Goal of research
The main goal of scientific research of the laboratory is to obtain and publish new results in the areas of algebraic and differential geometry.
A significant part of the methods used were previously proposed by the members of the research team in their scientific work.
Among these are:
active use of derived categories in algebro-geometric problems;
use of exceptional collections, semiorthogonal decompositions and homological projective duality;
use of methods of noncommutative geometry;
use of birational methods and the minimal model program;
use of geometric methods in representation theory;
use of homological methods.
Empirical base of research
Such concepts are not applicable for mathematics.
Results of research
Important results were obtained on the main topics announced for the year 2019:
Homological and motivic methods in noncommutative geometry
Geometric representation theory.
Research Fellow A. Kuznetsov (jointly with A. Perry) completely settled the question of homological projective duality for quadrics. This question is related to the classical projective duality that to any non-degenerate quadric in a projective space associates another non-degenerate quadric in the dual space (in mechanics, this is known as Legendre transform). Homological projective duality is a wide-ranging generalization of the classical duality that was suggested by A. Kuznetsov several years ago and allows one to give regular and uniform proofs of a lot of deep theorems about the structure of derived categories of coherent sheaves on algebraic varieties and semiorthogonal decompositions they carry. The pioneering work of A. Kuznetsov has gained wide acceptance worldwide, with dozens of researchers around the globe using homological projective duality in their work. However, even in the classical case of quadrics, homological projective duality is much more complicated, and up to now, there was no description of the homologically dual varieties.
Such a description has been obtained by A. Kuznetsov and A. Perry. It turned out that one has to enlarge the class of quadrics: one needs to consider not only hypersurfaces of degree two, but also two-sheeted covers of projective spaces ramified in such hypersurfaces. Moreover, one needs to distinguish the odd-dimensional and the even-dimensional case, and one also needs to leave the commutative world by considering sheaves of modules over certain Azumaya algebras of rank 4. Once all these generalizations have been done in the correct way, homological projective duality acquires its final form, and allows for a complete and explicit description. The work relies in an essential way on the technique of categorical joining introduced by A. Kuznesov earlier.
Laboratory Head D. Kaledin (jointly with Research Intern A. Konovalov and HSE graduate student K. Magidson) studied the relationship between non-commutative version of the Hodge-to-de Rham spectral sequence degeneration for smooth and proper DG-algebras over a field of characteristic 0, and topological notions such as those of an algebra over the sphere spectrum. The degeneration itself has been proved by D. Kaledin some time ago, via the method of Deligne and Illusie that relies on reduction to positive characteristic. However, the corresponding degeneration theorem in positive characteristic is false in full generality ‒ one has to impose conditions on the algebra (that then have to be checked separately). One of the conditions is numerical, and its meaning up to now was quite unclear. In the new paper, it has been completely clarified. It turned out that the numerical condition can be replaced by a meaningful one: one needs to require that the algebra lifts to an algebra over the sphere spectrum. This allowed to considerably simplify and streamline D. Kaledin's original proof: the reduction procedure is done over the sphere spectrum right away, and all the additional conditions are satisfied automatically. This leads to possible generalizations, for example, to Z/2Z-graded DG-algebras. Moreover, it has made evident that the degeneration theorem, while purely algebraic, really needs in its proof notions from algebraic topology and stable homotopy theory.
Research Fellow M. Finkelberg (jointly with A. Braverman and H. Nakajima) accomplished the major step towards a mathematically rigourous definition of the so-called "Coulomb branches" or supersymmetric quantum field theories in dimension 1. Coulomb (and Higgs) branches are certain hyperkaehler varieties whose existence has been predicted by physicists on String Theory grounds, and that have a multitude of applications both in physics and in mathematics, especially in the Geometric Representation Theory. However, unlike the Higgs branch that has been well-understood for a long time, the Coulomb branch, until very recently, did not even have a rigourous mathematical definition. The first step towards such a definition has been done by H. Nakajima who predicted that holomorphic functions on the Coulomb branch correspond to certain critical cohomology classes of the infinite Grassmanian. This year, M. Finkelberg, A. Braverman and H. Nakajima constructed the crucial part of this correspondence, namely, the commutative associative product on Nakajima's cohomology space. To do this, they had in particular to reinterpret this space and identify critical cohomology with the usual cohomology of another and smaller variety. The product on this space is then introduced by a convolution construction. After this work, the only thing that remains to be done to finish Nakajima program is a construction of the Poisson bracket, and this is expected to be accomplished in near future.
In a series of papers, Research Intern K. Loginov has studied semistable one-parameter degenerations of smooth Fano varieties. A good understanding of one-parameter degenerations and their stability properties is crucial both for the moduli space construction, and for a wealth of purely geometric results on the structure of varieties and their families. A full classification of semistable degeneration of K3 surfaces done some time ago by V. Kulikov was a real breakthrough in K3 surface geometry. Kulikov classification is given in terms of an invariant that at a first glance looks quite crude ‒ the so-called dual complex of the special fiber. In the most interesting case of a maximal degeneration this complex is a triangulation of the two-sphere. From the modern point of view, the dual complex is understood to be the "essential skeleton" of the Berkovich space, a seemingly much finer invariant, and it is expected that for the maximal degenerations of Calabi-Yau varieties, the dual complexes are again triangulations of spheres. For varieties of other types there are no precise predictions but there is a general theorem: Kollar with coauthors has proved that the dual complex of a semistable degeneration of a rationally connected variety is contractible.
K. Loginov studied degenerations of the most classical among the rationally connected varieties, namely, of Fano varieties, and he proved an amazing theorem that is much stronger and much more precise than Kollar's result. It turns out that in any dimension, the dual complex of a semistable degeneration of a smooth Fano variety is homeomorphic to a simplex of dimension at most the dimension of the variety (with maximal dimension corresponding to the maximal degeneration). Loginov also studied the monodromy of maximal semistable degenerations, and obtained a wealth of concrete geometric results in dimensions two and three.
Deputy Laboratory Head M. Verbitsky (jointly with J. Solomon) has studied the Fukaya category of sympletic manifold that also carries a hyperkaehler structure. They used the hyperkaehler rotation method ‒ that is, passing to a sufficiently general complex structure among those induced by the quaternionic action ‒ to show that the part of Fukaya category that corresponds to holomorphic Lagrangian subvarieties is formal. In other words, all the quantum corrections to compositions of morphisms vanish. These results explain, in particular, the well-known fact that the quantum product in the cohomology of a hyperkaehler manifold carries no instanton corrections, and actually gives a wide-ranging generalization of this fact (where one considers the whole Fukaya category rather than only its center).