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Stefan Müller-Stach

Publications and source records attributed to Stefan Müller-Stach.

32 records · Page 2Linked to original sources
arXiv2010

Relative Chow-Kuenneth decompositions for morphisms of threefolds

We show that any nonconstant morphism of a threefold admits a relative Chow-Kuenneth decomposition. As a corollary we get sufficient conditions for threefolds to admit an absolute Chow-Kuenneth decomposition. In case the image of the morphism is a surface, this implies another proof of a theorem on the absolute Chow-Kuenneth decomposition for threefolds satisfying a certain condition, which was obtained by the first author with P. L. del Angel. In case the image is a curve, this improves in the threefold case a theorem obtained by the second author where the singularity of the morphism was assumed isolated and the condition on the general fiber was stronger.

math.AG↗
arXiv2010

A note on the unirationality of a moduli space of double covers

In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal A}_4$ to Siegel space. We show that there is a birational model of $\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof for the unirationality of ${\mathcal A}_4$.

math.AG↗
arXiv2010

Hodge classes associated to 1-parameter families of Calabi-Yau 3-folds

We use $L^2$-Higgs cohomology to determine the Hodge numbers of the parabolic cohomology $H^1(\bar S, j_*\V)$, where the local system $\V$ arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve $\bar S$. The method gives a way to predict the presence of algebraic 2-cycles in the total space of the family and is applied to some examples.

math.AG↗
arXiv2008

Differential equations associated to Families of Algebraic Cycles

We develop a theory of differential equations associated to families of algebraic cycles in higher Chow groups (i.e., motivic cohomology groups). This formalism is related to inhomogeneous Picard--Fuchs type differential equations. For families of K3 surfaces the corresponding non-linear ODE turns out to be symilar to Chazy's equation.

math.AG↗
arXiv2007

Chow--Kuenneth decomposition for special varieties

In this paper we investigate Murre's conjecture on the Chow--Künneth decomposition for two classes of examples. We look at the universal families of smooth curves over spaces which dominate the moduli space $\cM_g$, in genus at most 8 and show existence of a Chow--Künneth decomposition. The second class of examples include the representation varieties of a finitely generated group with one relation. This is done in the setting of equivariant cohomology and equivariant Chow groups to get equivariant Chow--Künneth decompositions.

math.AG↗
arXiv2006

Chow-Kunneth decomposition for universal families over Picard modular surfaces

We discuss the existence of an absolute Chow-Kuenneth decomposition for complete degenerations of families of Abelian threefolds with complex multiplication over a particular Picard Modular Surface studied by Holzapfel. In addition to the work of Gordon, Hanamura and Murre we use Relatively Complete Models in the sense of Mumford-Faltings-Chai of Picard Modular Surfaces in order to describe complete degenerations of families of abelian varieties. We furthermore prove vanishing results for cohomology groups of irreducible representations of certain arithmetic subgroups in SU(2,1) using the non--compact Simpson type correspondence between the $L^2$--Higgs cohomology of the underlying VHS and the $L^2$--de Rham cohomology resp. intersection cohomology of local systems.

math.AG↗
arXiv2006

The Milnor-Chow homomorphism revisited

The aim of this note is to give a simplified proof of the surjectivity of the natural Milnor-Chow homomorphism $ρ: K^M_n(A) \to CH^n(A,n)$ between Milnor $K$-theory and higher Chow groups for essentially smooth (semi-)local $k$-algebras $A$ with $k$ infinite. It implies the exactness of the Gersten resolution for Milnor $K$-theory at the generic point. Our method uses the Bloch-Levine moving technique and some properties of the Milnor $K$-theory norm for fields.

math.AG↗
arXiv2004

The Abel-Jacobi map for higher Chow groups

We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel-Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.

math.AG↗
arXiv2003

Picard-Fuchs equations, Integrable Systems, and higher Algebraic K-theory

This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable systems. The principle behind everything is a strong interaction between K-theory and Picard-Fuchs type differential equations via Abel-Jacobi maps. Our main result is an extension of a theorem of Donagi and Markman to our setup.

math.AG↗
arXiv2002

On K_1 and K_2 of algebraic surfaces

We study the higher Chow groups $CH^2(X,1)$ and $CH^3(X,2)$ of smooth, projective algebraic surfaces over a field of char 0. We develop a theoretical framework to study them by using so-called higher normal functions and higher infinitesimal invariants when the cycles are supported on normal crossing divisors. Then we investigate their structure on hypersurfaces in projective 3-space. This leads to vanishing or decomposbility results. On the other hand we find examples of these groups in degrees 4 and 5 where the result is a highly indecomposably (hence big) abelian group. Part of the examples are due to Alberto Collino.

math.AG↗
arXiv2001

The transcendental part of the regulator map for K_1 on a mirror family of K3 surfaces

We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K_1 of a mirror family of quartic K3 surfaces. The resulting multivalued function does not satisfy the hypergeometric differential equation of the periods and we conclude that the cycle is indecomposable for most points in the mirror family. The occurring inhomogenous Picard-Fuchs equation are related to Painlevé VI type differential equations.

math.AG↗
arXiv1996

Motives of uniruled 3-folds

We construct projectors in the ring of correspondences of a complex uniruled 3-fold $X$ which lift the Kuenneth components of the diagonal in singular cohomology and have other properties which were conjectured by J. Murre. Such Murre decompositions have been already obtained for curves, surfaces, abelian varieties and varieties with cell decompositions by the work of Manin, Shermenev, Beauville, Murre et.al.. In particular they define a natural filtration on the Chow groups of $X$ which was conjectured by Bloch and Beilinson. To do this we use Mori theory and construct projectors in the situation of a fiber space over a surface. These projectors may also be used in more general situations.

alg-geom↗
arXiv1995

${\bf C}^*$-extensions of tori, higher Chow groups and applications to incidence equivalence relations for algebraic cycles.

Let X be a smooth projective variety of dimension n. If $p+q=n+1$ then Bloch has defined a ${\bf G}_m$-biextension E over the product of the Chow groups $CH^p_0(X)$ and $CH^q_0(X)$ of homologically trivial cycles. We prove that E is the pullback of the Poincare biextension over the product of intermediate Jacobians in characteristic zero. This is used to study various equivalence relations for algebraic cycles. In particular we reprove Murres result that Griffiths conjecture holds for codimension two cycles, i.e. every codim. two cycle algebraically and incidence equivalent to zero has torsion Abel-Jacobi invariant.

alg-geom↗