Fedor Bogomolov

Fedor Bogomolov (Фёдор Алексеевич Богомолов) is an American and Russian mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at Steklov Institutein Moscow before he became a professor at Courant Institute.He is most famous for his pioneering work on
hyperkähler manifolds.

Born 26.09.1946 in Moscow, Bogomolov graduated from
Moscow State University, Faculty of Mechanics and Mathematics, and earned his doctorate ("candidate degree") in 1973, in Steklov Institute. His advisor was S. P. Novikov.

Geometry of Kähler manifolds

Bogomolov's Ph. D. is titled "Compact Kähler varieties".In his early papers [Bogomolov, F. A. "Manifolds with trivial canonical class." (Russian) Uspehi Mat. Nauk 28 (1973), no. 6 (174), 193--194.MathSciNet | id = 390301] , [Bogomolov, F. A."Kahler manifolds with trivial canonical class". (Russian)Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 11--21.MathSciNet | id = 338459 ] , [Bogomolov, F. A."The decomposition of Kahler manifolds with a trivial canonical class." (Russian)Mat. Sb. (N.S.) 93(135) (1974), 573--575, 630. MathSciNet | id = 345969] Bogomolov studied manifolds which were much later called
Calabi-Yau and hyperkaehler.He proved a famous decomposition theorem,which is a cornerstone of classification of manifoldswith trivial canonical class. It is re-proven nowusing Calabi-Yau theorem and Berger's classification of Riemannian holonomies, and lies in foundation of the modernString theory.

In late 1970-es and early 1980-ies Bogomolov studied thedeformation theory for manifolds with trivial canonical class( [Bogomolov, F. A."Hamiltonian Kahlerian manifolds". (Russian)Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101--1104.MathSciNet | id = 514769] , [Bogomolov, F. A., "Kahler manifolds with trivial canonical class," Preprint Institute des Hautes EtudesScientifiques 1981 p.1-32.] ).He discovered what is now known as Bogomolov-Tian-Todorov theorem,proving the smoothness and un-obstructedness of the deformationspace for hyperkaehler manifolds (in 1978 paper) and thenextended this to all Calabi-Yau manifolds in the 1981 IHES preprint.Some years later, this theorembecome the mathematical foundation for Mirror Symmetry.

While studying the deformation theory of hyperkaehlermanifolds, Bogomolov discovered what is now known as
Bogomolov-Beauville-Fujiki form on $H^2(M)$.Studying properties of this form, Bogomolov erroneouslyconcluded that compact hyperkaehler manifolds don'texist, with exception of a K3 surface, torusand their products. Almost 4 years passed sincethis publication before Fujiki found a counterexample.

Other works in algebraic geometry

Bogomolov's most-cited paper is "Holomorphic tensors and vector bundles on projective manifolds." [Bogomolov, F. A. "Holomorphic tensors and vector bundles on projective manifolds." (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 6, 1227--1287, 1439 MathSciNet | id = 0522939] He proved what is now known as Bogomolov-Miyaoka-Yau inequalityand defined a new, refined notion of stability for holomorphicvector bundles (Bogomolov stability). Bogomolov also proved thata stable bundle on a surface, restricted to a curve of sufficiently big degree, remains stable.

In another seminal paper, "Families of curves on a surface of general type" [Bogomolov, F. A."Families of curves on a surface of general type." (Russian)Dokl. Akad. Nauk SSSR 236 (1977), no. 5, 1041--1044. MathSciNet | id = 457450] , Bogomolov laid the foundations to the now popular approach tothe theory of diophantine equations through geometry of
hyperbolic manifolds and dynamical systems.In this paper Bogomolov proved that on any
surface of general type with $c\_1^2>c\_2$,there is only a finite number of curves of bounded genus.Some 25 years later, Michael McQuillan [McQuillan, MichaelDiophantine approximations and foliations. Inst. Hautes Etudes Sci. Publ. Math. No. 87 (1998), 121--174. MathSciNet | id = 99m:32028 ] extended this argumentto prove the famous Green-Griffiths conjecturefor such surfaces.

Another remarkable paper is "Classification of surfaces of class $VII\_0$ with $b\_\{2\}=0$", [Bogomolov, F. A."Classification of surfaces of class $VII\_0$ with $b\_\{2\}=0$" (Russian)Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 2, 273--288, 469. MathSciNet | id = 427325] Using affine structures on complex manifolds,Bogomolov made the first step in a famously difficult (and still unresolved) problem of classification ofsurfaces of Kodaira class VII. These are compactcomplex surfaces with $b\_2=1$. If they are in addition minimal, they are called "class $VII\_0$". Kodaira classifiedall compact complex surfaces exceptclass VII, which are still not understood,except the case $b\_\{2\}=0$ (Bogomolov)and $b\_\{2\}=1$ (A. Teleman, 2005, [ A. Teleman,"Donaldson Theory on non-Kahlerian surfaces and class VII surfaces with $b\_2=1$,"Invent. math. 162, 493-521, 2005. MathSciNet | id = 2006i:32020] )

Later career

Bogomolov obtained his Habilitation(Russian "Dr. of Sciences") in 1983.In 1994, he emigrated to U.S. and becamea full professor in Courant Institute. He is veryactive in algebraic geometry and number theory.In 2006, Bogomolov turned 60;two major conferences commemoratinghis birthday were held - [http://www.math.princeton.edu/~ytschink/.miami05/ one in University of Miami] , and [http://www.mi.ras.ru/~orlov/acts/rasp.html another in Moscow, Steklov Institute] .

References

External links

* [http://www.math.nyu.edu/faculty/bogomolo/index.html Official NYU home page]
*
* [http://www.polit.ru/science/2006/11/10/bogomolov.html Новые перспективы науки] Nov. 2, 2006, Bilingua club, Moscow.
* [http://www.polit.ru/analytics/2006/06/22/bogomol.html "Из научной интеллигенции можно сформировать "сословие экспертов"] An interview of Fedor Bogomolov (Olga Orlova for polit.ru)