In Memoriam: George R. Kempf, 1944-2002Author(s): David Mumford and Bernard ShiffmanSource: American Journal of Mathematics, Vol. 124, No. 6 (Dec., 2002)Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/25099152 .

Accessed: 19/12/2014 00:21

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access toAmerican Journal of Mathematics.

http://www.jstor.org

This content downloaded from 67.66.218.73 on Fri, 19 Dec 2014 00:21:01 AMAll use subject to JSTOR Terms and Conditions

IN MEMORIAM: GEORGE R. KEMPF

1944-2002

Photo courtesy of the Kempffamily

I met George in 1970 when he burst on the algebraic geometry scene with a

spectacular PhD thesis. His thesis gave a wonderful analysis of the singularities of the subvarieties Wr of the Jacobian of a curve C obtained by adding the curve

to itself r times inside its Jacobian. This was one of the major themes that he

pursued throughout his career: understanding the interaction of a curve with its

Jacobian and especially to the map from the r-fold symmetric product of the

curve to the Jacobian. In his thesis he gave a determinantal representation both

of Wr and of its tangent cone at all its singular points, which gives you a complete

understanding of the nature of these singularities. A major focus of his later work

in this area were the Picard bundles: the vector bundles on the Jacobian whose

projectivizations are r-fold symmetric powers of C, for r > 2g - 1. He unwound

many of the mysteries of these bundles.

As George's research evolved, our work became closely intertwined in mul

tiple ways. In particular, he worked on invariant theory and on abelian varieties,

This content downloaded from 67.66.218.73 on Fri, 19 Dec 2014 00:21:01 AMAll use subject to JSTOR Terms and Conditions

especially linear systems on abelian varieties. Maybe his result in invariant the

ory which I loved the most concerns the orbits in a representation of a reductive

group that are "unstable," i.e. have 0 in their closure. He gave a beautiful con

struction of one canonical worst subgroup Gm in G carrying the point to 0. I

had looked for this in awkward ways and found it in some cases, but he saw

what was really going on. This result had many corollaries and completed the

program in Geometric Invariant Theory in the best possible way. Later on, he

studied extensively the singularities of orbit spaces, showing in many cases that

they had only rational singularities; he also studied the effective construction of

rings of invariants, and thus of orbit spaces.

Perhaps the area in which we were closest was his work on linear sys tems and the equations defining abelian varieties. I wrote three papers on this in

1966-67, much inspired by hearing Igusa's lectures on theta functions. But I used

to joke that George was the only one in the world who actually read these papers.

Again, he went deeper than I with more persistence and the deft touch by which

I always recognized his work. He kept finding better and more satisfying rea

sons why abelian varieties are so wonderful. For example, there was his theorem

that their homogeneous coordinate rings A were, in his terminology, exactly that:

"wonderful." He defined "wonderful" to mean that all the modules Torf (k,k) are

purely of degree /. This turns out to be the secret cohomological key to answer

many questions. Another unexpected and lovely result was the one he dedicated

to me for my 50th birthday: that multiplication gives an isomorphism between the

tensor product of the vector space of rank 2 theta functions, genetically twisted, and the vector space of rank 4 theta functions.

One of the things that distinguished his work was the total mastery with

which he used higher cohomology. A paper which, I believe, every new student

of algebraic geometry should read, is his elementary proof of the Riemann-Roch

theorem on curves: "Algebraic Curves" in Crelle, 1977. That such an old result

could be treated with new insight was the work of a master.

I won't discuss his work on the cohomology of homogeneous spaces or

the representation theory of algebraic groups, which others know much better

than I. Instead, I want to conclude by saying that this love of the simple and

satisfying elegance which can be found in these abstract fields brought George and I together. One feels that, given the disease with which he struggled, this

mathematics was a constant stable light to which he returned, that centered him

when other things failed. We miss the light he shed for us.

David Mumford, September 2002

This content downloaded from 67.66.218.73 on Fri, 19 Dec 2014 00:21:01 AMAll use subject to JSTOR Terms and Conditions

From the Editor's desk:

George Kempf had a remarkable way of making any topic elegant and trans

parent. I attended his lectures on complex abelian varieties in 1988-George would

casually walk into the classroom carrying his coffee cup and then effortlessly launch into a beautiful treatment of abelian varieties complete with new insights

combining analytic and geometric viewpoints. (Those lectures were later incorpo rated into his excellent book on the subject.) His facility for exposition extended

to his teaching at all levels. Before his illness interfered with his ability to teach, I

had the pleasant experience when I was Department Chair of hearing undergrad uates who were not math majors tell me how much they enjoyed his introductory

algebra course.

An endowment has been set up in his name at the Johns Hopkins University. Donations may be made to the George R. Kempf Memorial Endowment, Depart ment of Mathematics, The Johns Hopkins University, 3400 N. Charles Street,

Baltimore, MD 21218. His family also suggests that, according with George's

wishes, donations be made to the National Alliance for Research in Schizophre nia and Depressive Illnesses (http://www.narsad.org) in hopes that a cure can be

found for the future.

Bernard Shiftman

This content downloaded from 67.66.218.73 on Fri, 19 Dec 2014 00:21:01 AMAll use subject to JSTOR Terms and Conditions