Stefan Müller-Stach

Period mappings and period domains

Part I. Basic Theory of the Period Map: 1. Introductory examples 2. Cohomology of compact Kahler manifolds 3. Holomorphic invariants and cohomology 4. Cohomology of manifolds varying in a family 5. Period maps looked at infinitesimally Part II. The Period Map: Algebraic Methods: 6. Spectral sequences 7. Koszul complexes and some applications 8. Further applications: Torelli theorems for hypersurfaces 9. Normal functions and their applications 10. Applications to algebraic cycles: Noris theorem Part III: Differential Geometric Methods: 11. Further differential geometric tools 12. Structure of period domains 13. Curvature estimates and applications 14. Harmonic maps and Hodge theory Appendix A. Projective varieties and complex manifolds Appendix B. Homology and cohomology Appendix C. Vector bundles and Chern classes.

Picard–Fuchs Equations for Feynman Integrals

We present a systematic method to derive an ordinary differential equation for any Feynman integral, where the differentiation is with respect to an external variable. The resulting differential equation is of Fuchsian type. The method can be used within fixed integer space-time dimensions as well as within dimensional regularisation. We show that finding the differential equation is equivalent to solving a linear system of equations. We observe interesting factorisation properties of the D-dimensional Picard–Fuchs operator when D is specialised to integer dimensions.

Negative curves on algebraic surfaces

We study curves of negative self-intersection on algebraic surfaces. Our main result shows there exist smooth complex projective surfaces X, related to Hilbert modular surfaces, such that X contains reduced, irreducible curves C of arbitrarily negative self-intersection C 2 . Previously the only known examples of surfaces for which C 2 was not bounded below were in positive characteristic, and the general expectation was that no examples could arise over the complex numbers. Indeed, we show that the idea underlying the examples in positive characteristic cannot produce examples over the complex number field, and thus our complex examples require a different approach.

Motives of Uniruled 3-Folds

J. Murre has conjectured that every smooth projective variety X of dimension d admits a decomposition of the diagonal δ=p0+...+p2d ∈ CHd(X × X) ⊗ Q such that the cycles pi are orthogonal projectors which lift the Künneth components of the identity map in étale cohomology. If this decomposition induces an intrinsic filtration on the Chow groups of X, we call it a Murre decomposition. In this paper we propose candidates for such projectors on 3-folds by using fiber structures. Using Mori theory, we prove that every smooth uniruled complex 3-fold admits a Murre decomposition.J. Murre has conjectured that every smooth projective variety X of dimension d admits a decomposition of the diagonal δ=p0+...+p2d ∈ CHd(X × X) ⊗ Q such that the cycles pi are orthogonal projectors which lift the Kunneth components of the identity map in etale cohomology. If this decomposition induces an intrinsic filtration on the Chow groups of X, we call it a Murre decomposition. In this paper we propose candidates for such projectors on 3-folds by using fiber structures. Using Mori theory, we prove that every smooth uniruled complex 3-fold admits a Murre decomposition.

On Chow motives of 3-folds

Let k be a field of characteristic zero. For every smooth, projective k-variety Y of dimension n which admits a connected, proper morphism f : Y → S of relative dimension one, we construct idempotent correspondences (projectors) πij(Y ) ∈ CHn(Y ×Y,Q) generalizing a construction of Murre. If n = 3 and the transcendental cohomology group H2 tr(Y ) has the property that H2 tr(Y,C) = f∗H2 tr(S,C) + Im(f∗H1(S,C) ⊗H1(Y,C) → H2 tr(Y,C)), then we can construct a projector π2(Y ) which lifts the second Künneth component of the diagonal of Y . Using this we prove that many smooth projective 3-folds X over k admit a Chow-Künneth decomposition ∆ = p0 + ...+p6 of the diagonal in CH3(X ×X,Q).

The transcendental part of the regulator map forK1 on a mirror family of K3-surfaces

ULLER-STACH Abstract We compute the transcendental part of the normal function corresponding to the Deligne class of a cycle in K1 of a mirror family of quartic K 3 surfaces. The re- sulting 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 equations are related to Painlev´ e VI-type differential equations. 1. The regulator map and Picard-Fuchs equations In this paper we study the first nonclassical higher K -group K 1.X/ for a smooth com- plex projective surface X. It was conjectured by H. Esnault around 1995 that certain elements in this group can be detected in the transcendental part of the Deligne coho- mology group H 3

Algebraic Cycle Complexes

We collect several basic properties of algebraic cycle complexes defined by Bloch, Friedlander, Suslin and Voevodsky, like moving lemmas, localization, homotopy invariance and Mayer-Vietoris exact sequences. We also explain a generalization of the theorem of Nesterenko/Suslin/Totaro from fields to smooth, semilocal algebras of geometric type over an infinite base field. After this survey we give a new cubical proof of Bloch/Nart’s elementary vanishing theorem in codimension one. Then we show how these results give rise to a framework in which we can study the relationship between motivic cohomology (higher Chow groups) of smooth varieties and Zariski cohomology with respect to Quillen or Milnor K-sheaves. Finally we indicate how moving lemmas can be used to derive properties of algebraic cycle complexes over fields and give several examples.

Relative Chow–Künneth decompositions for morphisms of threefolds

Abstract We show that any nonconstant morphism of a threefold admits a relative Chow–Künneth decomposition. As a corollary we get sufficient conditions for threefolds to admit an absolute Chow–Künneth decomposition. In case the image of the morphism is a surface, this implies another proof of a theorem on the absolute Chow–Künneth 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.

Otto Toeplitz: Algebraiker der unendlichen Matrizen

ZusammenfassungOtto Toeplitz ist ein Mathematiker, dessen Schicksal exemplarisch für die Vernichtung der jüdischen wissenschaftlichen Elite in Deutschland durch die Nationalsozialisten ist. Sein Einfluss in der Mathematik ist noch heute, besonders durch den Begriff der Toeplitzmatrizen, deutlich spürbar. Die kulturgeschichtliche Bedeutung von Toeplitz ist ebenso groß. Als Gründungsherausgeber von zwei Zeitschriften hat er sein Engagement für die Didaktik und die Wissenschaftsgeschichte untermauert. Wir geben einen Einblick in sein Schicksal und seine Gedankenwelt unter dem Einfluss von David Hilbert und Felix Klein.

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