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This volume contains the proceedings of an NSF/Conference Board of the Mathematical Sciences (CBMS) regional conference on Hodge theory, complex geometry, and representation theory, held on June 18, 2012, at the Texas Christian University in Fort Worth, TX. Phillip Griffiths, of the Institute for Advanced Study, gave 10 lectures describing now-classical work concerning how the structure of Shimura varieties as quotients of Mumford-Tate domains by arithmetic groups had been used to understand the relationship between Galois representations and automorphic forms. He then discussed recent breakthroughs of Carayol that provide the possibility of extending these results beyond the classical case. His lectures will appear as an independent volume in the CBMS series published by the AMS. This volume, which is dedicated to Phillip Griffiths, contains carefully written expository and research articles. Expository papers include discussions of Noether-Lefschetz theory, algebraicity of Hodge loci, and the representation theory of SL2(R). Research articles concern the Hodge conjecture, Harish-Chandra modules, mirror symmetry, Hodge representations of Q-algebraic groups, and compactifications, distributions, and quotients of period domains. It is expected that the book will be of interest primarily to research mathematicians, physicists, and upper-level graduate students.

In this paper we generalize the E\\mathcal {E} and N\\mathcal {N}-type resolutions used by Martin-Deschamps and Perrin for curves in P3\\mathbb {P}^{3} to subschemes of pure codimension in projective space, and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage. Via these resolutions, Rao’s correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves E\\mathcal {E} satisfying H∗1(E)=0H^{1}_{*}( \\mathcal {E})=0 and Ext1(E∨,O)=0\\mathcal {E}xt^{1}( \\mathcal {E}^{\\vee }, \\mathcal {O})=0. Further, these resolutions are used to extend the work of Martin-Deschamps and Perrin for Cohen-Macaulay curves in P3\\mathbb {P}^{3} to subschemes of pure codimension two in Pn\\mathbb {P}^{n}. In particular, even linkage classes of such subschemes satisfy the Lazarsfeld-Rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class.

If \\(\\mathcal E, \\mathcal F\\) are vector bundles of ranks \\(r-1,r\\) on a smooth fourfold \\(X\\) and \\(\\mathcal{Hom}(\\mathcal E,\\mathcal F)\\) is globally generated, it is well known that the general map \\(\\phi: \\mathcal E \\to \\mathcal F\\) is injective and drops rank along a smooth surface. Chang improved on this with a filtered Bertini theorem. We strengthen these results by proving variants in which (a) \\(\\mathcal F\\) is not a vector bundle and (b) \\(\\mathcal{Hom}(\\mathcal E,\\mathcal F)\\) is not globally generated. As an application, we give examples of even linkage classes of surfaces on \\(\\mathbb P^4\\) in which all integral surfaces are smoothable, including the linkage classes associated with the Horrocks-Mumford surface.

In this paper we determine the irreducible components of the Hilbert schemes H4,g of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g: there are roughly ∼(g2/24) of them, most of which are families of multiplicity structures on lines. We give deformations which show that these Hilbert schemes are connected. For g≤−3 we exhibit a component that is disjoint from the component of extremal curves and use this to give a counterexample to a conjecture of Aït-Amrane and Perrin.

Let A be the local ring at a point of a normal complex variety with completion R. Srinivas has asked about the possible images of the induced map from Cl A to Cl R over all geometric normal domains A with fixed completion R. We use Noether-Lefschetz theory to prove that all finitely generated subgroups are possible in some familiar cases. As a byproduct we show that every finitely generated abelian group appears as the class group of the local ring at the vertex of a cone over some smooth complex variety of each positive dimension.

We compute the divisor class group of the general hypersurface Y of a complex projective normal variety X of dimension at least four containing a fixed base locus Z. We deduce that completions of normal local complete intersection domains of finite type over the complex numbers of dimension \\(\\ge 4\\) are completions of UFDs of finite type over the complex numbers.

We show that if \\(D \\subset \\mathbb P^N\\) is obtained from a codimension two local complete intersection \\(C\\) by adding embedded points of multiplicity \\(\\leq 3\\), then \\(D\\) is a flat limit of \\(C\\) and isolated points. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus, show the smoothness of the Hilbert component whose general member is a plane curve union a point in \\(\\mathbb P^3\\), and construct a Hilbert component whose general member has an embedded point.

For the completion B of a local geometric normal domain, V. Srinivas asked which subgroups of Cl B arise as the image of the map from Cl A to Cl B on class groups as A varies among normal geometric domains with B isomorphic to the completion of A. For two dimensional rational double point singularities we show that all subgroups arise in this way. We also show that in any dimension, every normal hypersurface singularity has completion isomorphic to that of a geometric UFD. Our methods are global, applying Noether-Lefschetz theory to linear systems with non-reduced base loci.