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A well-known theorem of Kulikov, Persson and Pinkham states that a degeneration of a family of K3-surfaces with trivial monodromy can be completed to a smooth family. We give a simple example that an analogous statement does not hold for Calabi–Yau threefolds. [ABSTRACT FROM AUTHOR]

We show that a family of isolated complete intersection singularities (ICIS) with constant total Milnor number has no coalescence of singularities. This extends a well-known result of Gabriélov, Lazzeri and Lê for hypersurfaces. We use A'Campo's theorem to see that the Lefschetz number of the generic monodromy of the ICIS is zero when the ICIS is singular. We give a pair applications for families of functions on ICIS which extend also some known results for functions on a smooth variety. [ABSTRACT FROM AUTHOR]

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fiber bundles. They are known to have positive signature divisible by 4. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyze by studying the monodromy action for bundles of punctured curves. As a by-product, we obtain a classification of all fix-point-free automorphisms on curves of genus at most 9. [ABSTRACT FROM AUTHOR]

In a previous paper, the authors defined an equivariant version of the so-called Saito duality between the monodromy zeta functions as a sort of Fourier transform between the Burnside rings of an abelian group and of its group of characters. Here, a so-called enhanced Burnside ring B ̂ (G) of a finite group G is defined. An element of it is represented by a finite G -set with a G -equivariant transformation and with characters of the isotropy subgroups associated to all points. One gives an enhanced version of the equivariant Saito duality. For a complex analytic G -manifold with a G -equivariant transformation of it one has an enhanced equivariant Euler characteristic with values in a completion of B ̂ (G). It is proved that the (reduced) enhanced equivariant Euler characteristics of the Milnor fibers of Berglund–Hübsch dual invertible polynomials are enhanced dual to each other up to sign. As a byproduct, this implies the result about the orbifold zeta functions of Berglund–Hübsch–Henningson dual pairs obtained earlier. [ABSTRACT FROM AUTHOR]

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an n × n linear system of ODEs with an irregular singularity of Poincaré rank 1 at z = ∞ and Fuchsian singularity at z = 0 , holomorphically depending on parameter t within a polydisk in ℂ n centered at t = 0. The eigenvalues of the leading matrix at ∞ , which is diagonal, coalesce along a coalescence locus Δ contained in the polydisk. Under minimal vanishing conditions on the residue matrix at z = 0 , we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.] that isomonodromic deformations can be extended to the whole polydisk, including Δ , in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of Δ , where it simplifies. Conversely, if the t -dependent system is isomonodromic in a small domain contained in the polydisk not intersecting Δ , and if suitable entries of the Stokes matrices vanish, then Δ is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations. [ABSTRACT FROM AUTHOR]

We study the rigidity of three-dimensional representations of braid groups associated with finite primitive irreducible complex reflection groups in . In many cases, we show the rigidity. For rigid representations, we give explicit forms of the representations, which turns out to be the monodromy representations of uniformization equations of Saito-Kato-Sekiguchi [Uniformization systems of equations with singularities along the discriminant sets of complex reflection groups of rank three, Kyushu J. Math. 68 (2014) 181-221; On the uniformization of complements of discriminant loci, RIMS Kokyuroku 287 (1977) 117-137]. Invariant Hermitian forms are also studied. [ABSTRACT FROM AUTHOR]

We consider preheating in models in which the potential for the inflaton is given by a fractional power, as is the case in axion monodromy inflation. We assume a standard coupling between the inflaton field and a scalar matter field. We find that in spite of the fact that the oscillation of the inflaton about the field value which minimizes the potential is anharmonic, there is nevertheless a parametric resonance instability, and we determine the Floquet exponent which describes this instability as a function of the parameters of the inflaton potential. [ABSTRACT FROM AUTHOR]

Consider an arrangement of homogeneous hyperplanes in with complement . The (co)homology of with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto endowed with the geometric monodromy. It is still an open problem to understand in general the cohomology of the Milnor fiber, even for dimension 1. In Sec. 1, we show that all questions about the first homology group are detected by a precise group, which is a quotient ot the commutator subgroup of by the commutator of its length zero subgroup, which didn't appear in the literature before. In Sec. 2, we state a conjecture of -monodromicity for the first homology, which is of a different nature with respect to the known results. Let be the graph of double points of we conjecture that if is connected, then the geometric monodromy acts trivially on the first homology of the Milnor fiber (so the first Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of . We show the truth of the conjecture under some stronger hypotheses. [ABSTRACT FROM AUTHOR]

Fujita's second theorem for Kähler fibre spaces over a curve asserts, that the direct image of the relative dualizing sheaf splits as the direct sum , where is ample and is unitary flat. We focus on our negative answer [F. Catanese and M. Dettweiler, Answer to a question by Fujita on variation of Hodge structures, to appear in Adv. Stud. Pure Math.] to a question by Fujita: is semiample? We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group of a Del Pezzo surface of degree 5 (branched on the union of the lines of , which form a bianticanonical divisor), and endowed with a semistable fibration with only three singular fibres. The simplest such surfaces are the three ball quotients considered in [I. C. Bauer and F. Catanese, A volume maximizing canonical surface in 3-space, Comment. Math. Helv. 83(1) (2008) 387-406.], fibred over a curve of genus 2, and with fibres of genus 4. These examples are a larger class than the ones corresponding to Shimura curves in the moduli space of Abelian varieties. [ABSTRACT FROM AUTHOR]

We give a simplified definition of topological T-duality that applies to arbitrary torus bundles. The new definition does not involve Chern classes or spectral sequences, only gerbes and morphisms between them. All the familiar topological conditions for T-duals are shown to follow. We determine necessary and sufficient conditions for existence of a T-dual in the case of affine torus bundles. This is general enough to include all principal torus bundles as well as torus bundles with arbitrary monodromy representations. We show that isomorphisms in twisted cohomology, twisted K-theory and of Courant algebroids persist in this general setting. We also give an example where twisted K-theory groups can be computed by iterating T-duality. [ABSTRACT FROM AUTHOR]

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber-Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart. [ABSTRACT FROM AUTHOR]

For any discrete group Γ and any two-dimensional complex representation ρ of Γ, we introduce the notion of ρ-equivariant functions, and we show that they are parametrized by vector-valued modular forms. We also provide examples arising from the monodromy of differential equations. [ABSTRACT FROM AUTHOR]

A real 3- or 4-manifold has by definition an orientation preserving smooth involution acting on it. We consider Lefschetz fibrations of 4-dimensional manifolds-with-boundary and open book decompositions on their boundary in the existence of a real structure. We prove that there is a real open book which cannot be filled by a real Lefschetz fibration, although it is filled by non-real Lefschetz fibrations. [ABSTRACT FROM AUTHOR]

Fix a prime Í, and let Fq be a finite field with q = 1 (mod t) elements. If Z > 2 and q 1, we show that asymptotically ((. -- \)2 /2Í2 of the elliptic curves E/Fq with complete rational ¿-torsion are such that E/{P) does not have complete rational i-torsion for any point P G E(¥q) of order t For t = 2 the asymptotic density is 0 or 1/4, depending whether q = 1 (mod 4) or 3 (mod 4). We also show that for any i, if E/¥q has an IFq-rational point R of order I2, then E/(ÍR) always has complete rational ¿-torsion. [ABSTRACT FROM AUTHOR]

We show that totally real elliptic Lefschetz fibrations admitting a real section are classified by their "real loci" which can be encoded in terms of a combinatorial object that we call a necklace diagram. By means of necklace diagrams, we obtain an explicit list of certain classes of totally real elliptic Lefschetz fibrations. [ABSTRACT FROM AUTHOR]

This paper looks at the L-function of the kth symmetric power of the $\overline{\mathbb{Q}}_\ell$-sheaf Aif over the affine line $\mathbb{A}^1_{\mathbb{F}_q}$ associated to the generalized Airy family of exponential sums. Using ℓ-adic techniques, we compute the degree of this rational function as well as the local factors at infinity. Using p-adic techniques, we study the q-adic Newton polygon of the L-function. [ABSTRACT FROM AUTHOR]

We formulate and prove a generalization of Zariski–van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz–Zariski–van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties. [ABSTRACT FROM AUTHOR]

Let S[n] be the Hilbert scheme of length n subschemes of a K3 surface S. H2(S[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL[H*(S[n],ℤ)] generated by monodromy operators, and let Mon2 be its image in OH2(S[n],ℤ). We prove that Mon2 is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon2 does not surject onto OH2(S[n],ℤ)/(±1), when n - 1 is not a prime power. As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S[n]. The weight 2 Hodge structure on H2(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2ρ(n - 1) - 1 distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1. The second main result states, that if a monodromy operator acts as the identity on H2(S[n],ℤ), then it acts as the identity on Hk(S[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon2, if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6). [ABSTRACT FROM AUTHOR]

We describe open book decompositions of links of simple surface singularities that support the corresponding unique Milnor fillable contact structures. The open books we describe are isotopic to Milnor open books. [ABSTRACT FROM AUTHOR]

We say that a complex analytic space, X, is an intersection cohomology manifold if and only if the shifted constant sheaf on X is isomorphic to intersection cohomology; with field coefficients, this is quickly seen to be equivalent to X being a homology manifold. Given an analytic function f on an intersection cohomology manifold, we describe a simple relation between V(f) being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of f. We then describe how the Sebastiani–Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Finally, as an easy application, we obtain restrictions on the cohomology of the Milnor fiber of a hypersurface with a special type of one-dimensional critical locus. [ABSTRACT FROM AUTHOR]

Let X ⊂ ℙ2m+1 be a projective variety with isolated singularities, complete intersection of a smooth hypersurface of degree k, with a smooth hypersurface F of degree n > k. Denote by NSm(F) and NSm(X) the mth Néron–Severi groups. We prove that if rkNSm(F) = 1, then rkNSm(X) = 1. Moreover, we prove that if $F \in |\mathcal{I}_{X,\mathbb{P}^{2m+1}}(n)|$ is general and n > max{k, 2m + 1}, then the natural map NSm(X) ⊗ ℚ → NSm(F) ⊗ ℚ is surjective. When X is a threefold, we deduce that X is factorial if and only if rkNS2(F) = 1. This allows us to prove the existence of factorial threefolds with many singularities. [ABSTRACT FROM AUTHOR]

In a former work, recalling what the Markoff theory is, we summarized some existing links with the group GL(2, ℤ) of 2 × 2 matrices. We also quoted the relation with conformal punctured toruses. The monodromy representation of the Poincaré group of such a torus was considered. Here we explicit the corresponding solution of the associated Riemann-Hilbert problem, and the resulting Fuchs differential equation. We precisely describe how the calculus runs. The main result is the description of a complete family of Fuchs differential equations with, as the monodromy group, the free group with two generators. We also identify a link with some eigenvalues of a Laplacian. The introduction explains the links that we see with information and computation theory (classical or quantum). [ABSTRACT FROM AUTHOR]

This primarily expository article collects together some facts from the literature about the monodromy of differential equations on a p-adic (rigid analytic) annulus, though often with simpler proofs. These include Matsuda's classification of quasi-unipotent ∇-modules, the Christol–Mebkhout construction of the ramification filtration, and the Christol–Dwork Frobenius antecedent theorem. We also briefly discuss the p-adic local monodromy theorem without proof. [ABSTRACT FROM AUTHOR]

A surface bundle determines a monodromy representation recording the twisting of the fiber under transport around a closed path in the base space. The fascinating relation between these monodromy representations and the Thurston classification of surface automorphisms will be studied. In this note we deal with a simple and interesting case: the fibrations of Fadell and Neuwirth. [ABSTRACT FROM AUTHOR]