On certain numerical invariants of algebraic varieties with application to abelian varieties

Abstract

1 2. Effective and algebraic cycles.5 3. Integrals of the second kind.12 4. Periods of integrals of the first kind.15 Chapter II.Invariants p and a of algebraic varieties.§1.Poincaré's normal functions for algebraic surfaces.17 2. Algebraic cycles of a surface.Fundamental theorem.19 3. Extension to any V¿.29 4. Some applications.38 5. Varieties representing the groups of points of other varieties.44 Part II.Abelian Varieties.Chapter I. General properties.§1.A summary of some fundamental theorems and definitions.46 2. Connectivity of varieties of rank one.52 3. Intermediary functions.57 4. Relation p = 1 + k.Theorem of Appell-Humbert.69 5. Formulas for the arithmetic genera.75 Chapter II.Abelian varieties with complex multiplication.§1.Generalities.77 2. Complex multiplication with irreducible characteristic equation.85 3. The characteristic equation is irreducible and Abelian.90 4. Characteristic equation of type [ / ( a ) ]' = 0, r > 1.(a) Generalities.95 5. Characteristic equation of type [ / ( a ) ]' = 0. (6) Two important special cases 102 6. Pure matrices of genus two or three.112 Chapter III.Abelian varieties with cyclic group and varieties of rank > 1; images of their involutions.§1.Varieties of rank one with cyclic group.118 2. Multiple points of Abelian varieties of rank > 1. 121 3. Connectivity of varieties of rank > 1. SOLOMON LEFSCHETZ [July Chapter IV.A class of algebraic curves with cyclic group and their Jacobi varieties.§1. Integrals of the first kind of the curves y* -Hj_f (x -a,-)"* (q odd prime) ... 2. Curves with three critical points.3. Curves with four critical points.4. Curves with r + 2 > 4 critical points.'In a monograph entitled UAnalysis Situs et la Géométrie Algébrique in course of publication in the Borel Series (Paris, Gauthiers-Villars), I have undertaken a thorough and systematic study of the topology of algebraic surfaces and varieties.Many new and interesting results, discovered since this memoir was written, will be found there.(Added in 1922).of simplification we shall denote the hypersurfaces C^, C*, by C0, Ci.We propose to prove the following three propositions: (a) Any ¿-cycle of Vd ( i < d ) is homologous to a cycle wholly within a C, and when i < d -1, the two cycles bound at the same time in their respective manifolds.(6) Any d-cycle is the sum of two others of which one is wholly within a C and the other is composed of a d-dimensional manifold contained in Co plus the loci of certain (d -1 )-cycles of Cu when u describes the lines a0ai.(c) The ith connectivity indices, i S d -2, of Vd and C are equal.For linear cycles (a) and (c) have already been proved by Picard, Castelnuovo, and Enriques, for two-cycles of algebraic surfaces by Poincaré, whose method adapted to the general case shall be used here.*We shall also apply his notations = , «, and say with him: "A is homologous to B (or A ~ß), modulo M ", to indicate that A -B bounds in M.

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