Fedor Bogomolov

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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 Institute in 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.

1 Geometry of Kähler manifolds
2 Other works in algebraic geometry
3 Later career
4 Reference
5 External links

[edit] Geometry of Kähler manifolds

Bogomolov's Ph. D. is titled "Compact Kähler varieties". In his early papers ^[1], ^[2], ^[3] 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 manifolds with trivial canonical class. It is re-proven now using Calabi-Yau theorem and Berger's classification of Riemannian holonomies, and lies in foundation of the modern String theory.

In late 1970-es and early 1980-ies Bogomolov studied the deformation theory for manifolds with trivial canonical class (^[4], ^[5]). He discovered what is now known as Bogomolov-Tian-Todorov theorem, proving the smoothness and un-obstructedness of the deformation space for hyperkaehler manifolds (in 1978 paper) and then extended this to all Calabi-Yau manifolds in the 1981 IHES preprint. Some years later, this theorem become the mathematical foundation for Mirror Symmetry.

While studying the deformation theory of hyperkaehler manifolds, Bogomolov discovered what is now known as Bogomolov-Beauville-Fujiki form on $H 2 (M)$ . Studying properties of this form, Bogomolov erroneously concluded that compact hyperkaehler manifolds don't exist, with exception of a K3 surface, torus and their products. Almost 4 years passed since this publication before Fujiki found a counterexample.

[edit] Other works in algebraic geometry

Bogomolov's most-cited paper is "Holomorphic tensors and vector bundles on projective manifolds." ^[6] He proved what is now known as Bogomolov-Miyaoka-Yau inequality and defined a new, refined notion of stability for holomorphic vector bundles (Bogomolov stability). Bogomolov also proved that a 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"^[7], Bogomolov laid the foundations to the now popular approach to the 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^[8] extended this argument to prove the famous Green-Griffiths conjecture for such surfaces.

Another remarkable paper is "Classification of surfaces of class $V I I 0$ with $b 2 = 0$ ",^[9] Using affine structures on complex manifolds, Bogomolov made the first step in a famously difficult (and still unresolved) problem of classification of surfaces of Kodaira class VII. These are compact complex surfaces with $b 2 = 1$ . If they are in addition minimal, they are called class $V I I 0$ . Kodaira classified all compact complex surfaces except class VII, which are still not understood, except the case $b 2 = 0$ (Bogomolov) and $b 2 = 1$ (A. Teleman, 2005, ^[10])

[edit] Later career

Bogomolov obtained his Habilitation (Russian "Dr. of Sciences") in 1983. In 1994, he emigrated to U.S. and became a full professor in Courant Institute. He is very active in algebraic geometry and number theory. In 2006, Bogomolov turned 60; two major conferences commemorating his birthday were held - one in University of Miami, and another in Moscow, Steklov Institute.

[edit] Reference

^ Bogomolov, F. A. Manifolds with trivial canonical class. (Russian) Uspehi Mat. Nauk 28 (1973), no. 6 (174), 193--194.MR 390301
^ Bogomolov, F. A. Kahler manifolds with trivial canonical class. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 11--21.MR 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. MR 345969
^ Bogomolov, F. A. Hamiltonian Kahlerian manifolds. (Russian) Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101--1104.MR 514769
^ Bogomolov, F. A., Kahler manifolds with trivial canonical class, Preprint Institute des Hautes Etudes Scientifiques 1981 p.1-32.
^ 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 MR 0522939
^ Bogomolov, F. A. Families of curves on a surface of general type. (Russian) Dokl. Akad. Nauk SSSR 236 (1977), no. 5, 1041--1044. MR 457450
^ McQuillan, Michael Diophantine approximations and foliations. Inst. Hautes Etudes Sci. Publ. Math. No. 87 (1998), 121--174. MR 99m:32028
^ Bogomolov, F. A. Classification of surfaces of class $V I I 0$ with $b 2 = 0$ (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 2, 273--288, 469. MR 427325
^ A. Teleman, Donaldson Theory on non-Kahlerian surfaces and class VII surfaces with $b 2 = 1$ , Invent. math. 162, 493-521, 2005. MR 2006i:32020