Fedor Alekseyevich Bogomolov (born 26 September 1946) (Фёдор Алексеевич Богомолов) is a Russian and American
mathematician, known for his research in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
number theory. Bogomolov worked at the
Steklov Institute
Steklov Institute of Mathematics or Steklov Mathematical Institute (russian: Математический институт имени В.А.Стеклова) is a premier research institute based in Moscow, specialized in mathematics, and a part o ...
in
Moscow before he became a professor at the
Courant Institute in
New York
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. 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, at the Steklov Institute. His doctoral advisor was
Sergei Novikov.
Geometry of Kähler manifolds
Bogomolov's Ph.D. thesis was entitled ''Compact Kähler varieties''. In his early papers Bogomolov studied the manifolds which were later called
Calabi–Yau and
hyperkähler. He proved a
decomposition theorem In mathematics, especially algebraic geometry, the decomposition theorem of Beilinson, Bernstein and Deligne or BBD decomposition theorem is a set of results concerning the cohomology of algebraic varieties. It was originally conjectured by Gelfand ...
, 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 physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and interac ...
.
In the late 1970s and early 1980s Bogomolov studied the
deformation theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesim ...
for manifolds with trivial canonical class. 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 hyperkähler manifolds, Bogomolov discovered what is now known as the Bogomolov–Beauville–Fujiki form on
. Studying properties of this form, Bogomolov erroneously concluded that compact hyperkaehler manifolds do not exist, with the exception of
K3 surfaces, tori, and their products. Almost four years passed since this publication before Akira Fujiki found a counterexample.
Other works in algebraic geometry
Bogomolov's paper on "Holomorphic tensors and vector bundles on projective manifolds" proves what is now known as the
Bogomolov–Miyaoka–Yau inequality, and also proves that a stable bundle on a surface, restricted to a curve of sufficiently big degree, remains stable. In "Families of curves on a surface of general type", Bogomolov laid the foundations to the now popular approach to the theory of
diophantine equations through geometry of
hyperbolic manifolds
In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyperbolic surfaces and hyperbolic 3-manifolds, resp ...
and
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
. In this paper Bogomolov proved that on any
surface of general type In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in ...
with
, there is only a finite number of curves of bounded genus. Some 25 years later,
Michael McQuillan extended this argument to prove the famous Green–Griffiths conjecture for such surfaces. In "Classification of surfaces of class
with
", 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
. If they are in addition minimal, they are called ''class
''.
Kunihiko Kodaira classified all compact complex surfaces except class VII, which are still not understood, except the case
(Bogomolov) and
(Andrei Teleman, 2005).
Other works in arithmetic geometry
Bogomolov has contributed to several aspects of arithmetic geometry. He posed the
Bogomolov conjecture In mathematics, the Bogomolov conjecture is a conjecture, named after Fedor Bogomolov, in arithmetic geometry about algebraic curves that generalizes the Manin-Mumford conjecture in arithmetic geometry. The conjecture was proved by Emmanuel Ull ...
about small points. Twenty years ago he contributed a proof (among many proofs) of the geometric
Szpiro's conjecture
In number theory, Szpiro's conjecture relates to the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known ''abc'' conjecture. It is named for Lucien Szpiro, who formulated it in ...
which appears to be the nearest to
Shinichi Mochizuki's claimed proof of the arithmetic Szpiro conjecture.
Later career
Bogomolov obtained his Habilitation (Russian ''"Dr. of Sciences"'') in 1983. In 1994, he emigrated to the United States and became a full professor at the Courant Institute. He is very active in algebraic geometry and number theory. From 2009 till March 2014 he served as the Editor-in-Chief of the
Central European Journal of Mathematics. Since 2014 he serves as the Editor-in-Chief of the European Journal of Mathematics. Since 2010 he is the academic supervisor of the
HSE Laboratory of algebraic geometry and its applications.
Bogomolov has extensively contributed to the revival of Russian mathematics. Three major international conferences commemorating his 70th birthday were held in 2016: at the
Courant Institute, the
University of Nottingham, and the
Higher School of Economics in Moscow.
References
External links
Official NYU home page*
{{DEFAULTSORT:Bogomolov, Fedor
1946 births
20th-century Russian mathematicians
21st-century Russian mathematicians
Living people
Soviet mathematicians
Academic staff of the Higher School of Economics
Courant Institute of Mathematical Sciences faculty
Algebraic geometers
Moscow State University alumni