Memoirs of the American Mathematical Society 2014; 145 pp; softcover Volume: 231 ISBN10: 0821898574 ISBN13: 9780821898574 List Price: US$79 Individual Members: US$47.40 Institutional Members: US$63.20 Order Code: MEMO/231/1088
 The authors study the complex geometry and coherent cohomology of nonclassical MumfordTate domains and their quotients by discrete groups. Their focus throughout is on the domains \(D\) which occur as open \(G(\mathbb{R})\)orbits in the flag varieties for \(G=SU(2,1)\) and \(Sp(4)\), regarded as classifying spaces for Hodge structures of weight three. In the context provided by these basic examples, the authors formulate and illustrate the general method by which correspondence spaces \(\mathcal{W}\) give rise to Penrose transforms between the cohomologies \(H^{q}(D,L)\) of distinct such orbits with coefficients in homogeneous line bundles. Table of Contents  Introduction
 Geometry of the Mumford Tate domains
 Homogeneous line bundles over the Mumford Tate domains
 Correspondence and cycle spaces; Penrose transforms
 The Penrose transform in the automorphic case and the main result
 Bibliography
