Domingo Toledo

Harmonic mappings of Kähler manifolds to locally symmetric spaces

© Publications mathématiques de l’I.H.É.S., 1989, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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.

Projective varieties with non-residually finite fundamental group

© Publications mathématiques de l’I.H.É.S., 1993, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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].

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.

Bounded negativity of self-intersection numbers of Shimura curves in Shimura surfaces

Shimura curves on Shimura surfaces have been a candidate for counterexamples to the bounded negativity conjecture. We prove that they do not serve this purpose: there are only finitely many whose self-intersection number lies below a given bound. Previously, this result has been shown in [BHK+13] for compact Hilbert modular surfaces using the Bogomolov-Miyaoka-Yau inequality. Our approach uses equidistribution and works uniformly for all Shimura surfaces.

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.

On Fundamental Groups of Class VII Surfaces

The purpose of this note is to obtain a restriction on the fundamental groups of nonelliptic compact complex surfaces of class VII in Kodaira’s classification [9]. We recall that these are the compact complex surfaces with first Betti number one and no nonconstant meromorphic functions. This seems to be the class of compact complex surfaces whose structure is least understood. The first and simplest examples are the general Hopf surfaces [9], III. Then there are various classes of examples found by Inoue [5,6], and which have been studied in more detail in [11]. The only known topological restriction beyond the first Betti number is that intersection form in two-dimensional homology is negative definite. There seems to be little known as to how wide this class of surfaces is. We prove the following theorem.

Quotients of non-classical ag domains are not algebraic

A flag domain D = G/V for G a simple real non-compact group G with compact Cartan subgroup is non-classical if it does not fiber holomorphically or anti-holomorphically over a Hermitian symmetric space. We prove that any two points in a non-classical domain D can be joined by a finite chain of compact subvarieties of D. Then we prove that for F an infinite, finitely generated discrete subgroup of G, the analytic space F\D does not have an algebraic structure.

Maps Between Complex Hyperbolic Surfaces

We study examples of surjective holomorphic maps between complex hyperbolic surfaces that are not covering maps. The induced homomorphisms on the fundamental groups have infinite kernel, thus are examples of the nonstandard homomorphisms found by Mostow. We also study the singularities of these maps.