Narrow Results

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.

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.

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.

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 2^{ρ(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 H^k(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).

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.

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.

Fujita’s second theorem for Kähler fibre spaces over a curve asserts, that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $V=A\oplus Q$, where $A$ is ample and $Q$ 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 $V$ 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 ${(\mathbb{Z}/n)}^2$ of a Del Pezzo surface $Z$ of degree 5 (branched on the union of the lines of $Z$, 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.

We study the rigidity of three-dimensional representations of braid groups associated with finite primitive irreducible complex reflection groups in $\mathrm{\text{GL}}(3,ℂ)$. 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 Kokyuroku287 (1977) 117–137]. Invariant Hermitian forms are also studied.

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 $9$.

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.

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.

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).

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.

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.

Consider an arrangement ${𝒜}$ of homogeneous hyperplanes in ${ℂ}^n,$ with complement ${ℳ}({𝒜})$. The (co)homology of ${ℳ}({𝒜})$ with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto ${ℂ}^{\ast },$ 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 ${\pi }_1({ℳ}({𝒜}))$ by the commutator of its length zero subgroup, which didn’t appear in the literature before.

In Sec. 2, we state a conjecture of $a$-monodromicity for the first homology, which is of a different nature with respect to the known results. Let $\mathrm{\Gamma }$ be the graph of double points of ${𝒜}:$ we conjecture that if $\mathrm{\Gamma }$ 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 ${𝒜}$. We show the truth of the conjecture under some stronger hypotheses.

Let X ⊂ ℙ^2m+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 NS_m(F) and NS_m(X) the mth Néron–Severi groups. We prove that if rkNS_m(F) = 1, then rkNS_m(X) = 1. Moreover, we prove that if is general and n > max{k, 2m + 1}, then the natural map NS_m(X) ⊗ ℚ → NS_m(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.

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.

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.

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 $\widehat{B}(G)$ 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 $\widehat{B}(G)$. 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.

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.