SCIENTIFIC PROGRAM

This will be introductory lectures to S. Kondo's lectures in this workshop. I will explain the theory of periods on algebraic K3 surfaces which allows one to construct the moduli space of lattice polarized K3 surfaces as an arithmetic quotient of a symmetric hermitian domain of type IV. An additional structure on transcendental cycles of K3 surfaces allows one to embed an arithmetic quotient of a complex ball into the type IV domain quotient. I will give some examples how the arithmetic complex reflection groups from Deligne-Mostow list arise as monodromy groups of K3 surfaces and give some examples when new groups of this sort arise in this way. Some of these constructions will be discussed in detail in Kondo's lectures.

Complex solutions of systems of partial differential equations are multivalued in general. For nice Fuchsian systems on a manifold $X$ the multivalence can be described by a discrete group lattice $\Gamma$, called monodromy group, acting on the uniformizing domain $\mathbb{B}$, say a hermitian symmetric one. On special subvarieties $W$ the restricted system may have the same uniformizing quality with a subdomain $\mathbb{D}$ of $\mathbb{B}$ and discrete lattice $\Gamma_\mathbb{D} \subset \Gamma$ acting on $\mathbb{D}$. There are nice cases, where $\mathbb{D}$ is a geodesic subvariety of $\mathbb{B}$. This property goes down to the embedding $W \supseteq \mathbb{D}/\Gamma \subset \mathbb{B}/\Gamma \subseteq X$ we startet with.

Following also an idea of M. Uludag we define - a littlebit finer than usual - an \emph{orbital variety} $\underline{X}$ as orbifold $X$ (in the usual sense) together with a well-choosen weight map $w:\: X \to \mathbb{N}_+ \cup \{\infty\}$. Moreover we define \emph{relative} (or \emph{embedded}) orbital varieties $\underline{W}$. In both, the absolute and relative, cases we introduce also several types of morphisms compatible with the structures. With these notions we are able to give general definitions for \emph{absolute} and \emph{relative orbital invariants}.

With this language we are able to present relations between them, genera-lizing Hirzebruch-Mumford Proportionality to arithmetic groups with torsion. For Picard and Hilbert modular spaces the orbital invariants can be expressed by special values of Dedekind zeta functions. In complex dimension 2 (and I guess also in higher dimensions) they can be also described by explicitly known universal rational functions in the geometric data and weights, as well in the absolute as in the relative (embedded curve) cases. Elliptic modular forms of Hirzebruch-Zagier and Kudla-Cogdell type are interpreted as relative orbital invariants. This result is useful for a purely geometric determination of them after fine surface classifications.

1. Combinatorics
2. Algebras (Orlik-Solomon algebra)
3. Cohomology algebra
4. The NBC-complex
5. The nbc-set

I shall give complex ball uniformizations of the moduli spaces of del Pezzo surfaces via periods of K3 surfaces. {\nobreakspace}Furthermore I shall mention a relation with Deligne-Mostow's complex ball uniformization of the moduli of points on ${\mathbb P}^1$, and an application of Borcherds theory of automorphic forms on type IV symmetric domain to our situation.

Dunkl systems, Hypergeometric functions associated to Dunkl systems Projective orbifold completions (of Fubini-Study, Euclidean and ball quotient type), Deligne-Mostow's theory as a special case. Modular interpretation of some special cases

It is known that the complement of the Whitehead link admits a hyperbolic structure. This space can be expressed as $\mathbb H^3/W$, {\nobreakspace}where $\mathbb H^3$ is the real 3-dimensional upper half space, and {\nobreakspace}the discreate subgroup {\nobreakspace}$W$ of $GL_2(\mathbb C)$, generated by two elements {\nobreakspace}$\left(\begin{array}{cc}1&i\\ {\nobreakspace}0&1\end{array}\right)$ and {\nobreakspace}$\left(\begin{array}{cc}1&0\\1+i&1\end{array}\right),$ {\nobreakspace}is isomorphic to the fundamental group of the complement of {\nobreakspace}the Whitehead link. {\nobreakspace}I shall represent the quotient space $\mathbb H^3/W$ by {\nobreakspace}constructing invariant functions on $\mathbb H^3$ with respect to $W$ {\nobreakspace}in terms of theta functions on the bounded symmetric domain of type $I_{2,2}$.

The lectures are based on a selection of topics from the author's paper ``Resonant hypergeometric systems and mirror symmetry''
(alg-geom/9711002) streamlined, updated and extended with ideas from more recent research (including modular aspects).

The linear representation of the fundamental group of the domain of definition of Lauricella's hypergeometric function and its faithfulness.

Definitions of orbifolds, orbifold geometry in complex dimension 1, Theorems of Fox-Bundgaard-Nielsen on orbifold uniformization in dimension 1. Orbifold geometry in dimension 2, Orbifolds over the Apollonius configuration, ball quotient orbifold towers. Some examples of K3 orbifolds. Orbifold geometry in higher dimensions. Braid groups of an orbifold.

For the Gauss hypergeometric differential equations with certain rational parameters, Schwarz maps are just the classical triangle functions. Under which conditions do they take algebraic values at algebraic arguments? We will see that the integral representation of hypergeometric functions provides an interpretation of Schwarz maps as period quotients on families of abelian varieties, and the number theoretic question has at least partial answers given on the one hand by automorphic functions and on the other hand by modern transcendence theory. The techniques explained here generalize to other kinds of Schwarz maps as well (with apparent singularities or in several variables). The lectures will cover old and recent joint work with H. Shiga and in part with T. Tsutsui.

In the 20th century, various generalizations of the Schwarz map (of the hypergeometric differential equation) are studied, under the condition that the exponents are real. I would like to propose a study of the Schwarz map when the exponents are not necessarily real.

We will discuss discrete groups acting on real hyperbolic space, emphasizing the isometry groups of Z-lattices of signature (n, 1). We will cover Vinberg's method of determining when such groups are generated by reflections, and how to understand the groups when they are. Then we will study some particular discrete groups acting on hyperbolic three-space, namely the ones uniformizing the components of the moduli space of real sextics in P1, and the the one uniformizing their union. This is very closely analogous to work of Carlson, Toledo and the speaker on moduli of real cubic surfaces, and displays all the interesting new phenomena found in that moduli space. The difference is that three dimensions are visualizable, so the constructions and the final answer are easier to understand than in the cubic-surface case. In order to explain the construction we will also discuss the uniformization of the moduli space of 6-tuples in the complex projective line by the complex 3-ball, which was discovered by Deligne-Mostow.