On mumfords construction of degenerating abelian varieties valery alexeev and iku nakamura received may 8, 1998 abstract. In 1976, the author published the first volume under the title lgebraic geometry. Mumford describes the analytic methods and results applicable when the ground field k is the complex field c and discusses the schemetheoretic methods and results used to deal with inseparable isogenies when the ground field k has characteristic p. Pdf a survey around the hodge, tate and mumfordtate. An abelian variety is a projective algebraic variety that is also an algebraic group with a group law which is commutative. The book is a sequel to algebraic geometry i published by springerverlag in 1976. Let x be a smooth projective variety over a finitely generated field k of characteristic0 and fix an embedding k \subset \mathbbc. Mumford,s conjecture and homotopy theory 1 abstract the thesis is an exposition on mumfords conjecture and the proof of the conjecture. The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of mumfords abelian varieties. Lectures on curves on an algebraic surface david mumford with a section by g. Honey and mumford learning styles were developed by peter honey and alan mumford, based upon the work of kolb, and they identified four distinct learning styles or preferences. Let us suppose that mumfordtate and hodge conjectures hold for an abelian.
Shimura curves of mumfords type have been extensively studied. David mumford, red book of varieties and schemes cf. We prove that a 1dimnl family of abelian varieties with an ample sheaf defining principal polarization can. The mumford tate conjecture is first proved for cm abelian varieties by h. The mumfordtate conjecture now it is time to put 1 and 1 together. In this note we give another proof of this result and extend it to cm motives. On mumfords families of abelian varieties rutger noot abstract in mum69, mumford constructs families of abelian varieties which are parametrized by shimura varieties but which are not of pel type. We notably determine, for each bre of such a family over a num.
In order to construct the mumford example, and to study the hilbert scheme of conics, some technical results are essential. We show that ribet sections are the only obstruction to the validity of the relative maninmumford conjecture for one dimensional families of semiabelian surfaces. We show that ribet sections are the only obstruction to the validity of the relative manin mumford conjecture for one dimensional families of semi abelian surfaces. Pdf this survey describe hodge, tate and mumfordtate conjectures for abelian varieties. Relative manin mumford for semi abelian surfaces d. Ramanujam and on the mordell weil theorem by yuri manin.
The red book of varieties and schemes david mumford springer. For a onedimensional family of abelian varieties equipped with principal theta divisors a canonical limit is constructed as a pair consisting of a reduced projective variety and a cartier divisor on it. Mar 11, 2017 we establish a conjecture of mumford characterizing rationally connected complex projective manifolds in several cases. On the newton polygons of abelian varieties of mumfords type. A very explicit book about algebraic curves and their jacobians. Pdf introduction to abelian varieties download ebook for. In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. We have developed an algebraic formalism of qhodge structures. Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. It was david mumford, who at first started the project of writing a textbook on. On the newton polygons of abelian varieties of mumford s type. Mumfords famous red book gives a simple readable account of the basic. I 291 data for l with respect to the morphism x y xk. The hard part, the rigidity lemma, is proved for complete algebraic varieties on page 43 of mumford s abelian varieties.
Geometric invariant theory by mumfordfogarty the first edition was published in 1965, a second, enlarged edition appeared in 1982 is the standard reference on applications of invariant theory to the construction of moduli spaces. Complex projective varieties, springerverlag, 1976. In mathematics, mumfords compactness theorem states that the space of compact riemann surfaces of fixed genus g 1 with no closed geodesics of length less than some fixed. The notes have been brought to the present form in collaboration with t. Two issues concerning a specific set of learning styles, described by honey and mumford 1986, are.
The mumfordtate conjecture is first proved for cm abelian varieties by h. The red book of varieties and schemes, mimeographed notes from harvard mathematics department, 1967. Projective spaces trivial edit other varieties whose cohomology is generated by algebraic cycles more or less trivial edit. Download limit exceeded you have exceeded your daily download allowance. Around hodge, tate and mumford tate conjectures on abelian varieties victoria cantoralfarf an advisor. All content in this area was uploaded by david mumford on feb 15, 2014.
Rationally connected varietieson a conjecture of mumford. It covers the material in the red book in more depth with several more topics added. This book contains what mumford had then intended to be volume ii. Mumford mentions in a footnote that complete algebraic varieties are automatically irreducible, and he later seems to assume without much explanation that they are connected. The use of theta functions, particularly since mumford s work, has been an important tool in the study of abelian varieties and invertible sheaves on them. Thecohomology ofthe moduli space ofabelian varieties arxiv. Milne august 6, 2012 abstract this the original tex.
We establish a conjecture of mumford characterizing rationally connected complex projective manifolds in several cases. Around hodge, tate and mumfordtate conjectures on abelian. It is a well known fact that its integral closure s. Let x be a complex abelian fourfold of mumfordtype and let v h 1 x. Abelian fourfold of mumfordtype and kugasatake varieties. Mumford s famous red book gives a simple readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra. The mumfordtate conjecture for products of abelian varieties. Around hodge, tate and mumfordtate conjectures on abelian varieties victoria cantoralfarf an advisor. X is a rank r free commutative group, and x xs is a constant group scheme, the. In the literature, the tate module usually means something slightly di erent. Abelian varieties with adic galois representation of mumford.
We define the notion of a special subvariety y in x of shimura type, and formulate necessary. Ramanujam and on the mordellweil theorem by yuri manin. This third, revised edition has been long awaited for by the. What is a curve and how explicitly can we describe them. Pdf on the newton polygons of abelian varieties of mumford. Algebraic geometry ii texts and readings in mathematics. On abelian automorphism groups of mumford curves aristides kontogeorgis and victor rotger abstract.
The complex mumfordtate group of x is isogenous to sl2 3. Mumfordtate varieties are generalizations of shimura varieties. Mumfordtate groups and abelian varieties introduction. Complex projective varieties where the corrections concerned the wiping out of some misprints, inconsistent notations, and other slight inaccuracies. On mumford s construction of degenerating abelian varieties valery alexeev and iku nakamura received may 8, 1998 abstract. The interaction of an ample line bundle with the group structure on an abelian variety is the subject of the first paper below as well as volume three of my tata lectures on theta. In mathematics, mumfords compactness theorem states that the space of compact riemann surfaces of fixed genus g 1 with no closed geodesics of length less than. Introduction to abelian varieties and the mumfordtate. The structure of the ring of endomorphisms of an abelian variety is discussed. May 5, 2010 the aim of the algebraic geometry 2 course is to introduce one of algebraic geometrys central ideas, moduli spaces, and develop it in one relatively simple setting. These are the learning approaches that individuals naturally prefer and they recommend that in order to maximise. This book is intended for selfstudy or as a textbook for graduate students.
Really, we know more, in fact, let r gq be the ring generated by such thetanullwerte. Abelian varieties remains the definitive account of the subject the classical theory is beautifully intertwined with the modern theory, in a way which sharply illuminates both it will remain for the foreseeable future a. In the main it follows mumford s book 16 except that most results are stated relative to an arbitrary base field, some additional results are proved, and etale cohomology is included. David mumford and tadao oda division of applied mathematics. To this end, we will mostly follow mumfords classic book 4. As far as i know, all nontrivial known cases of the mumfordtate conjecture more or less depend on the mumfordtate conjecture for abelian varieties. If we require b to be square free, then b is uniquely determined by d. In 1, mumford 1983conjectured that the rational cohomology of classifying space of mapping class group of riemann surfaces with sufficient large genus is a polynomial algebra on classes h of dimension 2i. Some evidence supporting the mumfordtate conjecture is provided by the fact that if x is an abelian variety over a number eld such that one associated adic representation is of mumfords type, then all associated galois representations are of this kind, cf. Complex projective varieties classics in mathematics reprint of the 1st.
A study of the hilbertmumford criterion for the stability of. Download abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the onedimensional ease. We recover information about the hodge structure of x using representations of the lie algebras sl 2 3 and sp 8 acting on v. Mumfords famous red book gives a simple readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and.
These are appendices on tates theorem on endomorphisms of abelian varieties over finite fields by c. In the main it follows mumford s book 16 except that most results are stated relative to an arbitrary base field, some additional. Leadership skill requirements across organizational levels troy v. David mumford was awarded the 2007 ams steele prize for mathematical exposition.
Now back in print, the revised edition of this popular study gives a systematic account of the basic results about abelian varieties. On mumford s construction of degenerating abelian varieties. I think this is a surprising theory because it leads to very explicit equations both for individual abelian varieties and for their moduli space. Get your kindle here, or download a free kindle reading app. However, thanks to a wonderful effort by tadao oda, we can now publish on this web site, for free distribution just click on the red, a penultimate draft for the second volume we do not anticipate an ultimate draft nor a third volume. We use rigid analytic uniformization by schottky groups to give a bound for the order of the abelian subgroups of the automorphism group of a. Knapp, advanced algebra, digital second edition east setauket, ny. Pdf on mumfords construction of degenerating abelian. Introduction these are notes for a lecture in elham izadis 2006 vigre seminar on the hodge conjecture. Question concerning mumfords abelian varieties stack exchange. Algebraic geometry of topological spaces i cortinas, guillermo and thom, andreas, acta mathematica, 2012. Abelian varieties bryden cais a canonical reference for the subject is mumfords book 6, but mumford generally works over an algebraically closed. Introduction to abelian varieties and the mumfordtate conjecture.
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