Friedrich Hirzebruch

Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 57 Chapter 1 : The Intersection Behaviour of the Curves T N . . . . . . 60 1.1. Special Points . . . . . . . . . . . . . . . . . . . . . 60 1.2. Modules in Imaginary Quadratic Fields . . . . . . . . . . 68 1.3. The Transversal Intersections of the Curves T N . . . . . . . 74 1.4. Contributions from the Cusps . . . . . . . . . . . . . . 78 1.5. Self-Intersections . . . . . . . . . . . . . . . . . . . . 82 Chapter 2: Modular Forms Whose Fourier Coefficients Involve Class Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1. The Modular Form ~oo(z ) . . . . . . . . . . . . . . . . 88 2.2. The Eisenstein Series of Weight 3 . . . . . . . . . . . . . 91 2.3. A Theta-Series Attached to an Indefinite Quadratic Form . . 96 2.4. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . I00 Chapter 3: Modular Forms with Intersection Numbers as Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.1. Modular Forms of Nebentypus and the Homology of the Hilbert Modular Surface . . . . . . . . . . . . . . . . . . . . t03 3.2. The Relationship to the Doi-Naganuma Mapping . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Spin-Manifolds and Group Actions

Let X be a compact oriented differentiable n-dimensional manifold (all manifolds are without boundary except in § 4) on which a Riemannian metric is introduced. Let Q be the principal tangential SO (n)-bundle of X.

Analytic cycles on complex manifolds

LET X be a complex manifold, Y a closed irreducible k-dimensional complex analytic subspace of X. Then Y defines or “carries” a 2k-dimensional integral homology class J’ of X, although the precise definition of y presents technical difficu1ties.S A finite formal linear combination 1 niYI with ni integers and Yi as above is called a complex analytic cycle, and the corresponding homology class c ni)vi is called a complex analytic homology class. If an integral cohomology class u corresponds under Poincart duality to a complex analytic homology class we shall say that u is a complex analytic cohomology class. The purpose of this paper is to show that a complex analytic cohomology class u satisfies certain topological conditions, independent of the complex structure of X. These conditions are that certain cohomology operations should vanish on U, for example Sq3u = 0: they are all torsion conditions. We also produce examples to show that these conditions are not vacuous even in the restricted classes of (a) Stein manifolds and (b) projective algebraic manifolds.

Some problems on differentiable and complex manifolds

A conference with the title Fiber bundles and differential geometry was held at Cornell University from May 3 to May 7, 1953.* It was supported by a grant from the National Science Foundation. The purpose of the present paper is to record those problems presented at the conference which concern differentiable, almost-complex, complex, and algebraic manifolds. Several of the problems have been solved since the conference. These solutions gave rise to new problems which we will discuss. This report is divided into three sections: differentiable manifolds, almostcomplex manifolds, complex manifolds. The last part contains also the problems on algebraic varieties. Since some problems of the first two sections were motivated by theorems on algebraic varieties, there are occasional references in Sections 1 and 2 to Section 3.1. The author wishes to thank Professors K. Kodaira and D. C. Spencer heartily for many discussions and valuable suggestions during the period of preparation of this report.

Elliptic genera, involutions, and homogeneous spin manifolds

We study the normalized elliptic genera Φ(X)=ϕ(X)/εk/2 for 4k-dimensional homogeneous spin manifolds X and show that they are constant as modular functions. The basic tool is a reduction formula relating Φ(X) to that of the self-intersection of the fixed point set of an involution γ on X. When Φ(X) is a constant it equals the signature of X. We derive a general formula for sign(G/H), G⊃H compact Lie groups, and determine its value in some cases by making use of the theory of involutions in compact Lie groups.

Elliptic Genera of Level N for Complex Manifolds

My lecture at the Como Conference was a survey on the theory of elliptic genera as developed by Ochanine, Landweber, Stong and Witten. A good global reference are the Proceedings of the 1986 Princeton Conference [1]. In this contribution to the Proceedings of the Como Conference I shall not reproduce my lecture, but rather sketch a theory of elliptic genera of level N for compact complex manifolds which I presented in the last part of my course at the University of Bonn during the Wintersemester 1987/88. For a natural number N > 1 the elliptic genus of level N of a compact complex manifold M of dimension d is a modular form of weight d for the group Γ1(N). In the cusps of Γ1(N) the genus degenerates either to χy(M)/(1+y)d where -y is an Nth-root of unity different from 1 or to χ(M,Kk/N) where K is the canonical line bundle and 0 < k < N. Here \({\chi _y}\left( M \right) = \sum\limits_{p = 0}^d {{\chi ^p}} \left( M \right){y^p} \) with \({\chi ^p}\left( M \right) = \chi \left( {M,{\Omega ^p}} \right) = \sum\limits_{q = 0}^d {{{\left( { - 1} \right)}^q}{h^{p,q}}} \) is the χy-genus introduced in [13] and χ(M.Kk/N) is the genus with respect to the characteristic power series

Involutionen auf Mannigfaltigkeiten

\frac{x}{{1 - {e^{ - x}}}}\cdot{e^{ - {{(k/N)}^ \bullet }x}}

Centennial of the German Mathematical Society

which equals the holomorphic Euler number of M with coefficients in the line bundie Lk provided K = LN (see [13]).