Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

2017 ◽  
Author(s):  
Claire Voisin
Keyword(s):  
Recent Work ◽  
Chow Groups ◽  
Ring Structure ◽  
Complete Intersections ◽  
Algebraic Varieties ◽  
Chow Ring ◽  
Hodge Structures ◽  
Chow Rings ◽  
Geometric Methods ◽  
The Right

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

scholarly journals Generalized vanishing theorems for Yukawa couplings in heterotic compactifications

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Lara B. Anderson ◽  
James Gray ◽  
Magdalena Larfors ◽  
Matthew Magill ◽  
Robin Schneider
Keyword(s):  
Vector Bundles ◽  
Geometric Approach ◽  
Low Energy ◽  
Geometric Features ◽  
Vanishing Theorems ◽  
Vanishing Theorem ◽  
Alternative Derivation ◽  
Yukawa Couplings ◽  
Geometric Methods ◽  
Higher Dimensional

Abstract Heterotic compactifications on Calabi-Yau threefolds frequently exhibit textures of vanishing Yukawa couplings in their low energy description. The vanishing of these couplings is often not enforced by any obvious symmetry and appears to be topological in nature. Recent results used differential geometric methods to explain the origin of some of this structure [1, 2]. A vanishing theorem was given which showed that the effect could be attributed, in part, to the embedding of the Calabi-Yau manifolds of interest inside higher dimensional ambient spaces, if the gauge bundles involved descended from vector bundles on those larger manifolds. In this paper, we utilize an algebro-geometric approach to provide an alternative derivation of some of these results, and are thus able to generalize them to a much wider arena than has been considered before. For example, we consider cases where the vector bundles of interest do not descend from bundles on the ambient space. In such a manner we are able to highlight the ubiquity with which textures of vanishing Yukawa couplings can be expected to arise in heterotic compactifications, with multiple different constraints arising from a plethora of different geometric features associated to the gauge bundle.

scholarly journals Preface to Symposium on Computational Geometric Methods in Multibody System Dynamics

2010 ◽  
Author(s):  
Zdravko Terze ◽  
Andreas Müller ◽  
Theodore E. Simos ◽  
George Psihoyios ◽  
Ch. Tsitouras
Keyword(s):  
System Dynamics ◽  
Multibody System ◽  
Multibody System Dynamics ◽  
Geometric Methods

An Assessment of Geometric Methods in Trajectory Synthesis for Shape Creating Operations

1998 ◽  
Author(s):  
Radha Sarma
Keyword(s):  
Rapid Prototyping ◽  
Functional Properties ◽  
Electrical Discharge ◽  
Electrical Discharge Machining ◽  
Trajectory Generation ◽  
Manufacturing Operations ◽  
Geometric Methods

Abstract This paper presents an assessment of the steps involved in trajectory synthesis (e.g., trajectory layout, trajectory generation, trajectory spacing, trajectory postprocessing, and trajectory physics) for shape creating manufacturing operations, e.g., rapid prototyping, milling, electrical discharge machining. The rationale for this paper is that the trajectory plays an important role in determining the productivity as explained below. Shape creating manufacturing operations are those that make use of a “tool” which operates on a “workpiece” to create the desired shape. The tool operates on the workpiece as dictated by the trajectory of the tool, resulting in the manufactured shape. Often, the geometric and functional properties of the end product are dependent on the trajectory of the tool. Moreover, the trajectory dictates the accuracy of the end product and the time taken to manufacture the product. Recognizing the important contribution of the trajectory in shape creating operations, this paper focuses on trajectory synthesis.

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