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We consider the problem of counting the number of varieties in a family over Q with a rational point. We obtain lower bounds for this counting problem for some families over P1, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher-dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics. [ABSTRACT FROM AUTHOR]

We establish the non-commutative analogue of Grothendieck's standard conjecture D for the differential graded category of G-equivariant matrix factorizations associated to an isolated hypersurface singularity where G is a finite group. [ABSTRACT FROM AUTHOR]

Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito's theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (f, X), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the concept of ρ -regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor tube fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map G with respect to the levels of a function ρ , which guarantees ρ -regularity. Consequently, both conditions are crucial for the presence of fibration structures. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts. [ABSTRACT FROM AUTHOR]

In Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) it was proved the existence of fibrations à la Milnor (in the tube and in the sphere) for real analytic maps f : (R n , 0) → (R k , 0) , where n ≥ k ≥ 2 , with non-isolated critical values. In the present article we extend the existence of the fibrations given in Cisneros-Molina et al. (São Paulo J Math Sci, 2023. https://doi.org/10.1007/s40863-023-00370-y) to differentiable maps of class C ℓ , ℓ ≥ 2 , with possibly non-isolated critical value. This is done using a version of Ehresmann fibration theorem for differentiable maps of class C ℓ between smooth manifolds, which is a generalization of the proof given by Wolf (Michigan Math J 11:65–70, 1964) of Ehresmann fibration theorem. We also present a detailed example of a non-analytic map which has the aforementioned fibrations. [ABSTRACT FROM AUTHOR]

This is a survey article, in which we explore how the presence of singularities affects the geometry and topology of complex projective hypersurfaces. [ABSTRACT FROM AUTHOR]

By applying an inverse Kaluza-Klein procedure, we provide explicit coordinate expressions for Riemannian metrics on two homeomorphic but not diffeomorphic spheres in seven dimensions. We identify Milnor's bundles, among which ten out of the fourteen exotic seven-spheres appear (ignoring orientation), with non-principal bundles having homogeneous fibres. Then, we use the techniques in [1] to obtain a general ansatz for the coordinate expression of a metric on the total space of any Milnor's bundle. The ansatz is given in terms of a metric on S4, a metric on S3 (which can smoothly vary throughout S4), and a connection on the principal SO(4)-bundle over S4. As a concrete example, we present explicit formulae for such metrics for the ordinary sphere and the Gromoll-Meyer exotic sphere. Then, we perform a non-abelian Kaluza-Klein reduction to gravity in seven dimensions, according to (a slightly simplified version of) the metric ansatz above. We obtain the standard four-dimensional Einstein-Yang-Mills system, for which we find solutions associated with the geometries of the ordinary sphere and of the exotic one. The two differ by the winding numbers of the instantons involved. [ABSTRACT FROM AUTHOR]

We give an example of a nonzero odd degree element of the classifying space of a connected Lie group such that all higher Milnor operations vanish on it. It is a counterexample of a conjecture of Kono and Yagita [Trans. Amer. Math. Soc. 339 (1993), pp. 781–798]. [ABSTRACT FROM AUTHOR]

Copyright of Annales Mathematiques Blaise Pascal is the property of Laboratoire mathematiques Blaise Pascal - CNRS UMR 6620 and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Papadima and Suciu proved an inequality between the ranks of the cohomology groups of the Aomoto complex with finite field coefficients and the twisted cohomology groups, and conjectured that they are actually equal for certain cases associated with the Milnor fiber of the arrangement. Recently, an arrangement (the icosidodecahedral arrangement) with the following two peculiar properties was found: (i) the strict version of Papadima-Suciu's inequality holds, and (ii) the first integral homology of the Milnor fiber has a non-trivial 2-torsion. In this paper, we investigate the relationship between these two properties for double covering spaces. We prove that (i) and (ii) are actually equivalent. [ABSTRACT FROM AUTHOR]

We study polynomials with complex coefficients which are nondegenerate in two senses, one of Kouchnirenko and the other with respect to its Newton polyhedron, through data on contact loci and motivic nearby cycles. Introducing an explicit description of these quantities we can answer in part the question concerning the motivic nearby cycles of restriction functions in the context of Newton nondegenerate polynomials. Furthermore, in the nondegeneracy in the sense of Kouchnirenko, we give calculations on cohomology groups of the contact loci. [ABSTRACT FROM AUTHOR]

Parametrized motion planning algorithms [1] have high degree of flexibility and universality, they can work under a variety of external conditions, which are viewed as parameters and form part of the input of the algorithm. In this paper we analyse the parameterized motion planning problem in the case of sphere bundles. Our main results provide upper and lower bounds for the parametrized topological complexity; the upper bounds typically involve sectional categories of the associated fibrations and the lower bounds are given in terms of characteristic classes and their properties. We explicitly compute the parametrized topological complexity in many examples and show that it may assume arbitrarily large values. [ABSTRACT FROM AUTHOR]

A triangulated category is said to be indecomposable if it admits no nontrivial semiorthogonal decompositions. We introduce a definition of a noncommutatively stably semiorthogonally indecomposable (NSSI) scheme. This property implies, among other things, that each connected closed subscheme has indecomposable derived category of coherent sheaves and that if Y is NSSI, then for any variety X all semiorthogonal decompositions of X x Y are induced from decompositions of X. We prove that any scheme which admits an affine morphism to an abelian variety is NSSI and that the total space of a fibration over a NSSI base with NSSI fibers is also NSSI. We apply this indecomposability to deduce that there are no phantom subcategories in some varieties, including surfaces C x ℙ¹, where C is any smooth proper curve of positive genus. [ABSTRACT FROM AUTHOR]

Given the germ of an analytic function on affine space with a smooth critical locus, we prove that the constancy of the reduced cohomology of the Milnor fiber in lowest possible non-trivial degree off a codimension two subset of the critical locus implies that the vanishing cycles are concentrated in lowest degree and are constant. [ABSTRACT FROM AUTHOR]

A log generic hypersurface in P n with respect to a birational modification of P n is by definition the image of a generic element of a high power of an ample linear series on the modification. A log very-generic hypersurface is defined similarly but restricting to line bundles satisfying a non-resonance condition. Fixing a log resolution of a product f = f 1 ... f p of polynomials, we show that the monodromy conjecture, relating the motivic zeta function with the complex monodromy, holds for the tuple (f 1 , ... , f p , g) and for the product fg, if g is log generic. We also show that the stronger version of the monodromy conjecture, relating the motivic zeta function with the Bernstein–Sato ideal, holds for the tuple (f 1 , ... , f p , g) and for the product fg, if g is log very-generic. Even the case f = 1 is intricate, the proof depending on nontrivial properties of Bernstein–Sato ideals, and it singles out the class of log (very-) generic hypersurfaces as an interesting class of singularities on its own. [ABSTRACT FROM AUTHOR]

In this paper, we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of the moduli of elliptically fibred K3 surfaces and is closely related to the root lattices of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positivedimensional ambi-typical strata, that is, strata which are both Shimada root strata and monodromy strata. We also discuss the relationship with moduli spaces of latticepolarised K3 surfaces. The appendix by M. Kirschmer contains computational results about the 1-dimensional ambi-typical strata. [ABSTRACT FROM AUTHOR]

Motivated by experimental measurements indicating deviations from the Standard Model predictions, we discuss F-theory-inspired models, which, in addition to the three chiral generations, contain a vector-like complete fermion family. The analysis takes place in the context of SU(5) x U(1)' GUT embedded in an E8 covering group, which is associated with the (highest) geometric singularity of the elliptic fibration. In this context, the U(1)' is a linear combination of four abelian factors subjected to the appropriate anomaly cancellation conditions. Furthermore, we require universal U(1)' charges for the three chiral families and different ones for the corresponding fields of the vector-like representations. Under the aforementioned assumptions, we find 192 models that can be classified into five distinct categories with respect to their specific GUT properties. We exhibit representative examples for each such class and construct the superpotential couplings and the fermion mass matrices. We explore the implications of the vector-like states in low-energy phenomenology, including the predictions regarding the B-meson anomalies. The rôle of R-parity violating terms appearing in some particular models of the above construction is also discussed. [ABSTRACT FROM AUTHOR]

In this partly expository paper we discuss conditions for the global injectivity of C2 semi-algebraic local diffeomorphisms f:Rn→Rn. In case n>2, we consider the foliations of Rn defined by the level sets of each n−2 projections of f, i.e., the maps Rn→Rn−2 obtained by deleting two coordinate functions of f. It is known that if the set of non-proper points of f has codimension greater than or equal to 2 and the leaves of the above-defined foliations are simply connected, then f is bijective. In this work we relate this simply connectedness with the notion of locally trivial fibrations. Then some computable regularity conditions at infinity ensuring such simply connectedness are presented. Further, we provide an equivalent statement of the Jacobian conjecture by using fibrations. By means of examples we prove that the results presented here are different from a previous result based on a spectral hypothesis. Our considerations are also applied to discuss the behaviour of some conditions when f is composed with linear isomorphisms: this is relevant due to some misunderstandings appearing in the literature. [ABSTRACT FROM AUTHOR]

We prove homological mirror symmetry for Milnor fibers of simple singularities in dimensions greater than 1, which are among the log Fano cases of Conjecture 1.5 of the authors' manuscript "Homological mirror symmetry for Milnor fibers via moduli of A∞-structures" [arXiv: 1806.04345]. The proof is based on a relation between matrix factorizations and Calabi-Yau completions. As an application, we give an explicit computation of the Hochschild cohomology group of the derived n-preprojective algebra of a Dynkin quiver for any n ≥ 1, and the symplectic cohomology group of the Milnor fiber of any simple singularity in any dimension greater than 1. [ABSTRACT FROM AUTHOR]

We present local classification results for isolated singularities of functions with respect to a Nambu structure (multi-vector field) of maximal degree, in a neighbourhood of a smooth point of its degeneracy hypersurface. The results depend on a logarithmic version of the Brieskorn-Sebastiani theorem, which guarantees the finiteness and freeness of the corresponding deformation module. This relates the functional moduli of the classification problem with the integrals of logarithmic forms along the vanishing cycles of the complement of the Milnor fibres of the restriction of the function on the degeneracy hypersurface of the Nambu structure, inside the Milnor fibres of the function itself. [ABSTRACT FROM AUTHOR]

We describe a theory of packing hyperboloid “diabolic” domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay–twist analogues of blue phases. We also discuss how these diabolic domains may be subject to “superluminal boosts,” yielding defects analogous to shock waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space. [ABSTRACT FROM AUTHOR]

We study the topological K-theory spectrum of the dg singularity category associated to a weighted projective complete intersection. We calculate the topological K-theory of the dg singularity category of a weighted projective hypersurface in terms of its affine Milnor fiber and monodromy operator, and, as an application, we obtain a lift of the Atiyah-Bott-Shapiro construction to the level of spectra. [ABSTRACT FROM AUTHOR]

Inspired by Denef-Loeser's identity of the Euler characteristic with compact supports of the contact loci with the Lefschetz numbers of a complex singularity, we study sheaf cohomology groups of contact loci of complex nondegenerate singularities. Moreover, also for these singularities, we obtain a motivic analogue of Lê Dũng Tráng's work on a monodromy relation of a complex singularity and its restriction to a generic hyperplane. [ABSTRACT FROM AUTHOR]

We define local fibration structures for real map germs with strictly positive dimensional discriminant: a local fibration structure over the complement of the discriminant, and a complete local fibration structure which includes the stratified discriminant into the picture. We provide new classes of map germs endowed with such local fibration structures. [ABSTRACT FROM AUTHOR]

This note corrects a computation in [3, Lemma 2.5] reformulating the criterion for " Disc f g ¯ ⊂ { 0 } " in terms of Disc (f , g). [ABSTRACT FROM AUTHOR]

Milnor's fibration theorem is about the geometry and topology of real and complex analytic maps near their critical points, a ubiquitous theme in mathematics. As such, after 50 years, this has become a whole area of research on its own, with a vast literature, plenty of different viewpoints, a large progeny, and connections with many other branches of mathematics. In this work we revisit the classical theory in both the real and complex settings, and we glance at some areas of current research and connections with other important topics. The purpose of this article is twofold. On the one hand, it should serve as an introduction to the topic for nonexperts, and on the other hand, it gives a wide perspective of some of the work on the subject that has been and is being done. It includes a vast literature for further reading. [ABSTRACT FROM AUTHOR]

It is due to Milnor that there is no Herman ring for any rational map with two critical values. In this paper, we are concerned with if there exist Herman rings for rational maps with three critical values. We first prove that such rational maps cannot have Herman rings of period 1, 2, or 3. Then we prove all regularly ramified rational maps have no Herman rings in their Fatou sets. [ABSTRACT FROM AUTHOR]

Abstract We describe an algorithm computing the monodromy and the pole order filtration on the Milnor fiber cohomology of any reduced projective plane curve C. The relation to the zero set of Bernstein–Sato polynomial of the defining homogeneous polynomial for C is also discussed. When C has some non-weighted homogeneous singularities, then we have to assume that a conjecture holds in order to get some of our results. In all the examples computed so far this conjecture holds. [ABSTRACT FROM AUTHOR]

Abstract We define and study opfibrations of V -enriched categories when V is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary category B , there is an equivalence of 2-categories between V -enriched opfibrations over the free V -category on B , and pseudofunctors from B to the 2-category of V -categories. This generalizes the classical (Set -enriched) Grothendieck correspondence. [ABSTRACT FROM AUTHOR]

An, 4-cellular space is a space built from a space A and its suspensions, analogous to the way that CW-complexes are built from S'° and its suspensions. The A-cellular approximation of a space X is an A-cellular space CWAX, which is closest to X among all. A-cellular spaces. The A-complexity of a space X is an ordinal number that quantifies how difficult it is to build an, 4-cellular approximation of X. In this paper, we study spaces with low complexity. In particular, we show that if A is a sphere localized at a set of primes then the A.-complexity of each space X is at most 1. [ABSTRACT FROM AUTHOR]

Abstract We introduce a class of F-theory vacua which may be viewed as a specialization of the so-called E 6 fibration, and construct a weak coupling limit associated with such vacua which we view as the 'square' of the Sen limit. We find that while Sen's limit is naturally viewed as an orientifold theory, the universal tadpole relation which equates the D3 charge between the associated F-theory compactification and the limit we construct suggests that perhaps the limiting theory is in fact an oriented theory compactified on the base of the F-theory elliptic fibration. [ABSTRACT FROM AUTHOR]

Abstract We study configuration space integral formulas for Milnor's homotopy link invariants, showing that they are in correspondence with certain linear combinations of trivalent trees. Our proof is essentially a combinatorial analysis of a certain space of trivalent "homotopy link diagrams" which corresponds to all finite type homotopy link invariants via configuration space integrals. An important ingredient is the fact that configuration space integrals take the shuffle product of diagrams to the product of invariants. We ultimately deduce a partial recipe for writing explicit integral formulas for products of Milnor invariants from trivalent forests. We also obtain cohomology classes in spaces of link maps from the same data. [ABSTRACT FROM AUTHOR]

Abstract Milnor–Thurston homology also known as "measure homology" is a generalisation of singular homology where chains are defined to be measures (possibly supported on infinitely many singular simplices). In this paper we prove that the canonical homomorphism from singular to Milnor–Thurston homology is injective in dimension zero for metrizable spaces. [ABSTRACT FROM AUTHOR]

We present some techniques that can be used to factorize the monodromy of certain Milnor open books. We also describe a class of Milnor open books for which we can explicitly express the monodromy as a product of Dehn twists. [ABSTRACT FROM AUTHOR]

We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that: – Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism. – Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow-downs from a special resolution (so-called the maximal resolution) of the singularity, which is an analogue of the one-to-one correspondence between the irreducible components of the reduced versal deformation space and the so-called P -resolutions of a quotient surface singularity. [ABSTRACT FROM AUTHOR]

An inverse limit of a sequence of covering spaces over a given space X is not, in general, a covering space over X but is still a lifting space , i.e. a Hurewicz fibration with unique path lifting property. Of particular interest are inverse limits of finite coverings (resp. finite regular coverings), which yield fibrations whose fibre is homeomorphic to the Cantor set (resp. profinite topological group). To illustrate the breadth of the theory, we present in this note some curious examples of lifting spaces that cannot be obtained as inverse limits of covering spaces. [ABSTRACT FROM AUTHOR]

This paper presents a notion of Lusternik–Schnirelmann category for small categories, which is an invariant under homotopy equivalences based on natural transformations. We focus on the relationship between this categorical Lusternik–Schnirelmann category and the classical one via the classifying space. We provide a combinatorial method to calculate the classical Lusternik–Schnirelmann category of the classifying space of a finite acyclic category, taking the barycentric subdivision into account. Moreover, we establish the product inequality for fibered and cofibered categories as an analogue of the inequality of the classical Lusternik–Schnirelmann category for Hurewicz fibrations. [ABSTRACT FROM AUTHOR]

Let L be a cyclic L ∞ -algebra of dimension 3 with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic L ∞ -algebra structure on the cohomology H ⁎ ( L ) . The inner product plus the higher products of the cyclic L ∞ -algebra defines a superpotential function f on H 1 ( L ) . We associate with an analytic Milnor fiber for the formal function f and define the Euler characteristic of L is to be the Euler characteristic of the étale cohomology of the analytic Milnor fiber. In this paper we prove a Thom–Sebastiani type formula for the Euler characteristic of cyclic L ∞ -algebras. As applications we prove the Joyce–Song formulas about the Behrend function identities for semi-Schur objects in the derived category of coherent sheaves over Calabi–Yau threefolds. A motivic Thom–Sebastiani type formula and a conjectural motivic Joyce–Song formula for the motivic Milnor fiber of cyclic L ∞ -algebras are also discussed. [ABSTRACT FROM AUTHOR]

We find natural and convenient conditions which allow us to produce classes of genuine real map germs with Milnor tube fibration, either with Thom regularity or without it. [ABSTRACT FROM AUTHOR]

Given X a path connected space and g : X → R a local fibration on its image, we prove that g satisfies a kind of deformation and consequently we obtain the path connectedness of its level sets. Then we provide global injectivity and invertibility theorems for local homeomorphisms f : X → R n . These generalize known analytical results such as those given by Balreira and by Silva and Teixeira. [ABSTRACT FROM AUTHOR]

The aim of this article is to develop the theory of Kähler differentials on algebroid irreducible plane curves defined over algebraically closed fields of positive characteristic and to compare it with the theory over C . We emphasize here the study of the properties of the set of values of Kähler differentials, since they were an important invariant for the analytic classification of branches (cf. [8] ), hoping that they will play an important role in positive characteristic too. At the end, we give a counterexample in positive characteristic for the result of Zariski that asserts that if a complex algebroid irreducible plane curve has coincident Tjurina and Milnor numbers, then the curve is equivalent to a monomial curve. [ABSTRACT FROM AUTHOR]

Let f : X → B be a locally non-trivial relatively minimal fibration of curves of genus g ≥ 2 . We obtain a lower bound of the slope λ ( f ) increasing with the gonality of the general fiber of f . In particular, we show that λ ( f ) ≥ 4 provided that f is non-hyperelliptic and g ≥ 16 . [ABSTRACT FROM AUTHOR]

We show that the forgetful functor from the category of braided regular crossed modules to the category of regular (or whiskered) groupoids is a fibration and also a cofibration. [ABSTRACT FROM AUTHOR]

We show that the topological complexity of the nonorientable surfaces of genus ≥4 is four. [ABSTRACT FROM AUTHOR]

We prove a special case of a conjecture of Dao–Kurano concerning the vanishing of Hochster's theta pairing. The proof uses Adams operations on both topological K -theory and perfect complexes with support. [ABSTRACT FROM AUTHOR]

Let A be a commutative ring containing the rationals. Let S be a multiplicatively closed subset such that 1 ∈ S and 0 ∉ S , T a cone in A such that S ⊂ T and I an ideal in A . Then ρ S , T I = { a | s a 2 m + t ∈ I 2 m for some m ∈ N , s ∈ S and t ∈ T } is an ideal. For a commutative ring the collection of non-reduced orders (total cones) is a fibration of the real spectrum. Both concepts carry information regarding multiple solutions in the constructible set associated with I , T and S . When the ring is a real regular domain, a non-reduced Nullstellensatz is presented that extends the real Nullstellensatz and relates these concepts. The notion of real multiplicity is proposed and examined for elements that are either positive definite (PD) or positive semi-definite (PSD) on the real spectrum. [ABSTRACT FROM AUTHOR]