Fedor Bogomolov

Fedor Bogomolov.

Fedor Alekseyevich Bogomolov (born 26 September 1946) (Фёдор Алексеевич Богомолов) is a Russian and American 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 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 Sergei Novikov.

Geometry of Kähler manifolds

Bogomolov's Ph. D. was entitled Compact Kähler varieties. In his early papers[1][2][3] Bogomolov studied the manifolds which were later called Calabi–Yau and hyperkaehler. He proved a decomposition theorem, used for the classification of manifolds with trivial canonical class. It has been re-proven using the Calabi–Yau theorem and Berger's classification of Riemannian holonomies, and is foundational for modern string theory.

In the late 1970s and early 1980s 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 became 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 four years passed since this 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."[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, McQuillan[8] extended this argument to prove the famous Green–Griffiths conjecture for such surfaces.

Another remarkable paper is "Classification of surfaces of class VII_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 VII_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]

Later career

Bogomolov obtained his Habilitation (Russian "Dr. of Sciences") in 1983. In 1994, he emigrated to the U.S. and became a full professor at the 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 at the University of Miami, and another in Moscow, Steklov Institute. From 2009 till March 2014 he served as the Editor-in-Chief of Central European Journal of Mathematics. From 2010 he is the director of the HSE Laboratory of algebraic geometry and its applications.

References

  1. ↑ Bogomolov, F. A. Manifolds with trivial canonical class. (Russian) Uspehi Mat. Nauk 28 (1973), no. 6 (174), 193–194. MR 390301
  2. ↑ Bogomolov, F. A. Kähler manifolds with trivial canonical class. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 11–21 MR 338459
  3. ↑ Bogomolov, F. A. The decomposition of Kähler manifolds with a trivial canonical class. (Russian) Mat. Sb. (N.S.) 93(135) (1974), 573–575, 630. MR 345969
  4. ↑ Bogomolov, F. A. (1978). "[Hamiltonian Kählerian manifolds]". Doklady Akademii Nauk SSSR (in Russian) 243 (5): 1101–1104. MR 514769.
  5. ↑ Bogomolov, F. A., Kähler manifolds with trivial canonical class, Preprint Institute des Hautes Etudes Scientifiques 1981 pp. 1–32.
  6. ↑ 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
  7. ↑ Bogomolov, F. A. (1977). Семейства кривых на поверхности общего типа [Families of curves on a surface of general type]. Doklady Akademii Nauk SSSR (in Russian) 236 (5): 1041–1044. MR 457450.
  8. ↑ McQuillan, Michael Diophantine approximations and foliations. Inst. Hautes Etudes Sci. Publ. Math. No. 87 (1998), 121–174. MR 99m:32028
  9. ↑ 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. MR 427325
  10. ↑ A. Teleman, Donaldson Theory on non-Kählerian surfaces and class VII surfaces with b_2=1, Invent. math. 162, 493–521, 2005. MR 2006i:32020

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