Abstract: Complex differential geometry intervenes in diophantine problems through several factors. First, if one considers holomorphic families of varieties, the problem of determining whether there exist only finitely many sections can be studied from a complex geometric point of view. But it also turns out (conjecturally at the moment) that the property of being Mordellic for a projective variety can be characterized in terms of purely complex differential geometric invariants, or complex analytic invariants. For instance, I conjectured that a projective variety X defined over a subfield of C finitely generated over the rationals is Mordellic if and only if every holomorphic map of C into X(C) is constant. It is known in many cases that certain projective varieties have this holomorphic property, but except for curves of genus ≧ 2 or subvarieties of abelian varieties which do not contain translations of abelian subvarieties of dimension > 0 (Fallings’ theorems) or varieties derived from those by products or unramified coverings or quotients, it is not known that they are Mordellic. Thus one obtains complex analytic criteria for a variety to be Mordellic. Similarly, in Chapter IX, we shall get quantitative diophantine criteria by inequalities at one absolute value.
Publication Year: 1991
Publication Date: 1991-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
Access and Citation
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot