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We study Hilbert schemes of quadrics of dimension $k \in \{0,1,2,3\}$ on smooth Gushel-Mukai varieties $X$ of dimension $n \in \{2,3,4,5,6\}$ by relating them to the relative Hilbert schemes of linear subspaces of dimension $k + 1$ of a certain family, naturally associated with $X$, of quadrics of dimension $n - 1$ over the blowup of $\mathbf{P}^5$ at a point., Comment: 65 pages

We give a proof of Mukai's Theorem on the existence of certain exceptional vector bundles on prime Fano threefolds. To our knowledge this is the first complete proof in the literature. The result is essential for Mukai's biregular classification of prime Fano threefolds, and for the existence of semiorthogonal decompositions in their derived categories. Our approach is based on Lazarsfeld's construction that produces vector bundles on a variety from globally generated line bundles on a divisor, on Mukai's theory of stable vector bundles on K3 surfaces, and on Brill--Noether properties of curves and (in the sense of Mukai) of K3 surfaces., Comment: 38 pages

We classify all 1-nodal degenerations of smooth Fano threefolds with Picard number 1 (both nonfactorial and factorial) and describe their geometry. In particular, we describe a relation between such degenerations and smooth Fano threefolds of higher Picard rank and with unprojections of complete intersection varieties., Comment: 79 pages; v2: 1-cuspidal factorial Fano threefolds included into the classification; 83 pages

Using the technique of categorical absorption of singularities we prove that the nontrivial components of the derived categories of del Pezzo threefolds of degree $d \in \{2,3,4,5\}$ and crepant categorical resolutions of the nontrivial components of the derived categories of nodal del Pezzo threefolds of degree $d = 1$ can be smoothly deformed to the nontrivial components of the derived categories of prime Fano threefolds of genus $g = 2d + 2 \in \{4,6,8,10,12\}$. This corrects and proves the Fano threefolds conjecture of the first author from [Kuz09], and opens a way to interesting geometric applications, including a relation between the intermediate Jacobians and Hilbert schemes of curves of the above threefolds. We also describe a compactification of the moduli stack of prime Fano threefolds endowed with an appropriate exceptional bundle and its boundary component that corresponds to degenerations associated with del Pezzo threefolds., Comment: 39 pages; v4: final version

We give two simple geometric constructions of a smooth family of projective varieties with central fiber isomorphic to the horospherical variety of type $\mathrm{G}_2$ and all other fibers isomorphic to the isotropic orthogonal Grassmannian $\mathrm{OGr}(2,7)$ and discuss briefly the derived category of this family., Comment: 7 pages

In this paper we investigate homologically finite-dimensional objects in the derived category of a given small dg-enhanced triangulated category. Using these we define reflexivity, hfd-closedness, and the Gorenstein property for triangulated categories, and discuss crepant categorical contractions. We illustrate the introduced notions on examples of categories of geometric and algebraic origin and provide geometric applications. In particular, we apply our results to prove a bijection between semiorthogonal decompositions of the derived category of a singular variety and the derived category of its smoothing with support on the central fiber., Comment: Final version; accepted to Selecta

We study the properties of a system biorthogonal to a complete and minimal system of exponentials in $L^2(E)$, where $E$ is a finite union of intervals, and show that in the case when $E$ is a union of two or three intervals the biorthogonal system is also complete., Comment: 15 pages

We introduce the notion of categorical absorption of singularities: an operation that removes from the derived category of a singular variety a small admissible subcategory responsible for singularity and leaves a smooth and proper category. We construct (under appropriate assumptions) a categorical absorption for a projective variety $X$ with isolated ordinary double points. We further show that for any smoothing $\mathcal{X}/B$ of $X$ over a smooth curve $B$, the smooth part of the derived category of $X$ extends to a smooth and proper over $B$ family of triangulated subcategories in the fibers of $\mathcal{X}$., Comment: 41 pages; v2, v3: minor improvements, v4: revised according to the referee's report, to appear in EPIGA, v5: published version

We study del Pezzo varieties, higher-dimensional analogues of del Pezzo surfaces. In particular, we introduce ADE classification of del Pezzo varieties, show that in type A the dimension of non-conical del Pezzo varieties is bounded by $12 - d - r$, where $d$ is the degree and $r$ is the rank of the class group, and classify maximal del Pezzo varieties., Comment: 62 pages, LaTeX, final version, to appear in Izvestiya

We construct $S$-linear semiorthogonal decompositions of derived categories of smooth Fano threefold fibrations $X/S$ with relative Picard rank $1$ and rational geometric fibers and discuss how the structure of components of these decompositions is related to rationality properties of $X/S$., Comment: 55 pages; v2: minor modifications

Hybrid opto, electro, and mechanical systems operating at several GHz offer extraordinary opportunities for the coherent control of opto-electronic excitations down to the quantum limit. We introduce here a monolithic platform for GHz semiconductor optomechanics based on electrically excited phonons guided along the spacer of a planar microcavity (MC) embedding quantum well (QW) emitters. The MC spacer bound by cleaved lateral facets acts as an embedded acoustic waveguide (WG) cavity with a high quality factor ($Q\sim10^5$) at frequencies well beyond 6~GHz, along which the acoustic modes live over tens of $\mu$s. The strong acoustic fields and the enhanced optomechanical coupling mediated by electronic resonances induce a huge modulation of the energy (in the meV range) and strength (over 80\%) of the QW photoluminescence (PL), which, in turn, becomes a sensitive local phonon probe. Furthermore, we show the coherent coupling of acoustic modes at different sample depths, thus opening the way for phonon-mediated coherent control and interconnection of three-dimensional epitaxial nanostructures., Comment: 5 Figures

A system of three non-identical Josephson junctions connected via an RLC circuit is considered. The method of Lyapunov exponents charts is used, which makes it possible to identify the main types of dynamics of the system and to analyze the dependence of its properties on parameters. The possibility of both two and three-frequency invariant tori is demonstrated. Saddle-node bifurcations of resonant tori are studied with the use of instantaneous Lyapunov exponents. The dependence of the charts on the type of coupling in the system is discussed., Comment: 17 pages, 9 figures, 1 table

We discuss recent developments in the study of semiorthogonal decompositions of algebraic varieties with an emphasis on their behaviour in families. First, we overview new results concerning homological projective duality. Then we introduce residual categories, discuss their relation to small quantum cohomology, and compute Serre dimensions of residual categories of complete intersections. After that we define simultaneous resolutions of singularities and describe a construction that works in particular for nodal degenerations of even-dimensional varieties. Finally, we introduce the concept of absorption of singularities which works under appropriate assumptions for nodal degenerations of odd-dimensional varieties., Comment: To appear in the Proceedings of the ICM 2022; 41 pages; comments are welcome

We describe in terms of spherical twists the Serre functors of many interesting semiorthogonal components, called residual categories, of the derived categories of projective varieties. In particular, we show the residual categories of Fano complete intersections are fractional Calabi--Yau up to a power of an explicit spherical twist. As applications, we compute the Serre dimensions of residual categories of Fano complete intersections, thereby proving a corrected version of a conjecture of Katzarkov and Kontsevich, and deduce the nonexistence of Serre invariant stability conditions when the degrees of the complete intersection do not all coincide., Comment: 53 pages, minor updates

We introduce the notion of hyperbolic equivalence for quadric bundles and quadratic forms on vector bundles and show that hyperbolic equivalent quadric bundles share many important properties: they have the same Brauer data; moreover, if they have the same dimension over the base, they are birational over the base and have equal classes in the Grothendieck ring of varieties. Furthermore, when the base is a projective space we show that two quadratic forms are hyperbolic equivalent if and only if their cokernel sheaves are isomorphic up to twist, their fibers over a fixed point of the base are Witt equivalent, and, in some cases, certain quadratic forms on intermediate cohomology groups of the underlying vector bundles are Witt equivalent. For this we show that any quadratic form over $\mathbb{P}^n$ is hyperbolic equivalent to a quadratic form whose underlying vector bundle has many cohomology vanishings; this class of bundles, called VLC bundles in the paper, is interesting by itself., Comment: 42 pages; v2: minor improvements

We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type of such threefolds we provide a unirationality criterion and prove stable non-rationality under additional assumptions., Comment: 36 pages, latex, final version, to appear in Journal of the Institute of Mathematics of Jussieu, a few typos are corrected

We introduce the notion of a simultaneous categorical resolution of singularities, a categorical version of simultaneous resolutions of rational double points of surface degenerations. Furthermore, we suggest a construction of simultaneous categorical resolutions which, in particular, applies to the case of a flat projective 1-dimensional family of varieties of arbitrarily high even dimension with ordinary double points in the total space and central fiber. As an ingredient of independent interest, we check that the property of a geometric triangulated category linear over a base to be relatively smooth and proper can be verified fiberwise. As an application we construct a smooth and proper family of K3 categories with general fiber the K3 category of a smooth cubic fourfold and special fiber the derived category of the K3 surface of degree 6 associated with a singular cubic fourfold., Comment: 23 pages; v4: a few typos corrected; to appear in Math. Z

For the derived category of the Cayley plane, which is the cominuscule Grassmannian of Dynkin type $\mathrm{E}_6$, a full Lefschetz exceptional collection was constructed by Faenzi and Manivel. A general hyperplane section of the Cayley plane is the coadjoint Grassmannian of Dynkin type $\mathrm{F}_4$. We show that the restriction of the Faenzi-Manivel collection to such a hyperplane section gives a full Lefschetz exceptional collection, providing the first example of a full exceptional collection on a homogeneous variety of Dynkin type $\mathrm{F}$. We also describe the residual categories of these Lefschetz collections, confirming conjectures of the second and third named author for the Cayley plane and its hyperplane section. The latter description is based on a general result of independent interest, relating residual categories of a variety and its hyperplane section., Comment: Final version. To appear in Transformation Groups

We discuss birational properties of Mukai varieties, i.e., of higher-dimensional analogues of prime Fano threefolds of genus $g \in \{7,8,9,10\}$ over an arbitrary field $\mathsf{k}$ of zero characteristic. In the case of dimension $n \ge 4$ we prove that these varieties are $\mathsf{k}$-rational if and only if they have a $\mathsf{k}$-point except for the case of genus $9$, where we assume $n \ge 5$. Furthermore, we prove that Mukai varieties of genus $g \in \{7,8,9,10\}$ and dimension $n \ge 5$ contain cylinders if they have a $\mathsf{k}$-point. Finally, we prove that the embedding $X \hookrightarrow \mathrm{Gr}(3,7)$ for prime Fano threefolds of genus $12$ is defined canonically over any field and use this to give a new proof of the criterion of rationality., Comment: 34 pages, latex

We study the upper and the lower densities of complete and minimal Gabor Gaussians systems. In contrast to the classical lattice case when they are both equal to $1$, we prove that the lower density may reach $0$ while the upper density may vary at least from $\frac{1}{\pi}$ to $e$. In the case when the upper density exceeds $1$, we establish a sharp inequality relating the upper and the lower densities., Comment: 16 pages, 1 figure

Optical parametric oscillations (OPOs) - a non-linear process involving the coherent coupling of an optically excited two particle pump state to a signal and an idler states with different energies - is a relevant mechanism for optical amplification as well as for the generation of correlated photons. OPOs require states with well-defined symmetries and energies: the fine-tuning of material properties and structural dimensions to create these states remains a challenge for the realization of scalable OPO-based functionalities in semiconductor nanostructures. Here, we demonstrate a pathway towards this goal based on the control of confined microcavity exciton-polaritons modulated by the spatially and time varying dynamical potentials produced by a surface acoustic waves (SAW). The exciton-polariton are confined in um-sized intra-cavity traps fabricated by structuring a planar semiconductor microcavity during the epitaxial growth process. OPOs in these structures benefit from the enhanced non-linearities of confined systems. We show that SAW fields induce state-dependent and time-varying energy shifts, which enable the energy alignment of the confined levels with the appropriate symmetry for OPO triggering. Furthermore, the dynamic acoustic tuning, which is fully described by a theoretical model for the modulation of the confined polaritons by the acoustic field, compensates for fluctuations in symmetry and dimensions of the confinement potential thus enabling a variety of dynamic OPO regimes. The robustness of the acoustic tuning is demonstrated by the synchronous excitation of an array of confined OPOs using a single acoustic beam, thus opening the way for the realization of scalable non-linear on-chip systems., Comment: Main text: 10 pages, 7 figures Supplementary: 6 pages, 3 figures

Polaritons enable the resonant coupling of excitons and photons to vibrations in the application-relevant super high frequency (SHF, 3-30 GHz) domain. We introduce a novel platform for coherent optomechanics based on the coupling of exciton-polaritons and electrically driven SHF longitudinal acoustic phonons confined within the spacer region of a planar Bragg microcavity. An intrinsic property of the microcavity platform is the back-feeding of phonons via reflections at the sample boundaries, which enables frequency x quality factors products exceeding 10^14 Hz as well as huge modulation amplitudes of the optical transition energies (up to 8 meV). We show that the modulation is dominated by the phonon-induced energy shifts of the excitonic polariton component, thus leading to an oscillatory transition between the regimes of weak and strong light-matter coupling. These results open the way for polariton-based coherent optomechanics in the non-adiabatic, side-band-resolved regime of coherent control.

We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with $A$. As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds., Comment: 48 pages. Latest addition to our series of articles on the geometry of Gushel-Mukai varieties; v2: minor stylistic improvements, results unchanged; v3: minor improvements; v4: final version, published in EPIGA

In our previous paper we suggested a conjecture relating the structure of the small quantum cohomology ring of a smooth Fano variety to the structure of its derived category of coherent sheaves. Here we generalize this conjecture, make it more precise, and support by the examples of (co)adjoint homogeneous varieties of simple algebraic groups of Dynkin types $A_n$ and $D_n$, i.e., flag varieties $Fl(1,n;n+1)$ and isotropic orthogonal Grassmannians $OG(2,2n)$; in particular we construct on each of those an exceptional collection invariant with respect to the entire automorphism group. For $OG(2,2n)$ this is the first exceptional collection proved to be full., Comment: 31 pages; v2: introduction clarified; v3: final version

We give necessary and sufficient conditions for unirationality and rationality of Fano threefolds of geometric Picard rank-1 over an arbitrary field of zero characteristic., Comment: 62 pages, latex, final version, to appear in AJM

Coherent super-high-frequency (SHF) vibrations provide an excellent tool for the modulation and control of excitations in semiconductors. Here, we investigate the piezoelectric generation and propagation of longitudinal bulk acoustic waves (LBAWs) with frequencies up to 20 GHz in GaAs crystals using bulk acoustic wave resonators (BAWRs) based on piezoelectric thin ZnO films. We show that the electro-acoustic conversion efficiency of the BAWRs depends sensitively on the sputtering conditions of the ZnO films. The BAWRs were then used for the study of the propagation properties of the LBAWs in GaAs in the frequency and temperature ranges from 1 to 20 GHz and 10 and 300 K, respectively, which have so far not been experimentally accessed. We found that the intrinsic acoustic absorption of GaAs in the temperature range from 80 K to 300 K is dominated by scattering with thermal phonons. At lower temperatures, in contrast, the intrinsic acoustic absorption saturates at a frequency-dependent value. Experiments carried out with different propagation lengths indicate that the saturation is associated with losses during reflection at the sample boundaries. We also demonstrate devices with high quality factor fabricated on top of acoustic Bragg-reflectors. The results presented here prove the feasibility of high-quality acoustic resonators embedding GaAs-based nanostructures, thus opening the way for the modulation and control of their properties by electrically excited SHF LBAWs., Comment: Research article

We introduce the notion of a categorical cone, which provides a categorification of the classical cone over a projective variety, and use our work on categorical joins to describe its behavior under homological projective duality. In particular, our construction provides well-behaved categorical resolutions of singular quadrics, which we use to obtain an explicit quadratic version of the main theorem of homological projective duality. As applications, we prove the duality conjecture for Gushel-Mukai varieties, and produce interesting examples of conifold transitions between noncommutative and honest Calabi-Yau threefolds., Comment: 48 pages. References updated.

We show that over an algebraically closed field of characteristic not equal to 2, homological projective duality for smooth quadric hypersurfaces and for double covers of projective spaces branched over smooth quadric hypersurfaces is a combination of two operations: one interchanges a quadric hypersurface with its classical projective dual and the other interchanges a quadric hypersurface with the double cover branched along it., Comment: 15 pages, minor updates

We describe the moduli stack of Gushel-Mukai varieties as a global quotient stack and its coarse moduli space as the corresponding GIT quotient. The construction is based on a comprehensive study of the relation between this stack and the stack of Lagrangian data; roughly speaking, we show that the former is a generalized root stack of the latter. As an application, we define the period map for Gushel-Mukai varieties and construct some complete nonisotrivial families of smooth Gushel-Mukai varieties. In an appendix, we describe a generalization of the root stack construction used in our approach to the moduli space., Comment: 51 pages

The origins of scientometrics (research metrics) were analysed and the lack of attention to elaboration of its methodology was emphasized. The approaches to evaluation of scientific activity outcome were considered and the tendency of transition from formal quantitative indicators to receiving expert conclusion on the basis of bibliometric indicators was noted. The principles of the Leiden manifesto of scientometrics were set out, keeping to which provides clear monitoring and support of science development, and favours establishing of the constructive dialog between scientific environment and society as well. The conceptual statements and peculiarities of practical realization of the informative and analytical system "Bibliometryka Ukraynskoyi Nauky" ("Bibliometrics of the Ukrainian Science") elaborated by the Vernadsky National Library of Ukraine, are being represented. The proposals on the formation of advisory councils, which are aimed to adopt conclusions on the effectiveness of research activity of institutions, were considered. The feasibility of building a common platform for expert evaluation of scientific studies for countries of the Eastern Partnership by initiating similar bibliometric projects in these countries and their further convergence is proved., Comment: 13 pages

We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras. We first explain how to induce a semiorthogonal decomposition of a surface $X$ with rational singularities from a semiorthogonal decomposition of its resolution. In the case when $X$ has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows to identify the components of the induced decomposition with derived categories of local finite dimensional algebras. Further, we present an obstruction in the Brauer group of $X$ to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of $X$. We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1., Comment: Exposition improved; Lemma 2.2 fixed