2014/11/03 by Jyh-Haur Teh, Teh, Jyh-Haur
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.AG #math.DG #math.MP
paper · pdf · doi:10.48550/arxiv.1411.0492
20 pages
arxiv created 2016/03/06 · arxiv updated 2016/03/08
By comparing Deligne complex and Aeppli-Bott-Chern complex, we construct a differential cohomology \widehatH^*(X, *, *) that plays the role of Harvey-Lawson spark group \widehatH^*(X, *), and a cohomology H^*ABC(X; \Z(*, *)) that plays the role of Deligne cohomology H^*D(X; \Z(*)) for every complex manifold X. They fit in the short exact sequence 0→ Hk+1ABC(X; \Z(p, q)) → \widehatHk(X, p, q) \oversetδ1→ Zk+1I(X, p, q) → 0 and \widehatH\bullet(X, \bullet, \bullet) possess ring structure and refined Chern classes, acted by the complex conjugation, and if some primitive cohomology groups of X vanish, there is a Lefschetz isomorphism. Furthermore, the ring structure of H\bulletABC(X; \Z(\bullet, \bullet)) inherited from \widehatH\bullet(X, \bullet, \bullet) is compatible with the one of the analytic Deligne cohomology H\bullet(X; \Z(\bullet)). We compute \widehatH^*(X, *, *) for X the Iwasawa manifold and its small deformations and get a refinement of the classification given by Nakamura.