Vyacheslav Shokurov
Encyclopedia
Vyacheslav Vladimirovich Shokurov, born on 18 May 1950, is a
Russia
n mathematician
best known for his research in
algebraic geometry
. The proof of the Noether–Enriques–Petri theorem,
the Cone theorem, the existence of a line on smooth Fano varieties
and, finally, the existence of log flips – these are several
of Shokurov's major contributions to the subject.
and Mathematics of Moscow State University
. Already as an undergraduate,
Shokurov showed himself to be a mathematician of outstanding talent.
In 1970, he proved the scheme analog of the Noether–Enriques–Petri
theorem, which later allowed him to solve a Schottky-type problem
for the polarized Prym varieties
, and to prove the existence
of a line on smooth Fano varieties.
Upon his graduation Shokurov entered the Ph.D. program in
Moscow State University under the supervision of Yuri Manin.
At this time Shokurov studied the geometry of Kuga varieties.
The results obtained in this area became the body of his thesis and
he was awarded his Ph.D. ("candidate degree") in 1976.
of algebraic varieties. After obtaining Ph.D. he worked in
Yaroslavl State Pedagogical University together with Zalman Skopec. It was
Skopec and another colleague, Vasily Iskovskikh,
who influenced considerably the development of mathematical
interests of Shokurov at that time. Iskovskikh, who was working
on the classification of three-dimensional smooth Fano varieties of
principal series, posed two classical problems to Shokurov: the
existence of a line on smooth Fano varieties and the smoothness of a
general element in the anticanonical linear system of any such
variety. Shokurov solved both of these problems for three-dimensional
Fano varieties and the methods which he introduced for this purpose
were later developed in the works of other mathematicians, who
generalized Shokurov's ideas to the case of higher-dimensional Fano
varieties, and even to the Fano varieties with (admissible)
singularities.
In 1983, Shokurov's paper
Prym varieties: theory and applications
was published. In it Shokurov brought to a completion
the work on solving the Schottki-type problem for Prym varieties
which originated in papers of Arnaud Beauville
and Andrey Tyurin.
Shokurov proved a criterion which allows to decide whether the
principally polarized Prym variety of a Beauville's pair, subject
to some stability conditions, is the Jacobian of some smooth curve.
As the main application this criterion provided the famous
Iskovskikh's criterion for rationality of a standard conic bundle
whose base is a smooth minimal rational surface.
development of the Minimal model program (MMP). In 1984 he
published a paper titled
On the closed cone of curves of algebraic 3-folds
where he proved that the negative part of the closed
cone of effective curves
on an algebraic 3-fold (with admissible
singularities) is locally polyhedral. A bit later, in 1985,
Shokurov published a paper titled
The nonvanishing theorem,
which became a cornerstone for the whole MMP as it was used in
the proofs of such fundamental theorems as the Cone theorem and the
Semi-ampleness theorem. Also in this paper, Shokurov proved
the termination of three-dimensional flips. And even though
he proved this only for three-dimensional varieties, most of
his techniques were later generalized by Yujiro Kawamata
to obtain similar results for varieties of any dimension.
The most innovative of Shokurov's ideas formed a basis for
a paper titled 3-fold log flips
where the existence of
three-dimensional flips (first proved by Mori
) was established
in a more general log setting. The inductive method and the singularity
theory of log pairs developed in the framework of that paper
allowed most of the paper's results to be later generalized
to arbitrary-dimensional varieties. Later on, in 2001,
Shokurov announced the proof of the existence of
4-dimensional log flips, whose complete version appeared in two books:
Flips for 3-folds and 4-folds
and Birational geometry: linear systems and finitely-generated algebras.
The most remarkable application of Shokurov's ideas concerning
the existence of log flips has led to the celebrated paper
Existence of minimal models for varieties of log general type
by Caucher Birkar, Paulo Cascini, Christopher Hacon
and James McKernan
.
Johns Hopkins University
in Baltimore
and a non-tenured faculty member
of the Steklov Mathematical Institute in Moscow
. He is actively
involved both in research and in teaching.
Russia
Russia
Russia or , officially known as both Russia and the Russian Federation , is a country in northern Eurasia. It is a federal semi-presidential republic, comprising 83 federal subjects...
n mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
best known for his research in
algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. The proof of the Noether–Enriques–Petri theorem,
the Cone theorem, the existence of a line on smooth Fano varieties
Fano variety
In algebraic geometry, a Fano variety, introduced by , is a non-singular complete variety whose anticanonical bundle is ample.Fano varieties are quite rare, compared to other families, like Calabi–Yau manifolds and general type surfaces....
and, finally, the existence of log flips – these are several
of Shokurov's major contributions to the subject.
Early years
In 1968 Shokurov became a student of the Faculty of Mechanicsand Mathematics of Moscow State University
Moscow State University
Lomonosov Moscow State University , previously known as Lomonosov University or MSU , is the largest university in Russia. Founded in 1755, it also claims to be one of the oldest university in Russia and to have the tallest educational building in the world. Its current rector is Viktor Sadovnichiy...
. Already as an undergraduate,
Shokurov showed himself to be a mathematician of outstanding talent.
In 1970, he proved the scheme analog of the Noether–Enriques–Petri
theorem, which later allowed him to solve a Schottky-type problem
Schottky problem
In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.-Geometric formulation:...
for the polarized Prym varieties
Prym variety
In mathematics, the Prym variety construction is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it was applied to an unramified double covering of a Riemann surface, and was used by W. Schottky and H. W. E...
, and to prove the existence
of a line on smooth Fano varieties.
Upon his graduation Shokurov entered the Ph.D. program in
Moscow State University under the supervision of Yuri Manin.
At this time Shokurov studied the geometry of Kuga varieties.
The results obtained in this area became the body of his thesis and
he was awarded his Ph.D. ("candidate degree") in 1976.
Work on birational geometry
V. V. Shokurov is most famous for his work on birational geometryof algebraic varieties. After obtaining Ph.D. he worked in
Yaroslavl State Pedagogical University together with Zalman Skopec. It was
Skopec and another colleague, Vasily Iskovskikh,
who influenced considerably the development of mathematical
interests of Shokurov at that time. Iskovskikh, who was working
on the classification of three-dimensional smooth Fano varieties of
principal series, posed two classical problems to Shokurov: the
existence of a line on smooth Fano varieties and the smoothness of a
general element in the anticanonical linear system of any such
variety. Shokurov solved both of these problems for three-dimensional
Fano varieties and the methods which he introduced for this purpose
were later developed in the works of other mathematicians, who
generalized Shokurov's ideas to the case of higher-dimensional Fano
varieties, and even to the Fano varieties with (admissible)
singularities.
In 1983, Shokurov's paper
Prym varieties: theory and applications
was published. In it Shokurov brought to a completion
the work on solving the Schottki-type problem for Prym varieties
which originated in papers of Arnaud Beauville
Arnaud Beauville
Arnaud Beauville is a French mathematician, whose research interest is algebraic geometry.A native of Bologne, Haute-Marne, Beauville earned his doctorate from Paris Diderot University in 1977, with a thesis regarding Prym varieties and the Schottky problem, under supervision of Jean-Louis...
and Andrey Tyurin.
Shokurov proved a criterion which allows to decide whether the
principally polarized Prym variety of a Beauville's pair, subject
to some stability conditions, is the Jacobian of some smooth curve.
As the main application this criterion provided the famous
Iskovskikh's criterion for rationality of a standard conic bundle
whose base is a smooth minimal rational surface.
Log Flips
Since the late 80's Shokurov began to contribute to thedevelopment of the Minimal model program (MMP). In 1984 he
published a paper titled
On the closed cone of curves of algebraic 3-folds
where he proved that the negative part of the closed
cone of effective curves
Cone of curves
In mathematics, the cone of curves of an algebraic variety X is a combinatorial invariant of much importance to the birational geometry of X.-Definition:Let X be a proper variety...
on an algebraic 3-fold (with admissible
singularities) is locally polyhedral. A bit later, in 1985,
Shokurov published a paper titled
The nonvanishing theorem,
which became a cornerstone for the whole MMP as it was used in
the proofs of such fundamental theorems as the Cone theorem and the
Semi-ampleness theorem. Also in this paper, Shokurov proved
the termination of three-dimensional flips. And even though
he proved this only for three-dimensional varieties, most of
his techniques were later generalized by Yujiro Kawamata
to obtain similar results for varieties of any dimension.
The most innovative of Shokurov's ideas formed a basis for
a paper titled 3-fold log flips
where the existence of
three-dimensional flips (first proved by Mori
Shigefumi Mori
-References:*Heisuke Hironaka, Fields Medallists Lectures, Michael F. Atiyah , Daniel Iagolnitzer ; World Scientific Publishing, 2007. ISBN 9810231172...
) was established
in a more general log setting. The inductive method and the singularity
theory of log pairs developed in the framework of that paper
allowed most of the paper's results to be later generalized
to arbitrary-dimensional varieties. Later on, in 2001,
Shokurov announced the proof of the existence of
4-dimensional log flips, whose complete version appeared in two books:
Flips for 3-folds and 4-folds
and Birational geometry: linear systems and finitely-generated algebras.
The most remarkable application of Shokurov's ideas concerning
the existence of log flips has led to the celebrated paper
Existence of minimal models for varieties of log general type
by Caucher Birkar, Paulo Cascini, Christopher Hacon
Christopher Hacon
Christopher Derek Hacon is a mathematician with British, Italian and US nationalities. In 2009 he was awarded the Cole Prize for outstanding contribution to algebra, along with James McKernan. He was also awarded the Clay Research Award...
and James McKernan
James McKernan
James McKernan is a British mathematician, and a professor of mathematics at MIT in the United States.McKernan was the joint winner of the Cole Prize in 2009, and joint recipient of the Clay Research Award in 2007. Both honors were received jointly with his colleague Christopher Hacon.-External...
.
Later career
Shokurov is presently a full professor atJohns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...
in Baltimore
Baltimore
Baltimore is the largest independent city in the United States and the largest city and cultural center of the US state of Maryland. The city is located in central Maryland along the tidal portion of the Patapsco River, an arm of the Chesapeake Bay. Baltimore is sometimes referred to as Baltimore...
and a non-tenured faculty member
of the Steklov Mathematical Institute in Moscow
Moscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
. He is actively
involved both in research and in teaching.