Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. ...This generalizes earlier results of Aluffi and Wu‐Zhou. The first application of our formula is a geometric description of Chern‐Mather classes of an arbitrary very affine variety, generalizing earlier results of Huh which held under the smooth and schön assumptions. As the second application, we prove an involution formula relating sectional maximum likelihood (ML) degrees and ML bidegrees, which was conjectured by Huh and Sturmfels in 2013.
We employ the formalism of vanishing cycles and perverse sheaves to introduce and study the vanishing cohomology of complex projective hypersurfaces. As a consequence, we give upper bounds for the ...Betti numbers of projective hypersurfaces, generalizing those obtained by different methods by Dimca in the isolated singularities case, and by Siersma–Tibăr in the case of hypersurfaces with a 1-dimensional singular locus. We also prove a supplement to the Lefschetz hyperplane theorem for hypersurfaces, which takes the dimension of the singular locus into account, and we use it to give a new proof of a result of Kato.
Nous utilisons le formalisme des cycles évanescents et des faisceaux pervers pour introduire et étudier la cohomologie évanescente des hypersurfaces projectives. Nous déduisons des majorants pour les nombres de Betti des hypersurfaces projectives, en généralisant ceux obtenus avec des méthodes différentes par Dimca dans le cas des singularités isolées, et par Siersma–Tibăr dans le cas des hypersurfaces avec lieu singulier de dimension 1. Nous prouvons aussi un complément au théorème de la section hyperplane de Lefschetz pour les hypersurfaces qui tient compte de la dimension du lieu singulier, et nous l’utilisons pour donner une nouvelle preuve du résultat de Kato.
Abstract
The Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf ...conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara.
Abstract We prove a Cohen-Dimca-Orlik-type theorem for rank one ${\mathbb{Z}}$-local systems on complex hyperplane arrangement complements. This settles a recent conjecture of S. Sugawara.
Abstract We extend the Hirzebruch–Milnor class of a hypersurface $X$ in an ambient complex algebraic manifold to the case where the normal bundle is nontrivial and $X$ cannot be defined by a global ...function, using the associated line bundle and the graded quotients of the monodromy filtration. The earlier definition requiring a global defining function of $X$ can be applied rarely to projective hypersurfaces with non-isolated singularities. Indeed, it is surprisingly difficult to get a one-parameter smoothing with total space smooth without destroying the singularities by blowing-ups (except certain quite special cases). As an application, assuming the singular locus is a projective variety, we show that the minimal exponent of a hypersurface can be captured by the spectral Hirzebruch–Milnor class, and higher du Bois and rational singularities of a hypersurface are detectable by the unnormalized Hirzebruch–Milnor class. Here the unnormalized class can be replaced by the normalized one in the higher du Bois case, but for the higher rational case, we must use also the decomposition of the Hirzebruch–Milnor class by the action of the semisimple part of the monodromy (which is equivalent to the spectral Hirzebruch–Milnor class). We cannot extend these arguments to the non-projective compact case by Hironaka’s example.
We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend the local-to-global ...divisibility results of Maxim and of Dimca and Libgober to the twisted setting. In the process, we also study the splitting fields containing the roots of the corresponding twisted Alexander polynomials.
Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating ...family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.
In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also ...asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.
Motivated by finding an effective way to compute the algebraic complexity of the nearest point problem for algebraic models, we introduce an efficient method for detecting the limit points of the ...stratified Morse trajectories in a small perturbation of any polynomial function on a complex affine variety. We compute the multiplicities of these limit points in terms of vanishing cycles. In the case of functions with only isolated stratified singularities, we express the local multiplicities in terms of polar intersection numbers.
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