Raymond Cheng, Department of Pure Mathematics, University of Waterloo
“Hodge and Lefschetz Decompositions”
Having discussed the Harmonic Theory underlying the Hodge decomposition, let us carry on to the case where the manifold we are working on is also Kahler. The Kahler metric makes compatible the naturally defined Laplacian operators, thereby enriching the cohomology de- composition we have seen on general compact Riemannian manifolds. Moreover, a Kahler form induces a map on cohomology—cupping with the form—which further refines the decomposi- tion of cohomology; this is the Lefschetz decomposition. This talk will discuss, and as much as reasonable, explain why these results are true.