Search results for " Geometry"

Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

2003

Let M(d,n) be the moduli stack of hypersurfaces of degree d > n in the complex projective n-space, and let M(d,n;1) be the sub-stack, parameterizing hypersurfaces obtained as a d fold cyclic covering of the projective n-1 space, ramified over a hypersurface of degree d. Iterating this construction, one obtains M(d,n;r). We show that M(d,n;1) is rigid in M(d,n), although the Griffiths-Yukawa coupling degenerates for d<2n. On the other hand, for all d>n the sub-stack M(d,n;2) deforms. We calculate the exact length of the Griffiths-Yukawa coupling over M(d,n;r), and we construct a 4-dimensional family of quintic hypersurfaces, and a dense set of points in the base, where the fibres ha…

Specialization of cycles and the K-theory elevator

2017

A general specialization map is constructed for higher Chow groups and used to prove a "going-up" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross and Schoen, and of the coordinate symbol on a genus-2 curve.

A rigidity theorem for Lagrangian deformations

2005

We consider deformations of singular Lagrangian varieties in symplectic manifolds. We prove that a Lagrangian deformation of a Lagrangian complete intersection is analytically rigid provided that this is the case infinitesimally. This result is given as a consequence of the coherence of the direct image sheaves of relative infinitesimal Lagrangian deformations.

Special Families of Curves, of Abelian Varieties, and of Certain Minimal Manifolds over Curves

2006

This survey article discusses some results on the structure of families f:V-->U of n-dimensional manifolds over quasi-projective curves U, with semistable reduction over a compactification Y of U. We improve the Arakelov inequality for the direct images of powers of the dualizing sheaf. For families of Abelian varieties we recall the characterization of Shimura curves by Arakelov equalities. For families of curves we recall the characterization of Teichmueller curves in terms of the existence of certain sub variation of Hodge structures. We sketch the proof that the moduli scheme of curves of genus g>1 can not contain compact Shimura curves, and that it only contains a non-compact Shimura c…

Irreducibility of Hurwitz spaces of coverings with one special fiber

2006

Abstract Let Y be a smooth, projective complex curve of genus g ⩾ 1. Let d be an integer ⩾ 3, let e = {e1, e2,..., er} be a partition of d and let | e | = Σi=1r(ei − 1). In this paper we study the Hurwitz spaces which parametrize coverings of degree d of Y branched in n points of which n − 1 are points of simple ramification and one is a special point whose local monodromy has cyclic type e and furthermore the coverings have full monodromy group Sd. We prove the irreducibility of these Hurwitz spaces when n − 1 + | e | ⩾ 2d, thus generalizing a result of Graber, Harris and Starr [A note on Hurwitz schemes of covers of a positive genus curve, Preprint, math. AG/0205056].

Segre and the Foundations of Geometry: From Complex Projective Geometry to Dual Numbers

2016

In 1886 Corrado Segre wrote to Felix Klein about his intention to study ‘geometrie projective pure’, completing and developing the work of von Staudt. He would continue this research project throughout the whole of his scientific life. In 1889, following a suggestion of Segre, Mario Pieri published his translation of the Geometrie der Lage, and from 1889 to 1890 Segre published four important papers, “Un nuovo campo di ricerche geometriche”, in which he completely developed complex projective geometry, considering new mathematical objects such as antiprojectivities and studying the Hermitian forms from a geometrical point of view with the related ‘hyperalgebraic varieties’. Segre developed …

Inductive synthesis of dot expressions

2005

We consider the problem of the synthesis of algorithms by sample computations. We introduce a formal language, namely, the so-called dot expressions, which is based on a formalization of the intuitive notion of ellipsis (‘...’). Whilst formally the dot expressions are simply a language describing sets of words, on the other hand, it can be considered as a programming language supporting quite a wide class of programs. Equivalence and asymptotical equivalence of dot expressions are defined and proved to be decidable. A formal example of a dot expression is defined in the way that, actually, it represents a sample computation of the program presented by the given dot expression. A system of s…

Counterexamples for unique continuation

1988

The Abel–Jacobi map for higher Chow groups

2006

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.

The Fubini and Tonelli Theorems for Product Local Systems

2010

The notion of product local system and of the Kurzweil-Henstock type integral related to a product local system is introduced. The main result is a version of the Fubini and Tonelli theorems for product local systems.