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Laurentiu Maxim

Publications and source records attributed to Laurentiu Maxim.

At least 19 recordsLinked to original sources
arXiv2026

On the topology of fibers of complex polynomial maps

We survey known results on the cohomology of fibers of complex polynomial maps, with particular emphasis on vanishing ranges, and establish new results on the vanishing cohomology of a polynomial at a bifurcation value. We also derive upper bounds for the first possibly nonvanishing Betti number of general and atypical fibers in terms of local singularity invariants. These results extend several theorems of Tib\u{a}r, Siersma, Dimca, and others from the case of isolated singularities, including singularities at infinity, to polynomial maps with arbitrary singularities.

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arXiv2025

Spectral sequences, Massey products and homology of covering spaces

We revisit the equivariant spectral sequence considered by Papadima-Suciu, and show that all its differentials are computed by higher order Massey products. As a first application, we extend to arbitrary field coefficients results of Pajitnov relating the size of Jordan blocks for the eigenvalue 1 part of the Alexander modules to the length of nonvanishing Massey products in cohomology. We also give computable upper bounds for the mod p Betti numbers of prime power cyclic covers, and resp. for the ranks of the cohomology groups with coefficients in a prime order rank one local system. Under suitable conditions, these bounds are improvements of the ones obtained by Papadima-Suciu. We also specialize these results to the case of hyperplane arrangement complements, showing, e.g., that vanishing of higher-order Massey products implies that the mod p Betti numbers of prime p tower cyclic covers are combinatorially determined.

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arXiv2025

Around a class version of the Hodge index theorem for singular varieties

We give an overview of recent developments around a characteristic class version of the Hodge index theorem for singular complex algebraic varieties. This was formulated by Brasselet-Schuermann-Yokura as a conjecture expressing the Goresky-MacPherson homology L-classes in terms of suitable Hodge-theoretic L-classes. Along the way, we clarify the relationship between several notions of L-classes appearing in the literature, but we also include many new cases for which the conjecture is true, e.g., all compact toric varieties, all (matroid) Schubert varieties, all Richardson and intersection varieties, all projective simply connected spherical varieties, and all compact complex algebraic surfaces and threefolds.

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arXiv2024

Weighted Ehrhart theory via mixed Hodge modules on toric varieties

We give a cohomological and geometrical interpretation for the weighted Ehrhart theory of a full-dimensional lattice polytope $P$, with Laurent polynomial weights of geometric origin. For this purpose, we calculate the motivic Chern and Hirzebruch characteristic classes of a mixed Hodge module complex $\mathcal{M}$ whose underlying cohomology sheaves are constant on the $\mathbb{T}$-orbits of the toric variety $X_P$ associated to $P$. Besides motivic coefficients, this also applies to the intersection cohomology Hodge module. We introduce a corresponding generalized Hodge $χ_y$-polynomial of the ample divisor $D_P$ on $X_P$. Motivic properties of these characteristic classes are used to express this Hodge polynomial in terms of a very general weighed lattice point counting and the corresponding weighted Ehrhart theory. We introduce, for such a mixed Hodge modules complex $\mathcal{M}$ on $X$, an Ehrhart polynomial $E_{P,\mathcal{M}}$ generalizing the Hodge polynomial of $\mathcal{M}$ and satisfying a reciprocity formula and a purity formula fitting with the duality for mixed Hodge modules. This Ehrhart polynomial and its properties depend only on a Laurent polynomial weight function on the faces $Q$ of $P$. In the special case of the intersection cohomology mixed Hodge module, the weight function corresponds to Stanley's $g$-function of the polar polytope of $P$, hence it depends only on the combinatorics of $P$. In particular, we obtain a combinatorial formula for the intersection cohomology signature.

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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 $f$.

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arXiv2022

On singular variants of the Singer-Hopf Conjecture

We propose singular variants 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 the conjecture 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.

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arXiv2020

Perverse sheaves on semi-abelian varieties

We give a complete (global) characterization of complex perverse sheaves on semi-abelian varieties in terms of their cohomology jump loci. Our results generalize Schnell's work on perverse sheaves on complex abelian varieties, as well as Gabber-Loeser's results on perverse sheaves on complex affine tori. We apply our results to the study of cohomology jump loci of smooth quasi-projective varieties, to the topology of the Albanese map, and in the context of homological duality properties of complex algebraic varieties.

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arXiv2019

Perverse sheaves on semi-abelian varieties -- a survey of properties and applications

We survey recent developments in the study of perverse sheaves on semi-abelian varieties. As concrete applications, we discuss various obstructions on the homotopy type of complex algebraic manifolds (expressed in terms of their cohomology jump loci), homological duality properties of complex algebraic manifolds, as well as new topological characterizations of semi-abelian varieties.

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arXiv2018

Homological congruence formulae for characteristic classes of singular varieties

For a pair $(f, g)$ of morphisms $f:X \to Z$ and $g:Y \to Z$ of (possibly singular) complex algebraic varieties $X,Y,Z$, we present congruence formulae for the difference $f_*T_{y*}(X) -g_*T_{y*}(Y)$ of pushforwards of the corresponding motivic Hirzebruch classes $T_{y*}$. If we consider the special pair of a fiber bundle $F \hookrightarrow E \to B$ and the projection $pr_2:F \times B \to B$ as such a pair $(f,g)$, then we get a congruence formula for the difference $f_*T_{y*}(E) -χ_y(F)T_{y*}(B)$, which at degree level yields a congruence formula for $χ_y(E) -χ_y(F)χ_y(B)$, expressed in terms of the Euler--Poincarv'e characteristic, Todd genus and signature in the case when $F, E, B$ are non-singular and compact. We also extend the finer congruence identities of Rovi--Yokura to the singular complex projective situation, by using the corresponding intersection (co)homology invariants.

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arXiv2018

Higher Order Degrees of Affine Plane Curve Complements

We study finiteness (and vanishing) properties of the higher order degrees associated to complements of complex affine plane curves with mild singularities at infinity. Our results impose new obstructions on the class of groups that can be realized as fundamental groups of affine plane curve complements. We also clarify the relationship between the higher order degrees and the multivariable Alexander polynomial of a non-irreducible plane curve.

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arXiv2018

Topology of subvarieties of complex semi-abelian varieties

We use the non-proper Morse theory of Palais-Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties, and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules, as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh's extension of Varchenko's conjecture to very affine manifolds, and provide a generalization of this result in the context of smooth closed subvarieties of semi-abelian varieties.

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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.

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arXiv2018

Thom-Sebastiani theorems for filtered D-modules and for multiplier ideals

We give a proof of the Thom-Sebastiani type theorem for holonomic filtered $D$-modules satisfying certain good conditions (including Hodge modules) by using algebraic partial microlocalization. By a well-known relation between multiplier ideals and $V$-filtrations of Kashiwara and Malgrange, the argument in the proof implies also a Thom-Sebastiani type theorem for multiplier ideals, which cannot be deduced from a already known proof of the Thom-Sebastiani theorem for mixed Hodge modules (since the latter gives only the information of graded pieces of multiplier ideals). We also sketch a more elementary proof of the Thom-Sebastiani type theorem for multiplier ideals (as communicated to us by M.~Mustaţǎ), which seems to be known to specialists, although it does not seem to be stated explicitly in the literature.

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