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Chris Peters

Publications and source records attributed to Chris Peters.

At least 19 recordsLinked to original sources
arXiv2026

Incidence equivalence, a survey

This is a survey of results on incidence equivalence, a notion introduced by P$.$Griffiths around 1970 when trying to extend the classical properties of the Abel-Jacobi map for curves. Using the intermediate jacobians and the associated Abel-Jacobi maps in higher dimension, a natural question came up: is the geometrically defined incidence equivalence relation the same as Abel-Jacobi equivalence, which is of transcendental nature? I give an overview of results related to this question and to several classical conjectures that are far from resolved, such as Grothendieck's generalized Hodge conjecture. The motivation for writing this survey came from a recently observed unexpected connection of Griffiths' question to the asymptotic behaviour of the archimedean height pairing in a geometric setting.

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arXiv2025

Motivic apsects of a remarkable class of Calabi-Yau threefolds

In this note we consider the motivic aspect of the middle cohomology of more than 200 classes of quasi-smooth Calabi--Yau threefolds inside weighted projective 4-space which come with an action of a cyclic group of even order. The action induces a self-dual Chow--K\"unneth decomposition. All but one component correspond to Fano threefolds. For these the generalized Hodge conjecture is known, but thanks to the nature of the decomposition we can give a direct proof for one of the components.

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arXiv2024

On isolated hypersurface singularities: algebra-geometric and symplectic aspects

These notes are based on a seminar which took place in the autumn of 2022 at the Mathematical Institute of the University of Leiden. Its goal was to understand the recent work of J. Evans and Y. Lekili on the symplectic cohomology of the Milnor fiber for specific classes of isolated singularities. This work uses inputs from several fields, notably from algebraic geometry, in particular singularity theory, and from symplectic geometry. The main aim of the notes is to make the work of J. Evans and Y. Lekili more accessible by explaining the main ideas from these fields and indicate how these play a role in this work.

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arXiv2023

A note on the primitive cohomology lattice of a projective surface

The isometry class of the intersection form of a compact complex surface can be easily determined from complex-analytic invariants. For projective surfaces the primitive lattice is another naturally occurring lattice. The goal of this note is to show that it can be determined from the intersection lattice and the self-intersection of a primitive ample class, at least when the primitive lattice is indefinite. Examples include the Godeaux surfaces, the Kunev surface and a specific Horikawa surface. There are also some results concerning (negative) definite primitive lattices, especially for canonically polarized surfaces of general type.

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arXiv2023

A remarkable class of elliptic surfaces of amplitude 1 in weighted projective space

Surfaces of amplitude 1 in ordinary projective space are of general type, but this need not be the case in weighted projective spaces. Indeed, there are 4 classes of quasi-smooth weighted hypersurfaces in $\mathbf{P}(1,2,a,b)$ of amplitude 1 with an elliptic pencil cut out by hyperplanes. Their moduli spaces are constructed, the monodromy of their universal families is determined as well as their period maps. These all turn out to be non-injective. We analyse the reason behind this, which for each type is different. For the two classes that give properly elliptic surfaces this leads to a mixed Torelli-type theorem as in the case of the Catanese-Kunev-Todorov surfaces. We added an application to certain compactifications of moduli spaces of surfaces of general type with $K^2=1$, $p_g=2$ and $q=0$, as well as detailed SageMath-calculations. The appendix written by Wim Nijgh shows that the general member of the type 1 and type 2 elliptic family has "trivial" Picard lattice, i.e. is spanned by fiber components and a multisection.

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arXiv2021

On complex surfaces with definite intersection form

A compact complex surface with positive definite intersection lattice is either the projective plane or a false projective plane. If the intersection lattice is negative definite, the surface is either a non-minimal secondary Kodaira surface, a non-minimal elliptic surface with $b_1=1$, or a class VII surface with $b_2>0$. In all cases the lattice is odd and diagonalizable over the integers.

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arXiv2020

Holomorphic Bisectional Curvature and Applications to Deformations and Rigidity for Variations of Mixed Hodge Structure

In this article, we prove a rigidity criterion for period maps of admissible variations of graded-polarizable mixed Hodge structure, and establish rigidity in a number of cases, including families of quasi-projective curves, projective curves with ordinary double points, the complement of the canonical curve in families of Kynev--Todorov surfaces, period maps attached to the fundamental groups of smooth varieties and normal functions.

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arXiv2017

A motivic study of generalized Burniat surfaces

Generalized Burniat surfaces are surfaces of general type with $p_g=q$ and Euler number $e=6$ obtained by a variant of Inoue's construction method for the classical Burniat surfaces. I prove a variant of the Bloch conjecture for these surfaces. The method applies also to the so-called Sicilian surfaces introduced by Bauer, Catanese and Frapporti. This implies that the Chow motives of all of these surfaces are finite-dimensional in the sense of Kimura.

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arXiv2017

On a motivic interpretation of primitive, variable and fixed cohomology

This note addresses the motivic nature of some classical cohomological results due to Lefschetz, namely the primitive decomposition (for the cohomology of smooth projective varieties), and, secondly, the splitting of the cohomology of a complete intersection into the "fixed" and "variable part".

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arXiv2017

On complete intersections in varieties with finite-dimensional motive

Let $X$ be a complete intersection inside a variety $M$ with finite dimensional motive and for which the Lefschetz-type conjecture $B(M)$ holds. We show how conditions on the niveau filtration on the homology of $X$ influence directly the niveau on the level of Chow groups. This leads to a generalization of Voisin's result. The latter states that if $M$ has trivial Chow groups and if $X$ has non-trivial variable cohomology parametrized by $c$-dimensional algebraic cycles, then the cycle class maps $A_k(X) \to H_{2k}(X)$ are injective for $k<c$. We give variants involving group actions which lead to several new examples with finite dimensional Chow motives.

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arXiv2017

Differential Geometry of the Mixed Hodge Metric

We investigate properties of the Hodge metric of a mixed period domain. In particular, we calculate its curvature and the curvature of the Hodge bundles. We also consider when the pull back metric via a period map is Kähler. Several applications in cases of geometric interest are given, such as for normal functions and biextension bundles.

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arXiv2017

Rigidity of Spreadings and Fields of Definition

Varieties without deformations are defined over a number field. Several old and new examples of this phenomenon are discussed such as Bely\u ı curves and Shimura varieties. Rigidity is related to maximal Higgs fields which come from variations of Hodge structure. Basic properties for these due to P. Griffiths, W. Schmid, C. Simpson and, on the arithmetic side, to Y. André and I. Satake all play a role. This note tries to give a largely self-contained exposition of these manifold ideas and techniques, presenting, where possible, short new proofs for key results.

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arXiv2017

On rigidity of locally symmetric spaces

In this note I generalize the classical results of Calabi-Vesentini to certain non-compact locally symmetric domains, namely those that are quotients of a hermitian symmetric domain by a neat arithmetic subgroup of the group of its holomorphic automorphisms.

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arXiv2013

Abelian Fourfolds of Weil type and certain K3 Double Planes

Double planes branched in 6 lines give a famous example of K3 surfaces. Their moduli are well understood and related to abelian fourfolds of Weil type. We compare these two moduli interpretations and in particular divisors on the moduli spaces. On the K3 side, this is achieved with the help of elliptic fibrations. We also study the Kuga-Satake correspondence on these special divisors.

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arXiv2012

Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory

We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also suggests to associate abelian varieties to polarized even weight Hodge structures. The latter construction can also be explained in terms of algebraic groups which might be useful from the point of view of Tannakian categories. The constructions depend on moduli much as in Teichmüller theory although the period maps in general are only real analytic. One of the nice features is how the index for certain differential operators canonically associated to the geometry of the situation (spin structure, complex structure etc.) leads to integrality of skew pairings on the topological K-group (coming from the Index Theorem) which then serves as a polarization for the jacobian.

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arXiv2009

Bloch-type conjectures and an example of a threefold of general type

The hypothetical existence of a good theory of mixed motives predicts many deep phenomena related to algebraic cycles. One of these, a generalization of Bloch's conjecture says that "small Hodge diamonds" go with "small Chow groups". Voisin's method (which produces examples with small Chow groups) is analyzed carefully to widen its applicability. A threefold of general type without 1- and 2-forms is exhibited for which this extension yields Bloch's generalized conjecture.

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