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 : U → C * be an algebraic map inducing an epimorphism in fundamental groups. The pullback of the universal cover of C * by f gives rise to an infinite cyclic cover U f of U . The action of the deck group Z on U f induces a Q[t, t −1 ]-module structure on H * (U f ; Q). We show that the torsion parts A * (U f ; Q) of the Alexander modules H * (U f ; Q) carry canonical Qmixed Hodge structures. Furthermore, we compare the resulting mixed Hodge structure on A * (U f ; Q) to the limit mixed Hodge structure on the generic fiber of f . Contents 1. Introduction 2. Preliminaries 2.1. Denotations and Assumptions 2.2. Alexander Modules 2.3. Monodromy action on Alexander Modules 2.4. Differential Graded Algebras 2.5. Mixed Hodge Structures and Complexes 2.6. Real Mixed Hodge Complexes on Smooth Varieties 2.7. Rational Mixed Hodge Complexes on Smooth Varieties 2.8. Limit Mixed Hodge Structure 3. Thickened Complexes 3.1. Thickened Complex of a Differential Graded Algebra

The parasitic chytrid fungus Batrachochytrium dendrobatidis (Bd ) can cause the lethal disease chytridiomycosis in amphibians and therefore may play a role in population declines. The yellow-bellied toad Bombina variegata suffered strong declines throughout western and northwestern parts of its range and is therefore listed as highly endangered for Germany and the federal state of Hesse. Whether chytridiomycosis may play a role in the observed local declines of this strictly protected anuran species has never been tested. We investigated 19 Hessian yellowbellied toad populations for Bd infection rates, conducted capture-mark-recapture studies in 4 of them over 2 to 3 yr, examined survival histories of recaptured infected individuals, and tested whether multi-locus heterozygosity of individuals as well as expected heterozygosity and different environmental variables of populations affect probabilities of Bd infection. Our results show high prevalence of Bd infection in Hessian yellow-bellied toad populations, but although significant decreases in 2 populations could be observed, no causative link to Bd as the reason for this can be established. Mass mortalities or obvious signs of disease in individuals were not observed. Conversely, we show that growth of Bd-infected populations is possible under favorable habitat conditions and that most infected individuals could be recaptured with improved body indices. Neither genetic diversity nor environmental variables appeared to affect Bd infection probabilities. Hence, genetically diverse amphibian specimens and populations may not automatically be less susceptible for Bd infection.

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.

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-Poincaré 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.Classically, much of the manifold theory, e.g., Morse theory, Lefschetz theorems, Hodge decompositions, and especially Poincaré Duality, is recovered in the singular stratified context if, instead of the usual (co)homology, one uses Goresky-MacPherson's intersection homology groups [12,13]. We recall here the definition of intersection homology of complex analytic (or algebraic) varieties, and discuss some preliminary results concerning the corresponding intersection homology Euler characteristic. For more details on intersection homology, the reader may consult, e.g., [9] and the references therein.Let X be a purely n-dimensional complex analytic (or algebraic) variety with a fixed Whitney stratification. All strata of X are of even real (co)dimension. By [11], X admits a triangulation which is compatible with the stratification, so X can also be viewed as a PL stratified pseudomanifold. Let (C * (X), ∂) denote the complex of finite PL chains on X, with Z-coefficients. The intersection homology groups of X, denoted IH i (X), are the homology groups of a complex of "allowable chains", defined by imposing restrictions on how chains meet the singular strata. Specifically, the chain complex (IC * (X), ∂) of allowable finite PL chains is defined as follows: if ξ ∈ C i (X) has support |ξ|, then ξ ∈ IC i (X) if, and only if, dim(|ξ| ∩ S) < i − s and dim(|∂ξ| ∩ S) < i − s − 1, for each stratum S of complex codimension s > 0. The boundary operator on allowable chains is induced from the usual boundary operator on chains of X. (The second condition above ensures that ∂ restricts to the complex of allowable chains.)

Zusammenfassung Rund 40 Prozent aller in Deutschland vorkommenden Pflanzen‐ und Tierarten (Insekten, Säugetiere, Fische, Vögel, Amphibien und Reptilien) sind gefährdet oder vom Aussterben bedroht, 3–4 Prozent gelten bereits als verschollen. Ein Vergleich mit anderen europäischen Ländern zeigt, dass die Gefährdungssituation in Deutschland auch aus internationaler Sicht sehr ernst zu nehmen ist. Um Naturschutz als Voraussetzung für eine global nachhaltige Entwicklung besser in der modernen Gesellschaft zu verankern, bedarf es (1.) eines Umdenkens und veränderten Handelns jedes Einzelnen durch Aufklärung, Kommunikation und Diskussion; (2.) einer Neuausrichtung der Politik und der vorhandenen Förderinstrumente; (3.) einer Stärkung von Biodiversitäts‐ und Naturschutzbelangen in und durch Hochschulen und Universitäten, sowie (4.) Naturschutzinvestitionen, um die Natur als Lebensgrundlage des Menschen (ökosystemische Dienstleistungen), sowie aufgrund ihres Eigenwertes zu erhalten. Naturschutz muss sich auch wirtschaftlich stärker lohnen, nicht nur für Landwirte, sondern jegliche naturschutzkonforme Landnutzung und die damit erbrachte naturschutzfachliche Leistung sollte honoriert werden. Damit die eigentlich gesetzten und klar definierten Ziele erreicht werden können, prognostiziert die Bund/Länder‐Arbeitsgemeinschaft Naturschutz, Landschaftspflege und Erholung (LANA), dass unter den derzeitigen Rahmenbedingungen in Deutschland mindestens 1,4 Mrd. Euro zusätzlich jährlich benötigt werden, was etwa 17 Euro pro Einwohner und Jahr entsprechen würde. Dieser vergleichsweise geringe Betrag würde mehr als eine Verdoppelung des in Deutschland verfügbaren Budgets zur Förderung der Biodiversität entsprechen.