregular dihedral cover and $B\subset Y$ smoothly embedded away from a cone singularity whose link is $K$, the second author gave a formula for the contribution $Ξ(K)$ to $σ(X)$ resulting from the non-smooth point. We extend the above results to the case where $Y$ is a {\it topological} four-manifold and $B$ is locally flat, away from the possible singularity. Owing to the presence of non-locally-flat points on $B$, $X$ in this setting is a stratified pseudomanifold, and we use the Intersection Homology signature of $X$, $σ_{IH}(X)$. For any knot $K$ whose determinant is not $\pm 1$, a homotopy ribbon obstruction is derived from $Ξ(K)$, providing a new technique to potentially detect slice knots that are not ribbon.

math.GT↗
arXiv2018

Algebraic Intersection Spaces

We define a variant of intersection space theory that applies to many compact complex and real analytic spaces $X$, including all complex projective varieties; this is a significant extension to a theory which has so far only been shown to apply to a particular subclass of spaces with smooth singular sets. We verify existence of these so-called algebraic intersection spaces and show that they are the (reduced) chain complexes of known topological intersection spaces in the case that both exist. We next analyze "local duality obstructions", which we can choose to vanish, and verify that algebraic intersection spaces satisfy duality in the absence of these obstructions. We conclude by defining an untwisted algebraic intersection space pairing, whose signature is equal to the Novikov signature of the complement in $X$ of a tubular neighborhood of the singular set.

math.AT↗
arXiv2018

On the signed Euler characteristic property for subvarieties of abelian varieties

We give an elementary proof of the fact that a pure-dimensional closed subvariety of a complex abelian variety has a signed intersection homology Euler characteristic. We also show that such subvarieties which, moreover, are local complete intersections, have a signed Euler-Poincare characteristic. Our arguments rely on the construction of circle-valued Morse functions on such spaces, and use in an essential way the stratified Morse theory of Goresky-MacPherson. Our approach also applies (with only minor modifications) for proving similar statements in the analytic context, i.e., for subvarieties of compact complex tori. Alternative proofs of our results can be given by using the general theory of perverse sheaves.

math.AT↗