Alexander I. Efimov

Homological mirror symmetry for punctured spheres

We prove that the wrapped Fukaya category of a punctured sphere (S with an arbitrary number of points removed) is equivalent to the triangulated category of singularities of a mirror Landau-Ginzburg model, proving one side of the homological mirror symmetry conjecture in this case. By investigating fractional gradings on these categories, we conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.

Maximal lengths of exceptional collections of line bundles

In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King’s conjecture for toric Fano varieties. More generally, we prove that for any constant c> 3 there exist infinitely many toric Fano varieties Y with Picard number three, such that the maximal length of exceptional collection of line bundles on Y is strictly less than c rkK0(Y ). To obtain varieties without full exceptional collections of line bundles, it suffices to put c =1 . On the other hand, we prove that for any toric nef-Fano DM stack Y with Picard number three, there exists a strong exceptional collection of line bundles on Y of length at least 3 rkK0(Y ). The constant 3 is thus maximal with this property.

Deformation theory of objects in homotopy and derived categories III: abelian categories

Abstract This is the third paper in a series. In Part I we developed a deformation theory of objects in homotopy and derived categories of DG categories. Here we show how this theory can be used to study deformations of objects in homotopy and derived categories of abelian categories. Then we consider examples from (noncommutative) algebraic geometry. In particular, we study noncommutative Grassmanians that are true noncommutative moduli spaces of structure sheaves of projective subspaces in projective spaces.

Deformation theory of objects in homotopy and derived categories II: Pro-representability of the deformation functor☆

Abstract This is the second paper in a series. In part I we developed deformation theory of objects in homotopy and derived categories of DG categories. Here we extend these (derived) deformation functors to an appropriate bicategory of artinian DG algebras and prove that these extended functors are pro-representable in a strong sense.

Some remarks on L-equivalence of algebraic varieties

In this short note we study the questions of (non-)L-equivalence of algebraic varieties, in particular, for abelian varieties and K3 surfaces. We disprove the original version of a conjecture of Huybrechts (Int J Math 16(1):13–36, 2005, Conjecture 0.3) stating that isogenous K3 surfaces are L-equivalent. Moreover, we give examples of derived equivalent twisted K3 surfaces, such that the underlying K3 surfaces are not L-equivalent. We also give examples showing that D-equivalent abelian varieties can be non-L-equivalent (the same examples were obtained independently in Ito et al. Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties. arXiv:1612.08497). This disproves the original version of a conjecture of Kuznetsov and Shinder (Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics. Sel Math New Ser. arXiv:1612.07193, Conjecture 1.6). We deduce the statements on (non-)L-equivalence from the very general results on the Grothendieck group of an additive category, whose morphisms are finitely generated abelian groups. In particular, we show that in such a category each stable isomorphism class of objects contains only finitely many isomorphism classes. We also show that a stable isomorphism between two objects X and Y with

Generalized non-commutative degeneration conjecture

{\text {End}}(X)=\mathbb {Z}

Influence of Nano-Scale Clusters on Gas Dynamics Parameters of Plasma Jet Created by Capillary Type Discharge

End(X)=Z implies that X and Y are isomorphic.

Coherent analogues of matrix factorizations and relative singularity categories

Two methods for dissociation of diatomic molecules based on nonperiodic excitation generated by feedback control mechanism are described and analyzed by computer simulation for classical and quantum-mechanical ensembles. The first method of control design uses nonlinear resonance curve of the system to fulfill the resonance conditions at any time of excitation. second method is based on the speed-gradient principle. Implementation of the proposed methods by pulse laser control is described. Efficiently of the proposed methods is demonstrated by the example of hydrogen fluoride (HF) molecule dissociation. Simulations confirmed that new methods are more efficient than the existing algorithms based on harmonic (monochromatic) and linear chirping excitation both for the model case of single molecule and for an ensembles of molecules. It is shown that, the dissociation rate for the quantum-mechanical ensemble is just a few percent slower that the one for the classical ensemble. It justifies using classical models for feedback control design in the dissociation problem.