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Anatoly Libgober

Publications and source records attributed to Anatoly Libgober.

At least 19 recordsLinked to original sources
arXiv2024

CAT(0) geometry of complex curve complements and families

Motivated by the question of whether braid groups are CAT(0), we investigate the CAT(0) behavior of fundamental groups of plane curve complements and certain universal families. If $C$ is the branch locus of a generic projection of a smooth, complete intersection surface to $\PP^2$, we show that $π_1(\PP^2\setminus C)$ is CAT(0). In the other direction, we prove that the fundamental group of the universal family associated with the singularities of type $E_6$, $E_7$, and $E_8$ is not CAT(0).

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arXiv2024

Instanton Floer homology and Milnor Fibers

In the early days of the Floer theory, Atiyah asked if there is a Milnor fiber description of the Floer homology of the links of singularities. We answer this question for the Brieskorn-Hamm complete intersection singularities. The resulting combinatorial formulas lead to an independent proof of the equality of the Casson invariants in the Donaldson and Seiberg-Witten theories. We use similar techniques to express the Heegaard Floer d-invariant of torus knots in terms of their Milnor fibers.

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arXiv2021

Free quotients of fundamental groups of smooth quasi-projective varieties

We consider the structure of classes of curves on a projective simply connected surface for which fundamental groups of the complements admit free quotients having rank greater than one with irreducible components belonging to a selected subset the effective cone of the surface. In particular we show a finiteness result for such classes if the ranks of free quotients of the fundamental groups with components in the subset of effective cone are sufficiently large.

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arXiv2021

Self-products of rationally elliptic spaces and inequalities between the ranks of homotopy and homology groups

We give a survey on recent results on inequalities between the ranks of homotopy and cohomology groups (resp., graded components of mixed Hodge structures on these groups) of rationally elliptic spaces (resp., quasi-projective varieties which are rationally elliptic). We also discuss a refinement of these results describing a new invariant of rationally elliptic spaces allowing to compare the ranks of homotopy and homology groups. This invariant is a specialization of an invariant $r\left ( P(t),Q(t);\varepsilon \right )$ of a pair $\left (P(t),Q(t) \right )$ of polynomials with non-negative integer coefficients, describing the range of variable $r$ such that $rP(t)<Q(t)^r$ for all $t \ge \varepsilon$. This range is related to the classical Lambert W-function $W(z)$.

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arXiv2021

Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties

We discuss inequalities between the values of \emph{homotopical and cohomological Poincaré polynomials} of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective varieties, we prove inequalities between the values of generating functions for the ranks of the graded pieces of the weight and Hodge filtrations of the canonical mixed Hodge structures on homotopy and cohomology groups. Several examples of such mixed Hodge polynomials and related inequalities for rationally elliptic quasi-projective algebraic varieties are presented. One of the consequences is that the homotopical (resp. cohomological) mixed Hodge polynomial of a rationally elliptic toric manifold is a sum (resp. a product) of polynomials of projective spaces. We introduce an invariant called \emph{stabilization threshold} $\frak{pp} (X;\varepsilon)$ for a simply connected rationally elliptic space $X$ and a positive real number $\varepsilon$, and we show that the Hilali conjecture implies that $\frak{pp} (X;1) \le 3$.

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arXiv2017

Motivic zeta functions and infinite cyclic covers

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) a rational function in $K_0({\rm Var}^{\hat μ}_{\mathbb{C}})[\mathbb{L}^{-1}]$, which we call {\it motivic infinite cyclic zeta function}, and show its birational invariance. Our construction is a natural extension of the notion of {\it motivic infinite cyclic covers} introduced by the authors, and as such, it generalizes the Denef-Loeser motivic Milnor zeta function of a complex hypersurface singularity germ.

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arXiv2016

Motivic infinite cyclic covers

We associate with an infinite cyclic cover of a punctured neighborhood of a simple normal crossing divisor on a complex quasi-projective manifold (assuming certain finiteness conditions are satisfied) an element in the Grothendieck ring $K_0({\rm Var}^{\hat μ}_{\mathbb{C}})$, which we call {\it motivic infinite cyclic cover}, and show its birational invariance. Our construction provides a unifying approach for the Denef-Loeser motivic Milnor fibre of a complex hypersurface singularity germ, and the motivic Milnor fiber of a rational function, respectively.

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arXiv2011

Sequences of LCT-polytopes

To r ideals on a germ of smooth variety X one attaches a rational polytope in the r-dimensional Euclidean space (the LCT-polytope) that generalizes the notion of log canonical threshold in the case of one ideal. We study these polytopes, and prove a strong form of the Ascending Chain Condition in this setting: we show that if a sequence P_m of such LCT-polytopes converges to a compact subset Q in the Hausdorff metric, then Q is equal to the intersection of all but finitely many of the P_m. Furthermore, Q is an LCT-polytope.

math.AG↗
arXiv2009

Multivariable Hodge theoretical invariants of germs of plane curves

We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinement of the zero sets of multivariable Alexander polynomials. In particular we identify some hyperplanes on which all polynomials in multivariable Bernstein ideal vanish.

math.AG↗
arXiv2007

Hodge genera of algebraic varieties, II

We study the behavior of Hodge-theoretic genera under morphisms of complex algebraic varieties. We prove that the additive $χ_y$-genus which arises in the motivic context satisfies the so-called ``stratified multiplicative property", which shows how to compute the invariant of the source of a proper surjective morphism from its values on various varieties that arise from the singularities of the map. By considering morphisms to a curve, we obtain a Hodge-theoretic analogue of the Riemann-Hurwitz formula. We also study the contribution of monodromy to the $χ_y$-genus of a smooth projective family, and prove an Atiyah-Meyer type formula for twisted $χ_y$-genera. This formula measures the deviation from multiplicativity of the $χ_y$-genus, and expresses the correction terms as higher-genera associated to cohomology classes of the quotient of the total period domain by the action of the monodromy group. By making use of Saito's theory of mixed Hodge modules, we also obtain formulae of Atiyah-Meyer type for the corresponding Hirzebruch characteristic classes.

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arXiv2005

Regular Functions Transversal at Infinity

We generalize and complete some of Maxim's recent results on Alexander invariants of a polynomial transversal to the hyperplane at infinity. Roughly speaking, and surprisingly, such a polynomial behaves both topologically and algebraically (e.g. in terms of the variation of MHS on the cohomology of its smooth fibers), like a homogeneous polynomial.

math.AG↗
arXiv2004

Discrete torsion, orbifold elliptic genera, and the chiral de Rham complex

Given a compact complex algebraic variety with an effective action of a finite group $G$, and a class $α\in H^2(G,U(1))$, we introduce an orbifold elliptic genus with discrete torsion $α$, denoted $Ell^α_{orb}(X,G, q, y)$. We give an interpretation of this genus in terms of the chiral de Rham complex attached to the orbifold $[X/G]$. If $X$ is Calabi-Yau and $G$ preserves the volume form, $Ell^α_{orb}(X,G, q, y)$ is a weak Jacobi form. We also obtain a formula for the generating function of the elliptic genera of symmetric products with discrete torsion.

math.AG↗
arXiv2003

Local topology of reducible divisors

We show that the universal abelian cover of the complement to a germ of a reducible divisor on a complex space $Y$ with isolated singularity is $(dimY-2)$-connected provided that the divisor has normal crossings outside of the singularity of $Y$. We apply this result to obtain a vanishing property for the cohomology of local systems of rank one and we also study vanishing in the case of local systems of higher rank. This second version contains a corrected proof of Corollary 4.1 from the first version.

math.AG↗
arXiv2002

McKay correspondence for elliptic genera

We establish a correspondence between orbifold and singular elliptic genera of a global quotient. While the former is defined in terms of the fixed point set of the action, the latter is defined in terms of the resolution of singularities. As a byproduct, the second quantization formula of Dijkgraaf, Moore, Verlinde and Verlinde is extended to arbitrary Kawamata log-terminal pairs.

math.AG↗
arXiv2002

Homotopy groups of complements and non-isolated singularities

We obtain sufficient conditions for the vanishing of higher homotopy groups of the complements to hypersurfaces in ${\mathbb C}^n$ in terms of the behavior at infinity and relate the monodromy of non isolated singularities to the position of singularities in generic plane sections.

math.AG↗
arXiv2001

Elliptic Genera of Singular Varieties

Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. Orbifold elliptic genus is given in terms of the fixed point sets of the action. We show that the generating function for this orbifold elliptic genus $\sum Ell_{orb}(X^n,Σ_n)p^n$ for symmetric groups $Σ_n$ acting on $n$-fold products coincides with the one proposed by Dijkgraaf, Moore, Verlinde and Verlinde. Two notions of elliptic genera are conjectured to coincide.

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