2004/04/20 by Justin Sawon, Sawon, Justin · 1 citation
Mathematics · #53C26 #53C80 #57M27 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C26 #msc:53C80 #msc:57M27
paper · pdf · doi:10.48550/arxiv.math/0404360
Cambridge PhD thesis (2000), 130 pages, 5 figures
arxiv created 2004/04/20 · arxiv updated 2009/12/01
We investigate invariants of compact hyperkähler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifold, arriving at a number. For certain graph homology classes, the invariants we get are Chern numbers, and in fact all characteristic numbers arise in this way. We use relations in graph homology to study and compare these hyperkähler manifold invariants. For example, we show that the norm of the Riemann curvature can be expressed in terms of the volume and characteristic numbers of the hyperkähler manifold. We also investigate the question of whether the Rozansky-Witten invariants give us something more general than characteristic numbers. Finally, we introduce a generalization of these invariants which incorporates holomorphic vector bundles into the construction.