James A. Carlson

Harmonic mappings of Kähler manifolds to locally symmetric spaces

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A Defect Relation for Equidimensional Holomorphic Mappings Between Algebraic Varieties

0. Introduction 1. Notations, terminology, and sign conventions (a) Line bundles and Chern classes (b) Currents and forms in C0 2. Construction of a volume form 3. A second main theorem for non-degenerate maps 4. The defect relation (preliminary form) 5. The first main theorem and defect relation 6. Variants and applications (a) Schottky-Landau theorems (b) Remarks on the case ci(L) + ci(Kv) ? 0 (c) Holomorphic mappings with growth conditions Bibliography

The complex hyperbolic geometry of the moduli space of cubic surfaces

Recall that the moduli space of smooth (that is, stable) cubic curves is isomorphic to the quotient of the upper half plane by the group of fractional linear transformations with integer coefficients. We establish a similar result for stable cubic surfaces: the moduli space is biholomorphic to a quotient of the compex 4-ball by an explict arithmetic group generated by complex reflections. This identification gives interesting structural information on the moduli space and allows one to locate the points in complex hyperbolic 4-space corresponding to cubic surfaces with symmetry, e.g., the Fermat cubic surface. Related results, not quite as extensive, were announced in alg-geom/9709016.

Period mappings and period domains

Part I. Basic Theory of the Period Map: 1. Introductory examples 2. Cohomology of compact Kahler manifolds 3. Holomorphic invariants and cohomology 4. Cohomology of manifolds varying in a family 5. Period maps looked at infinitesimally Part II. The Period Map: Algebraic Methods: 6. Spectral sequences 7. Koszul complexes and some applications 8. Further applications: Torelli theorems for hypersurfaces 9. Normal functions and their applications 10. Applications to algebraic cycles: Noris theorem Part III: Differential Geometric Methods: 11. Further differential geometric tools 12. Structure of period domains 13. Curvature estimates and applications 14. Harmonic maps and Hodge theory Appendix A. Projective varieties and complex manifolds Appendix B. Homology and cohomology Appendix C. Vector bundles and Chern classes.

Quadratic presentations and nilpotent Kähler groups

It has been known for at least thirty years that certain nilpotent groups cannot be Kahler groups, i.e., fundamental groups of compact Kahler manifolds. The best known examples are lattices in the three-dimensional real or complex Heisenberg groups. It is also known that lattices in certain other standard nilpotent Lie groups, e.g., the full group of upper triangular matrices and the free k-step nilpotent Lie groups, k > 1, are not Kahler. The Heisenberg case was known to J-P. Serre in the early 1960’s, and unified proofs of the above statements follow readily from Sullivan’s theory of minimal models [6],[15], [19], or from Chen’s theory of iterated integrals [4], [10], or from more recent developments such as [9].

Bounds on the dimension of variations of Hodge structure

We derive upper bounds on the dimension of a variation of Hodge structure of weight two and show that these bounds are sharp. Using them we exhibit maximal geometric variations of Hodge structure. Analogous results for higher weight are obtained in the presence of a nondegeneracy hypothesis, and variations coming from hypersurfaces are shown to be nondegenerate. Maximal geometric variations of higher weight are also constructed.

Polyhedral resolutions of algebraic varieties

We give a method for constructing relatively small smooth simplicial resolutions of singular projective algebraic varieties. For varieties of dimension n, at most n applications of the basic process yields a resolution of combinatorial dimension at most n. The object so obtained may be used to compute the mixed Hodge stucture of the underlying variety.

A complex hyperbolic structure for moduli of cubic surfaces

Abstract We show that the moduli space M of marked cubic surfaces is biholomorphic to ( B 4 − H )/Г, where B 4 is complex hyperbolic four-space, Γ is a specific group generated by complex reflections, and H is the union of reflection hyperplanes for Γ. Thus M has a complex hyperbolic structure, i.e., an (incomplete) metric of constant negative holomorphic sectional curvature.

Harmonic maps from compact Kähler manifolds to exceptional hyperbolic spaces

It has been conjectured that a lattice in a noncompact group of real rank one, other than SU(1,n), cannot be isomorphic to the fundamental group of a compact Kähler manifold; moreover, it is known to be true for SO(1,n). In this note it is shown that this conjecture also holds for the case of uniform lattices in F4(−20), the group of isometries of the Cayley hyperbolic plane. The result is a consequence of a classification theorem for harmonic maps between Kähler and Cayley hyperbolic manifolds.

Rigidity of harmonic maps of maximum rank

Thehomotopical rank of a mapf:M →N is, by definition, min{dimg(M) ¦g homotopic tof}. We give upper bounds for this invariant whenM is compact Kähler andN is a compact discrete quotient of a classical symmetric space, e.g., the space of positive definite matrices. In many cases the upper bound is sharp and is attained by geodesic immersions of locally hermitian symmetric spaces. An example is constructed (Section 9) to show that there do, in addition, exist harmonic maps of quite a different character. A byproduct is construction of an algebraic surface with large and interesting fundamental group. Finally, a criterion for lifting harmonic maps to holomorphic ones is given, as is a factorization theorem for representations of the fundamental group of a compact Kähler manifold. The technique for the main result is a combination of harmonic map theory, algebra, and combinatorics; it follows the path pioneered by Siu in his ridigity theorem and later extended by Sampson.