Đặc Trưng Toán Học: Lớp Đặc Trưng của Đa Tạp Singular và K-Lý Thuyết

OTES 3LRWU3UDJDF] (GLWRU "IRKHËUSER 6ERLAG "ASEL "OSTON "ERLIN s s %DITOR 0IOTR 0RAGACZ )NSTITUTE OF -ATHEMATICS OF THE 0OLISH !CADEMY OF 3CIENCES UL 3NIADECKICH  0/ "OX    7ARSZAWA 0OLAND E MAIL PPRAGACZ IMPANGOVPL  -ATHEMATICAL 3UBJECT #LASSIlCATION % # # # # , - . ! !XX $XX %XX 1 3XX 0 + ( ! #)0 CATALOGUE RECORD FOR THIS BOOK IS AVAILABLE FROM THE ,IBRARY OF #ONGRESS 7ASHINGTON $# 53! "IBLIOGRAPHIC INFORMATION PUBLISHED BY $IE $EUTSCHE "IBLIOTHEK $IE $EUTSCHE "IBLIOTHEK LISTS THIS PUBLICATION IN THE $EUTSCHE .ATIONALBIBLIOGRAlE DETAILED BIBLIOGRAPHIC DATA IS AVAILABLE IN THE )NTERNET AT HTTPDNBDDBDE )3".     "IRKHËUSER 6ERLAG "ASEL n "OSTON n "ERLIN 4HIS WORK IS SUBJECT TO COPYRIGHT !LL RIGHTS ARE RESERVED WHETHER THE WHOLE OR PART OF THE MATERIAL IS CONCERNED SPECIlCALLY THE RIGHTS OF TRANSLATION REPRINTING RE USE OF ILLUSTRATIONS RECITATION BROADCASTING REPRODUCTION ON MICROlLMS OR IN OTHER WAYS AND STORAGE IN DATA BANKS &OR ANY KIND OF USE PERMISSION OF THE COPYRIGHT OWNER MUST BE OBTAINED ¥  "IRKHËUSER 6ERLAG 0/ "OX  #(  "ASEL 3WITZERLAND 0ART OF 3PRINGER 3CIENCE "USINESS -EDIA 0RINTED ON ACID FREE PAPER PRODUCED FROM CHLORINE FREE PULP 4#& ' 0RINTED IN 'ERMANY )3".        WWWBIRKHAUSERCH Contents Preface. ix Piotr Pragacz Notes on the Life and Work of Alexander Grothendieck.

xxvii Paolo Aluffi Characteristic Classes of Singular Varieties. 31 Michel Brion Lectures on the Geometry of Flag Varieties. Grassmannians and flag varieties. Singularities of Schubert varieties.

The diagonal of a flag variety. Positivity in the Grothendieck group of the flag variety. 82 Anders Skovsted Buch Combinatorial K-theory. K-theory of Grassmannians.

The bialgebra of stable Grothendieck polynomials. Geometric specializations of stable Grothendieck polynomials. Alternating signs of the coefficients cw,µ. 102 vi Contents Haibao Duan Morse Functions and Cohomology of Homogeneous Spaces.

Computing homology: a classical method. Elements of Morse theory. Morse functions via Euclidean geometry. Morse functions of Bott-Samelson type.

132 Ali Ulas Ozgur Kisisel Integrable Systems and Gromov-Witten Theory. Completely integrable systems. Random matrices and enumeration of graphs. Witten’s conjecture and Kontsevich’s solution.

Some of the further developments. 159 Piotr Pragacz Multiplying Schubert Classes. Characteristic map and BGG-operators. Structure constants for Schubert classes.

A combinatorial proof of the Pieri formula. 173 Jörg Schürmann Lectures on Characteristic Classes of Constructible Functions. Calculus of constructible functions. Stratified Morse theory for constructible functions and Lagrangian cycles.

Characteristic classes of Lagrangian cycles. Verdier-Riemann-Roch theorem and Milnor classes. Appendix: Two letters of J. 199 Marek Szyjewski Algebraic K-theory of Schemes.

K-theory of fields. K•
of noetherian schemes. K• of certain varieties. 269 Contents vii Harry Tamvakis Gromov-Witten Invariants and Quantum Cohomology of Grassmannians.

296 Dédié à Alexandre Grothendieck Preface The articles in this volume1 are an outgrowth of seminars and schools of Impanga. Impanga is an algebraic geometry group operating since 2000 at the Institute of Mathematics of Polish Academy of Sciences in Warsaw. Besides seminars Impanga organized the following schools at the Banach Center in Warsaw: • Characteristic classes of singular varieties, April 2002, • Stratifications of moduli spaces, May 2002, • Schubert varieties, May 2003, and • Hommage à Grothendieck, January 2004. More information about the Impanga seminars and schools2 can be found on: http://www.pl/∼pragacz/impanga.htm Impanga also co-organized the school Algebraic geometry, algebra, and applications in Borovetz, Bulgaria (September 2003).

Let us describe briefly the contents of the lecture notes in this volume3. Paolo Aluffi discusses various characteristic classes generalizing classical Chern classes for nonsingular varieties: the classes of Mather, Schwartz-MacPher- son, and Fulton. A particular emphasis is put on concrete computations of these classes, often with the help of Segre classes. Michel Brion gives a comprehensive introduction to the geometry of flag va- rieties and Grassmannians.

A special emphasis is put on geometric properties of Schubert varieties and their resolutions: Bott-Samelson schemes. Vanishing prop- erties of line bundles over Schubert varieties are studied. Also Richardson varieties together with their applications are discussed. One of the main goals is to present a proof of Buch’s conjecture on the structure constants in the Grothendieck ring of a flag variety.

1 During the preparation of this volume, the Editor was partially supported by KBN grant No. 2 The first three schools were partially supported by EAGER. 3 The lecture notes by Aluffi and Schürmann stem from the first school, the notes by Brion, Buch, and Tamvakis from the third school, the article by Pragacz opening the volume from the fourth school, and the notes by Szyjewski from the school in Borovetz. The remaining contributions by Duan, Kisisel, and Pragacz come from the seminars of Impanga.

x Preface Anders Buch studies Grothendieck polynomials and their properties (follow- ing the work of Lascoux-Schützenberger and Fomin-Kirillov). They are used to show various combinatorial aspects of the Grothendieck ring of a Grassmannian and K-theoretic formulas for “quiver degeneracy loci”. Haibao Duan presents an elementary introduction to classical Morse theory, and shows its applications to homogeneous spaces. In particular, “Morse functions of Bott-Samelson type” are discussed together with Bott-Samelson cycles.

Some applications to Schubert calculus are also mentioned. The starting point of the article by Ali Kisisel are completely integrable sys- tems, in particular: KP and KdV hierarchies. Then the infinite Grassmannian of Sato and Toda hierarchy are described. Finally, through the studies of random matrices and enumeration of graphs, the article approaches the solution by Kont- sevich of Witten’s conjecture, as well as some aspects of Gromov-Witten theory.

Piotr Pragacz shows a way of computing the structure constants for multipli- cation of Schubert classes in the cohomology rings of generalized flag varieties G/P. Jörg Schürmann presents stratified Morse theory for constructible functions and its applications to characteristic classes of singular varieties. The point of view of characteristic classes of Lagrangian cycles is emphasized and a Verdier- type Riemann-Roch theorem is discussed. Marek Szyjewski gives an introduction to higher K-groups of Quillen (and also to those of Milnor).

Computations of higher K-groups of fields and of projective bundles, quadrics, and Severi varieties are presented. Some arithmetical aspects of the theory are also discussed. Harry Tamvakis studies quantum cohomology of homogeneous spaces, no- tably of various Grassmannians. It is shown that three-point genus zero Gromov- Witten invariants are equal to classical triple intersection numbers on homogeneous spaces of the same Lie type.

Quantum analogs of the Pieri and Giambelli formulas are also presented. We dedicate the whole volume to Alexander Grothendieck who remains for us an unsurpassed master in cohomological studies of algebraic varieties. The opening article by Piotr Pragacz discusses some aspects of his life and work. The Editor wishes to thank the authors for their scientific con- tributions.

He is grateful to Andrzej Weber for his help with the Impanga schools. Warm thanks go also to Dr. Thomas Hempfling from Birkhäuser-Verlag for his invitation to publish this material in Trends in Mathematics and for a pleasant cooperation during the preparation of this volume. Warszawa, October 2004 The Editor Trends in Mathematics: Topics in Cohomological Studies of Algebraic Varieties, xi–xxviii
c 2005 Birkhäuser Verlag Basel/Switzerland Notes on the Life and Work of Alexander Grothendieck* Piotr Pragacz** When I was a child I loved going to school.

The same instructor taught us reading, writing and arithmetic, singing (he played upon a little violin to accompany us), the archaeology of prehistoric man and the discovery of fire. I don’t recall anyone ever being bored at school. There was the magic of numbers and the magic of words, signs, and sounds. Grothendieck: Récoltes et Semailles Abstract.

This is a story of Alexander Grothendieck – a man who has changed the face of mathematics during some 20 years of his work on functional anal- ysis and algebraic geometry. Last year he turned 75. This paper, written in April 2004, is based on a talk presented at the Hommage à Grothendieck session of Impanga1 , held at the Banach Centre in Warsaw (January 2004). Alexander Grothendieck was born in Berlin in 1928.

His father, Alexander Shapiro (1890–1942) was a Russian Jew from a Hassidic town on a now Russian-Ukrainian- Belorussian border. He was a political activist – an anarchist involved in all the major European revolutions during 1905–1939. In the 20’s and 30’s he lived mostly in Germany, operating in the left-wing movements against more and more power- ful Nazis, and working as a street photographer. In Germany, he met Hamburg- born Hanka Grothendieck (1900–1957).

(The name Grothendieck comes from platt- deutsch, a Northern German dialect.) Hanka Grothendieck worked on and off as a ∗ Translated from the Polish by Janusz Adamus. This paper was originally published in Wiadomości Matematyczne (Ann. We thank the Editors of this journal for permission to reprint the paper. ∗∗ Partially supported by Polish KBN grant No.

1 Impanga is an algebraic geometry group, operating since 2000 at the Institute of Mathematics of the Polish Academy of Sciences. This session hosted the talks of: M. Chalupnik, Grothendieck topologies and étale cohomology, T. Maszczyk, Toposes and the unity of mathematics, J.

Gorski, Grothendieck stacks on Mazovia plains, O. Kȩdzierski, Why the derived categories?, A. Weber, The Weil conjectures, G. Banaszak, l-adic representations, P.

Krasoń, Mordell-Weil groups of Abelian varieties. xii Piotr Pragacz journalist, but her true passion was writing. On March 28, 1928 she gave birth to their son Alexander. During 1928–1933 Alexander lived with his parents in Berlin.

After Hitler’s rise to power, Alexander’s parents immigrated to France, leaving their son (for about 5 years) with the Heydorns, a surrogate family in Hamburg, where he went to a primary and secondary school. In 1939 Alexander joined his parents in France. His father was soon interned by the French Vichy police in the Vernet camp in the Pyrenees, and then handed out to the Nazis occupiers. He was murdered in the German concentration camp Auschwitz-Birkenau in 1942.

Very young Alexander Grothendieck Hanka and Alexander Grothendieck did not survive the occupation without problems. In the years 1940–1942 they were interned – as “undesirable dangerous foreigners” – in the Rieucros camp near Mende in southern France. Hanka was later transferred to the Gurs camp in the Pyrenees, whilst Alexander was allowed to continue his education in Collège Cévenol in a Cévennes Mountains town of Chambon-sur-Lignon in the southern Massif Central. The college, run by local Protestants under the leadership of Pastor Trocmé, was a sanctuary to many children (mainly Jews) whose lives were endangered during the war.

Already then, Alexander asked himself a question that showed the uniqueness of his mind: How to accurately measure the length of a curve, area of a surface, or volume of a solid? Continuing the reflection on these problems during his university studies in Montpellier (1945–1948), he independently obtained results equivalent to Lebesgue’s measure and integration theory. As expressed by J. Dieudonné in [D], the university in Montpellier – in Grothendieck’s days – wasn’t a “proper place” for studying great mathematical problems. In the fall of 1948 Grothendieck arrived in Paris, where he spent a year attending courses in the famous École Normale Supérieure (ENS), the “birthplace” of most of the French mathematical Notes on the Life and Work of Alexander Grothendieck xiii elite.

In particular, he took part in Cartan’s legendary seminar, that year devoted to algebraic topology.) Grothendieck’s interests, however, began focusing on functional analysis. Fol- lowing Cartan’s advice, in October 1949 he comes to Nancy, a centre of functional analysis studies, where J. Schwartz, and others run a seminar on Fréchet spaces and their direct limits. They encounter a number of problems which they are unable to solve, and suggest Grothendieck to try and attack them.

The result surpasses all expectations. In less than a year, Grothendieck manages to solve all the problems by means of some very ingenious constructions.