# Alexander EfimovRussian Academy of Sciences | RAS · Algebraic Geometry Section

Alexander Efimov

## About

11

Publications

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298

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## Publications

Publications (11)

Mirror symmetry for higher genus curves is usually formulated and studied in terms of Landau-Ginzburg models; however the critical locus of the superpotential is arguably of greater intrinsic relevance to mirror symmetry than the whole Landau-Ginzburg model. Accordingly, we propose a new approach to the A-model of the mirror, viewed as a trivalent...

We prove that the wrapped Fukaya category of a punctured sphere ($S^2$ 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 co...

In this paper, we provide a Hodge-theoretic interpretation of Laurent
phenomenon for general skew-symmetric quantum cluster algebras, using
Donaldson-Thomas theory for a quiver with potential. It turns out that the
positivity conjecture reduces to the certain statement on purity of monodromic
mixed Hodge structures on the cohomology with the coeffi...

It is well known that the 'Fukaya category' is actually an -precategory in the sense of Kontsevich and Soǐbel'man. This is related to the fact that, generally speaking, the morphism spaces are defined only for transversal pairs of Lagrangian submanifolds, and higher multiplications are defined only for transversal sequences of Lagrangian submanifol...

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 pa...

In the paper \cite{KS}, Kontsevich and Soibelman in particular associate to
each finite quiver $Q$ with a set of vertices $I$ the so-called Cohomological
Hall algebra $\cH,$ which is $\Z_{\geq 0}^I$-graded. Its graded component
$\cH_{\gamma}$ is defined as cohomology of Artin moduli stack of
representations with dimension vector $\gamma.$ The produ...

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 $c>\frac 34$ there exist infinitely many toric Fano varietie...

Following an idea of Kontsevich, we introduce and study the notion of formal completion of a compactly generated (by a set of objects) enhanced triangulated category along a full thick essentially small triangulated subcategory. In particular, we prove (answering a question of Kontsevich) that using categorical formal completion, one can obtain ord...

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.

This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module E over a DG category we define four deformation functors Defh(E), coDefh(E), Def(E), coDef(E). The first two functors describe the deformations (and co-deformations) of E in the homotop...

Katzarkov has proposed a generalization of Kontsevich's mirror symmetry
conjecture, covering some varieties of general type. Seidel \cite{Se} has
proved a version of this conjecture in the simplest case of the genus two
curve. Basing on the paper of Seidel, we prove the conjecture (in the same
version) for curves of genus $g\geq 3,$ relating the Fu...