Characteristic cohomology of the infinitesimal period relation

Abstract

The infinitesimal period relation (also known as Griffiths' transversality) is the system of partial differential equations constraining variations of Hodge structure. This paper presents a study of the characteristic cohomology associated with that system of PDE.Date: May 9, 2018. 2010 Mathematics Subject Classification. 14D07, 32G20. 58A15, 58A17. Key words and phrases. Variation of Hodge structure, infinitesimal period relation (Griffiths' transversality), characteristic cohomology, flag domain.(1) The notationĎ for G C /P comes from Hodge theory: we think ofĎ as the compact dual of a period domain (or, more generally, a Mumford-Tate domain).

Variations of Hodge structure and orbits in flag varieties

“…By [46,Theorem 4.1], the invariant characteristic cohomology ofĎ is generated by classes of horizontal Schubert varieties (HSV). Moreover, the homology class of any horizontal cycle inĎ may be expressed as a linear combination of horizontal Schubert classes [47,Theorem 4.7]. Applying a "horizontal twist" to Borel and Haefliger's question, we ask: when does the class (or a multiple thereof) of a singular HSV, such as (1.1), admit a smooth, horizontal, algebraic representative?…”