F. A. Bogomolov, S. O. Gorchinskiy, A. B. Zheglov, V. V. Nikulin, D. O. Orlov, D. V. Osipov, A. N. Parshin, V. L. Popov, V. V. Przyjalkowski, Yu. G. Prokhorov, M. Reid, A. G. Sergeev, D. V. Treschev, A. K. Tsikh, I. A. Cheltsov, E. M. Chirka, “Viktor Stepanovich Kulikov (on his 70th birthday)”, Russian Math. Surveys, 77:3 (2022), 555

Our colleague and friend, the prominent Russian mathematician Viktor Stepanovich Kulikov observed his 70th birthday on 13 April 2022.

He was born in Moscow; his father was a military officer, and his mother was a doctor. He graduated from the famous School no. 2, where such researchers as B. V. Shabat, O. V. Lokutsievskii, and E. M. Chirka were his first teachers of mathematics. In 1969 he enrolled in the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University, where he became a student of Igor’ Rostislavovich Shafarevich. Kulikov recalls favourably his being one of Shafarevich’s last students. After graduating from the Faculty of Mechanics and Mathematics at Moscow University, Kulikov started his postgraduate studies in the Steklov Mathematical Institute of Russian Academy of Sciences, and in 1977 he defended his Ph.D. thesis. Then he started his work in the Moscow Institute of Railway Engineering. He cherishes warm memories of that institute, where he taught for 23 years and worked his way up from a junior researcher to a professor and head of department. During that period of time he solved a number of important problem, wrote dozens of papers, and in 1992 defended his D.Sc. thesis.

In the 1990s, which were a hard time for the country and for mathematics in it, he stayed in Russia, and since 1997 he has been a researcher in the Steklov Institute, where he has continued his very intensive investigations and has established many remarkable results. He is the author of more than 85 research publications; the full list is available on his webpage¹ at the Math-Net.Ru web portal. This list includes a monograph, namely, a survey of Hodge theory written in conjunction with P. F. Kurchanov, which was the point of entrance to this important area of contemporary mathematics for many generations of researchers.

The star of Kulikov as a researcher rose extremely early and has been shining brightly ever since. In his Ph.D. thesis he proved that on K3 surfaces the period map is epimorphic. Not only the result itself, but also his approach to it were outstanding: the proof was based on techniques of modification of surface degenerations which he has invented and which have given a strong impetus to the further development of the theory of modifications in multidimensional birational geometry.

Since the early 1980s Kulikov investigated the fundamental groups of the complements to plane algebraic curves, general projections of algebraic surfaces, the properties of algebraic surfaces of general type, and other topics which have become classical by now. He obtained deep results in each of these directions. He investigated thoroughly the properties of the Alexander polynomials of plane algebraic curves and of the fundamental groups of the complements to hypersurfaces in affine spaces. He introduced the concept of a $C$-group, which is a group with a presentation which generalizes the Wirtinger presentation of knot groups. Kulikov proved that the fundamental groups of the complements to hypersurfaces in affine spaces are $C$-groups and any $C$-group can be realized as the fundamental group of the complement to a real submanifold of codimension $2$ in a sphere of dimension greater than $2$.

In 2000 Kulikov proved that the braid monodromy type of an embedded algebraic surface determines this surface up to a diffeomorphism. In 2004 he found a remarkable generalization of Burniat’s classical construction and thus gave new examples of surfaces of general type, which are now known as Kulikov surfaces. Subsequently, in 2018 these ideas led him to the discovery of remarkable examples of Godeaux surfaces that are the spectral surfaces of rings of commuting operators which admit no isospectral deformations.

In 2008 Kulikov proved Chisini’s conjecture stated in 1944 by O. Chisini, a famous Italian geometer. Kulikov showed that a smooth surface in a projective space is uniquely determined by the branch curve of its general linear projection onto the projective plane.

Together with V. M. Kharlamov, in 2002–2006 Kulikov constructed a series of examples of algebraic surfaces and higher-dimensional manifolds which answered a number of natural open questions in algebraic (as well as in symplectic) geometry. These included the first examples of algebraic surfaces and plane cuspidal curves that are not deformation equivalent to their complex conjugates (the surfaces in these examples are rigid with regard to deformations; for example, they include all fake projective planes) and also the first examples of algebraic surfaces defined over the field of real numbers such that the actions of complex conjugation on them are (orientation preserving) diffeomorphic, but they are not deformation equivalent as real surfaces. In addition, in 2008 Kulikov discovered an ingenious approach to the famous problem of non-rationality of a cubic fourfold, which was based on some conjectural properties of the Hodge structures of surfaces, and which received a strong resonance among the academic world.

In 2011–2017 Kulikov investigated the problem of finding the number of irreducible components of the Hurwitz space of covers of the projective line that have a fixed Galois group and a fixed monodromy type. He obtained a generalization of the classical Luroth–Clebsch–Hurwitz theorem stating that the Hurwitz spaces of general covers of fixed degree of the projective line with a fixed number of branch points are irreducible. In particular, he gave new strong criteria for the irreducibility of Hurwitz spaces of this type. To solve these questions Kulikov introduced entirely new techniques in group theory, namely, the concept of the factorization semigroup over a group; he investigated it thoroughly and then used it in a purely geometric context.

In 2012, jointly with F. A. Bogomolov, he investigated the diffeomorphism type of the complement to a line arrangement in the projective plane: when does the incidence matrices of an arrangement determine uniquely its diffeomorphism type? To answer this question they defined new non-trivial operations on the set of incidence matrices.

In 2018–2021 Kulikov found and investigated the connection between the set of rational Belyi pairs and the set of germs of finite morphisms of smooth algebraic surfaces defined over the field of complex numbers, and proved an analogue of Chisini’s conjecture for almost general covers of the projective plane.

There are several mathematical objects named after Viktor Kulikov, which are intensively investigated by mathematicians all over the world, for example, Kulikov degenerations of K3 surfaces and Kulikov surfaces.

Kulikov has always been very active in teaching: in the Moscow Institute of Railway Engineering he gave lectures and lessons which covered all branches of higher mathematics. In the Steklov Mathematical Institute he read a number of special courses and supervised research seminars at the Scientific and Educational Center of the institute. In 2009–2017 he organized seven all-Russian workshop conferences on algebraic geometry and complex analysis for young scientists (in the village of Lyutovo, Yaroslavl’ Oblast, and in Koryazhma, Arkhangel’sk Oblast). In 2001–2009 he was a member of the Counsel of Experts on Mathematics and Mechanics of the Higher Certification Commission. He is a member of the Dissertation Council D002.022.03 at the Steklov Mathematical Institute.

Maintaining communication with Viktor Kulikov does not just bear fruit from the standpoint of mathematics, but it is also a source of permanent pleasure. On his 70th birthday, all of us, his friends and colleagues, wish Viktor Stepanovich Kulikov to keep his ardour in life and research, prove many brilliant results and make new interesting discoveries.

Citation: F. A. Bogomolov, S. O. Gorchinskiy, A. B. Zheglov, V. V. Nikulin, D. O. Orlov, D. V. Osipov, A. N. Parshin, V. L. Popov, V. V. Przyjalkowski, Yu. G. Prokhorov, M. Reid, A. G. Sergeev, D. V. Treschev, A. K. Tsikh, I. A. Cheltsov, E. M. Chirka, “Viktor Stepanovich Kulikov (on his 70th birthday)”, Russian Math. Surveys, 77:3 (2022), 555–557

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\paper Viktor Stepanovich Kulikov (on his 70th birthday)

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