arXiv subjects

Claus Hertling

Publications and source records attributed to Claus Hertling.

At least 37 records · Page 2Linked to original sources

arXiv2018

Automorphisms with eigenvalues in $S^1$ of a ${\mathbb Z}$-lattice with cyclic finite monodromy

For any finite set $M\subset {\mathbb Z}_{\geq 1}$ of positive integers, there is up to isomorphism a unique ${\mathbb Z}$-lattice $H_M$ with a cyclic automorphism $h_M:H_M\to H_M$ whose eigenvalues are the unit roots with orders in $M$ and have multiplicity 1. The paper studies the automorphisms of the pair $(H_M,h_M)$ which have eigenvalues in $S^1$. The main result are necessary and sufficient conditions on the set $M$ such that the only such automorphisms are $\pm h_M^k,k\in{\mathbb Z}$. The proof uses resultants and cyclotomic polynomials. It is elementary, but involved. Special cases of the main result have been applied to the study of the automorphisms of Milnor lattices of isolated hypersurface singularities.

math.NT↗

arXiv2017

Potentials of a Frobenius like structure

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame which encompasses families of arrangements. The frame uses the notion of matroids. For the proof of the existence of the potentials, a power series ansatz is made. The proof that it works requires that certain decompositions of tuples of coordinate vector fields are related by certain elementary transformations. This is shown with a nontrivial result on matroid partition.

math.AG↗

arXiv2017

Real Seifert forms and polarizing forms of Steenbrink mixed Hodge structures

An isolated hypersurface singularity comes equipped with many different pairings on different spaces, the intersection form and the Seifert form on the Milnor lattice, a polarizing form for a mixed Hodge structure on a dual space, and a flat pairing on the cohomology bundle. This paper describes them and their relations systematically in an abstract setting. We expect applications also in other areas than singularity theory. A good part of the paper is elementary, but not well known: the classification of irreducible Seifert form pairs, the polarizing form on the generalized eigenspace with eigenvalue 1, an automorphism from a Fourier-Laplace transformation which involves the Gamma function and which relates Seifert form and polarizing form and a flat pairing on the cohomology bundle. New is a correction of a Thom-Sebastiani formula for Steenbrink's Hodge filtration in the case of singularities. It uses the Fourier-Laplace transformation. A special case is a square root of a Tate twist for Steenbrink mixed Hodge structures.

math.AG↗

arXiv2017

Conjectures on spectral numbers for upper triangular matrices and for singularities

Cecotti and Vafa proposed in 1993 a beautiful idea how to associate spectral numbers $α_1,...,α_n\in{\mathbb R}$ to real upper triangular $n\times n$ matrices $S$ with 1's on the diagonal and eigenvalues of $S^{-1}S^t$ in the unit sphere. Especially, $\exp(-2πiα_j)$ shall be the eigenvalues of $S^{-1}S^t$. We tried to make their idea rigorous, but we succeeded only partially. This paper fixes our results and our conjectures. For certain subfamilies of matrices their idea works marvellously, and there the spectral numbers fit well to natural (split) polarized mixed Hodge structures. We formulate precise conjectures saying how this should extend to all matrices $S$ as above. The idea might become relevant in the context of semiorthogonal decompositions in derived algebraic geometry. Our main interest are the cases of Stokes like matrices which are associated to holomorphic functions with isolated singularities (Landau-Ginzburg models). Also there we formulate precise conjectures (which overlap with expectations of Cecotti and Vafa). In the case of the chain type singularities, we have positive results. We hope that this paper will be useful for further studies of the idea of Cecotti and Vafa.

math.AG↗

arXiv2017

$μ$-constant monodromy groups and Torelli results for the quadrangle singularities and the bimodal series

This paper is a sequel to [He11] and [GH17]. In [He11] a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_μ^{mar}$ for marked singularities in one $μ$-homotopy class of isolated hypersurface singularities was established. It is an analogue of a Teichmüller space. It comes together with a $μ$-constant monodromy group $G^{mar}\subset G_{\mathbb{Z}}$. Here $G_{\mathbb{Z}}$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_μ^{mar}$ is connected. This is equivalent to $G^{mar}= G_{\mathbb{Z}}$. Also Torelli type conjectures were formulated. In [He11] and [GH17] $M_μ^{mar}, G_{\mathbb{Z}}$ and $G^{mar}$ were determined and all conjectures were proved for the simple, the unimodal and the exceptional bimodal singularities. In this paper the quadrangle singularities and the bimodal series are treated. The Torelli type conjectures are true. But the conjecture $G^{mar}= G_{\mathbb{Z}}$ and $M_μ^{mar}$ connected does not hold for certain subseries of the bimodal series.

math.AG↗

arXiv2016

Potentials of a Frobenius like structure and $m$ bases of a vector space

This paper proves the existence of potentials of the first and second kind of a Frobenius like structure in a frame which encompasses families of arrangements. Surprisingly the proof is based on the study of finite sets of vectors in a finite-dimensional vector space $V$. Given a natural number $m$ and a finite set $(v_i)$ of vectors we give a necessary and sufficient condition to find in the set $(v_i)$ $m$ bases of $V$. If $m$ bases in $(v_i)$ can be selected, we define elementary transformations of such a selection and show that any two selections are connected by a sequence of elementary transformations.

math.AG↗

arXiv2016

Regular F-manifolds: initial conditions and Frobenius metrics

A regular F-manifold is an F-manifold (with Euler field) (M, \circ, e, E), such that the endomorphism {\mathcal U}(X) := E \circ X of TM is regular at any p\in M. We prove that the germ ((M,p), \circ, e, E) is uniquely determined (up to isomorphism) by the conjugacy class of {\mathcal U}_{p} : T_{p}M \rightarrow T_{p}M. We obtain that any regular F-manifold admits a preferred system of local coordinates and we find conditions, in these coordinates, for a metric to be Frobenius. We study the Lie algebra of infinitesimal symmetries of regular F-manifolds. We show that any regular F-manifold is locally isomorphic to the parameter space of a Malgrange universal connection. We prove an initial condition theorem for Frobenius metrics on regular F-manifolds.

math.DG↗

arXiv2016

On the classification of quasihomogeneous singularities

The motivation for this paper are computer calculations of complete lists of weight systems of quasihomogeneous polynomials with isolated singularity at 0 up to rather large Milnor numbers. We review combinatorial characterizations of such weight systems for any number of variables. This leads to certain types and graphs of such weight systems. Using them, we prove an upper bound for the common denominator (and the order of the monodromy) by the Milnor number, and we show surprising consequences if the Milnor number is a prime number.

math.AG↗

arXiv2016

$μ$-constant monodromy groups and Torelli results for marked singularities, for the unimodal and some bimodal singularities

This paper is a sequel to [He7]. There a notion of marking of isolated hypersurface singularities was defined, and a moduli space $M_μ^{mar}$ for marked singularities in one $μ$-homotopy class of isolated hypersurface singularities was established. One can consider it as a global $μ$-constant stratum or as a Teichmüller space for singularities. It comes together with a $μ$-constant monodromy group $G^{mar}\subset G_Z$. Here $G_Z$ is the group of automorphisms of a Milnor lattice which respect the Seifert form. It was conjectured that $M_μ^{mar}$ is connected. This is equivalent to $G^{mar}= G_Z$. Also Torelli type conjectures were formulated. All conjectures were proved for the simple singularities and 22 of the exceptional unimodal and bimodal singularities. In this paper the conjectures are proved for the remaining unimodal singularities and the remaining exceptional bimodal singularities.

math.AG↗

arXiv2016

Reduced and nonreduced presentations of Weyl group elements

This paper is a sequel to work of Dynkin on subroot lattices of root lattices and to work of Carter on presentations of Weyl group elements as products of reflections. The quotients $L/L_1$ are calculated for all irreducible root lattices $L$ and all subroot lattices $L_1$. The reduced (i.e. those with minimal number of reflections) presentations of Weyl group elements as products of arbitrary reflections are classified. Also nonreduced presentations are studied. Quasi Coxeter elements and strict quasi Coxeter elements are defined and classified. An application to extended affine root lattices is given. A side result is that any set of roots which generates the root lattice contains a $Z$-basis of the root lattice.

math.RT↗

arXiv2016

Hermitian metrics on F-manifolds

An $F$-manifold is complex manifold with a multiplication on the holomorphic tangent bundle with a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metrics on $F$-manifolds from $tt^*$ geometry. It clarifies the logic between several notions. It also introduces a new {\it canonical} hermitian metric. Near irreducible points it makes the manifold almost hyperbolic. This holds for the singularity case and will hopefully lead to applications there.

math.DG↗

arXiv2015

The Painlevé III equation of type (0,0,4,-4), its associated vector bundles with isomonodromic connections, and the geometry of the movable poles

The paper is about a Painlevé III equation and its relation to isomonodromic families of vector bundles on P^1 with meromorphic connections. The purpose of the paper is two-fold: it offers a conceptual language for the geometrical objects underlying Painlevé equations, and it offers new results on a particular Painlevé III equation, which we denote by P_{III}(0,0,4,-4). This is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears very widely in geometry and physics. Complex multi-valued solutions on C^* are the natural context for most of the paper, but in the last three chapters real solutions on the positive real line (with or without singularities) are addressed. Results about the asymptotics of real solutions near 0 and near infinity are combined with results on the global geometry of the moduli spaces of initial data and monodromy data. This leads to a new global picture of all zeros and poles of all real solutions on the positive real line.

math.AG↗

arXiv2011

mu-constant monodromy groups and marked singularities

mu-constant families of holomorphic function germs with isolated singularities are considered from a global perspective. First, a monodromy group from all families which contain a fixed singularity is studied. It consists of automorphisms of the Milnor lattice which respect not only the intersection form, but also the Seifert form and the monodromy. We conjecture that it contains all such automorphisms, modulo {id,-id}. Second, marked singularities are defined and global moduli spaces for right equivalence classes of them are established. The conjecture on the group would imply that these moduli spaces are connected. The relation with Torelli type problems is discussed and a new global Torelli type conjecture for marked singularities is formulated. All conjectures are proved for the simple and 22 of the 28 exceptional singularities.

math.AG↗

arXiv2010

Examples of non-commutative Hodge structures

We show that if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semi-definite Hermitian form, then the associated integrable twistor structure (or TERP structure, or non-commutative Hodge structure) is pure and polarized.

math.AG↗

arXiv2009

Limits of families of Brieskorn lattices and compactified classifying spaces

We investigate variations of Brieskorn lattices over non-compact parameter spaces, and discuss the corresponding limit objects on the boundary divisor. We study the associated variation of twistors and the corresponding limit mixed twistor structures. We construct a compact classifying space for regular singular Brieskorn lattices and prove that its pure polarized part carries a natural hermitian structure and that the induced distance makes it into a complete metric space.

math.AG↗

arXiv2009

Frobenius manifolds, projective special geometry and Hitchin systems

We consider the construction of Frobenius manifolds associated to projective special geometry and analyse the dependence on choices involved. In particular, we prove that the underlying F-manifold is canonical. We then apply this construction to integrable systems of Hitchin type.

math.AG↗

arXiv2008

Curvature of classifying spaces for Brieskorn lattices

We study tt*-geometry on the classifying space for regular singular TERP-structures, e.g., Fourier-Laplace transformations of Brieskorn lattices of isolated hypersurface singularities. We show that (a part of) this classifying space can be canonically equipped with a hermitian structure. We derive an estimate for the holomorphic sectional curvature of this hermitian metric, which is the analogue of a similar result for classifying spaces of pure polarized Hodge structures.

math.AG↗

arXiv2008

Nilpotent orbits of a generalization of Hodge structures

We study a generalization of Hodge structures which first appeared in the work of Cecotti and Vafa. It consists of twistors, that is, holomorphic vector bundles on P^1, with additional structure, a flat connection on C^*, a real subbundle and a pairing. We call these objects TERP-structures. We generalize to TERP-structures a correspondence of Cattani, Kaplan and Schmid between nilpotent orbits of Hodge structures and polarized mixed Hodge structures. The proofs use work of Simpson and Mochizuki on variations of twistor structures and a control of the Stokes structures of the poles at zero and infinity. The results are applied to TERP-structures which arise via oscillating integrals from holomorphic functions with isolated singularities.

math.AG↗