The aim of this paper is to give a geometric interpretation of holomorphic and smooth Deligne cohomology. Before stating the main results we recall the definition and basic properties of Deligne cohomology.Let X be a smooth complex projective variety and let Ω r X be the sheaf of germs of holomorphic r-forms on X. The qth Deligne complex of X is the complex of sheaveswhere Z(q) = (2π √ −1) q Z ⊂ C, and Z(q) X is the constant sheaf on X associated with the group Z(q). The hypercohomology H * (X, Z(q) D ) of the complex Z(q) D is called the Deligne cohomology of X. Our basic reference for Deligne cohomology is [EV].One of the key properties of Deligne cohomology is that for every p ≥ 1 the groupof the group H p,p Z (X) of integral (p, p)-classes of X by the the pth intermediate Jacobian J p (X) of Griffiths. For p = 1 the group H 2 (M, Z(1) D ) is isomorphic to the first cohomology group H 1 (O * X ) of the sheaf O * X of germs of non-vanishing holomorphic functions on X, and the sequence (1) reduces in this case to the classical short exact sequenceIt is well known that the group H 1 (X, O * X ) is isomorphic to the group CH 1 (X) of divisors of X modulo rational equivalence, the Jacobian J(X) is isomorphic to the group CH 1 hom (X) of rational equivalence classes of homologous to 0 divisors of X, and H 1,1 Z (X) is the image of the cycle map CH 1 (X) → H 2 (X; Z). One would like to have a similar cohomological description of the groups CH p (X) of codimension p cycles of X modulo rational equivalence, for p > 1, together with an analogous to (2) short exact sequence completely describing the image H 2p alg (X; Z) and the kernel CH PAWE L GAJERDeligne cohomology can be thought as a step in this direction. Indeed, the cycle homomorphism (3) lifts to a homomorphismis the Abel-Jacobi homomorphism. Another important property of Deligne cohomology is the existence of characteristic classes, called "regulators",from the algebraic K-groups of X into the Deligne cohomology of X, which generalize the classical Chern classes of holomorphic vector bundles. Several important conjectures of arithmetic algebraic geometry are formulated in terms of these regu-The second degree Deligne cohomology groups H 2 (X; Z(q) D ) have the following geometric interpretations.• H 2 (X; Z(0) D ) is the ordinary second cohomology group H 2 (X; Z) of X that can be identified with the group of smooth principal C * -bundles over X.• H 2 (X; Z(1) D ) is isomorphic to the group H 1 (O * X ) of isomorphism classes of holomorphic principal C * -bundles over X.• H 2 (X; Z(2) D ) is isomorphic to the group of isomorphism classes of holomorphic principal C * -bundles over X with holomorphic connections.• For every q > 2 the group H 2 (X; Z(q) D ) is isomorphic to the group of isomorphism classes of holomorphic principal C * -bundles with flat connections over X.Thanks to the above description of the groups H 2 (X; Z(q) D ) the geometric structure of regulators, cycle homomorphisms, and Abel-Jacobi homomorphisms has been completely understood in the c...

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Abstract. We present a novel hierarchical force-directed method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Projecting high-dimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, efficient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a mid-range PC.

We present a novel hierarchical force-directed method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higher-dimensional embedding into a two or three dimensional subspace of E. Projecting high-dimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, efficient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a mid-range PC.

Abstract. This paper describes a system for Graph dRawing with Intelligent Placement, GRIP. The system is designed for drawing large graphs and uses a novel multi-dimensional force-directed method together with fast energy function minimization. The system allows for drawing graphs with tens of thousands of vertices in under a minute on a mid-range PC. To the best of the authors' knowledge GRIP surpasses the fastest previous algorithms. However, speed is not achieved at the expense of quality as the resulting drawings are quite aesthetically pleasing.

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