By Mark Green, Phillip Griffiths, Matt Kerr

The authors examine the advanced geometry and coherent cohomology of nonclassical Mumford-Tate domain names and their quotients by way of discrete teams. Their concentration all through is at the domain names $D$ which happen as open $G(\mathbb{R})$-orbits within the flag types for $G=SU(2,1)$ and $Sp(4)$, considered as classifying areas for Hodge constructions of weight 3. within the context supplied by means of those easy examples, the authors formulate and illustrate the overall strategy in which correspondence areas $\mathcal{W}$ provide upward thrust to Penrose transforms among the cohomologies $H^{q}(D,L)$ of precise such orbits with coefficients in homogeneous line bundles

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**Example text**

A Lagrangian ﬂag F is given by a ﬂag (0) ⊂ F1 ⊂ F2 ⊂ F3 ⊂ F4 = V, where F2 = F2⊥ , F3 = F1⊥ , the ⊥ dim Fj = j being with respect to Q. ˇ the set of The Lagrangian ﬂag is determined by F1 ⊂ F2 , and we denote by D 1 ˇ Lagrangian ﬂags. Then D → GL (V ) given by (F1 , F2 ) → F2 is a P -bundle. Upon choice of a reference ﬂag we have an identiﬁcation ˇ = GC /B D where B is a Borel subgroup of GC . The weight vector ﬂag is a Lagrangian ﬂag. We may think of GC as the set of Lagrange frames f• = (f1 , f2 , f3 , f4 ), meaning those where Q(fi , fj ) is the matrix 1 1 −1 −1 and where [f1 ] ⊂ [f1 , f2 ] ⊂ ˇ is [f1 , f2 , f3 ] ⊂ [f1 , f2 , f3 , f4 ] is a Lagrange ﬂag.

From a Hodge theoretic perspective one possibility is to realize D as a subdomain for polarized Hodge structures of weight n = 5 and with Hodge numbers h5,0 = h0,5 = 1, h4,1 = h1,4 = 0, h3,2 = h2,3 = 1. In fact, it will be more convenient to interpret D not as this Mumford-Tate domain, but rather as the Hodge ﬂags for the Mumford-Tate domain to be described now. The Siegel space H. Definition. We deﬁne H to the open set in GL (2, V ) consisting of Lagrange lines E ⊂ P3 with H < 0 on E. By reversing the sign of H we see that H is biholomorphic to the Siegel generalized upper plane Hg for g = 2, which is the Mumford-Tate domain for polarized Hodge structures of weight n = 1 and with Hodge number h1,0 = 2.

32 I. GEOMETRY OF THE MUMFORD-TATE DOMAINS Referring to ﬁgures 3 and 4 the position of a Lagrange quadrilateral relative to the real hyperquadric QH may be pictured as E13 E34 E24 p3 p4 p1 p2 E12 Figure 5 that is, the pictured Lagrangian lines Eij are of three types • E12 lies “inside” QH , meaning that H < 0 on the corresponding La˜12 in V ; grangian 2-plane E • E13 meets QH in a real circle; as a consequence H has signature (1, 1) ˜13 ; E24 has a similar property;32 on E ˜34 . • E34 lies “outside” QH , meaning that H > 0 on E There are eight orbits of the four Lagrange ﬂags in the above picture; thus we have (p1 , E12 ) and (p2 , E12 ) associated to E12 .