Fyodor Alexejewitsch Bogomolov

Fyodor Bogomolov

Fyodor Alexejewitsch Bogomolow ( Russian Фёдор Алексеевич Богомолов ; English transcription Fedor Alekseevich Bogomolov; born September 26, 1946 in Moscow ) is a Russian-American mathematician who deals with algebraic geometry.

Bogomolow studied at Lomonossow University and received his doctorate in 1973 (candidate title) at the Steklow Institute under Sergei Novikow ( compact Kähler manifolds ). In 1983 he completed his habilitation (Russian doctorate). In 1994 he went to the USA and became a professor at the Courant Institute of Mathematical Sciences of New York University .

Bogomolow studied Kahler manifolds and Hyperkähler manifolds (Riemannian manifolds with a symplectic group as holonomy group) in his early work . Bogomolow examined them as complex algebraic varieties (holomorphic symplectic manifolds) and gave criteria for these in 1974 in a decomposition theorem named after him for the fact that the holonomy group of compact holomorphic symplectic manifolds is symplectic. He later studied the deformation theory of Hyperkähler manifolds, which led to the Bogomolov-Tian-Todorov theorem (from which in particular it followed that the deformations of Calabi-Yau manifolds had no obstructions). This work forms one of the foundations of the mirror symmetry investigated in string theory

In 1978 he studied holomorphic vector bundles in projective space (where he introduced Bogomolov stability of vector bundles) and proved an inequality named after him, Shing-Tung Yau and Yōichi Miyaoka for the Chern numbers of compact complex surfaces.

In 1977 he proved that on algebraic surfaces of general type whose Chern numbers satisfy the inequality there are only a finite number of curves with restricted gender. The work was important for the later development of hyperbolic arithmetic algebraic geometry ( Paul Vojta , Serge Lang ) and was extended by Michael McQuillan in his proof of the conjecture by Green and Griffiths. ${\displaystyle c_{1}^{2}>c_{2}}$

In 1976 he published a paper on the still open problem of the classification of algebraic surfaces of Kodaira class VII.

A conjecture in Diophantine geometry is named after Bogomolow, which generalizes the Manin-Mumford conjecture and which was proven in 1998 by Emmanuel Ullmo and Shou-Wu Zhang . Both are about curves C from gender over number fields K and their Jacobi varieties J with an embedding X of C in J over K. The Manin-Mumford conjecture says that in the intersection of C with the torsion subgroup of J there are only finite many Points, the Bogomolov conjecture says that there are only a finite number of points in X with bounded Neron-Tate heights on J. ${\displaystyle g\geq 2}$

Bogomolow (2nd from left) in Shafarevich's apartment (right), with Helmut Koch (left), AN Todorov

In 1978 he was invited speaker at the International Congress of Mathematicians in Helsinki ( Unstable vector bundles and curves on surfaces ).

Fonts

• with Yuri Tschinkel (editor): Cohomological and geometric approaches to rationality problems - new perspectives, Birkhäuser, Progress in Mathematics 2009

Web links

• Website at New York University
• Conference in his honor at the Steklov Institute (Russian)

Individual evidence

1. ↑ so until 1978 Eugenio Calabi called
2. ^ The decomposition of Kähler manifolds with trivial canonical class (Russian), Mat. Sbornik (NS), Vol. 93, 1974, p. 135.
3. ↑ Kähler Manifolds with trivial canonical class, Preprint IHES 1981
4. ↑ Holomorphic tensors and vector bundles on projective manifolds (Russian), Izvestija Akad. Nauka SSSR Ser. Mat., Vol. 42, 1978, pp. 1227-1287, 1439
5. ^ Families of curves on surfaces of general type (Russian), Doklady Akad. Nauka SSR, Vol. 236, 1977, pp. 1041-1044.
6. ↑ Classification of areas of class VII0 with b2 = 0 (Russian), Izvestija Akad.Nauka SSSR Ser. Mat., Vol. 40, 1976, pp. 273-288, 469