arXiv subjects
Publications and source records attributed to Dmitri Orlov.
We give a construction of the previously announced family of three-dimensional noncommutative Fano minifolds of Euler degree $72$. We prove that their defining $\mathbb{Z}$-algebras are Koszul and Artin--Schelter regular, and compute the Hilbert series of their section algebras.
We introduce notions of non-commutative Fano varieties and non-commutative minifolds, and study the latter in dimension three. We prove that the Euler matrices of full exceptional collections generating geometric helices on these minifolds form five mutation classes, distinguished by their Euler degrees \(22,40,54,64,72.\) The first four classes are realized by (commutative) Fano threefolds \(X_{22},V_5,Q^3,\mathbb P^3,\) respectively. The fifth is realized by the non-commutative family \(M_{72}\) and cannot be realized by a full exceptional collection on a smooth projective threefold.
We construct explicit unirational del Pezzo surfaces of degree $1$ with arithmetic Picard rank one over $\mathbb{Q}$, $\mathbb{F}_5$, and $\mathbb{C}(t)$. Moreover, we prove that every smooth real geometrically rational surface is unirational over $\mathbb{R}$ if and only if it has a real point.
We study derived categories of coherent sheaves on abelian varieties. We give a criterion for the equivalence of the derived categories on two abelian varieties. We describe the autoequivalence group for the derived category of coherent sheaves of an abelian variety.
Gyrokinetic simulations are utilized to study effects of magnetic islands on the ion temperature gradient (ITG) turbulence in the KSTAR tokamak with resonant magnetic perturbations. Simulations show that the transport is controlled by the nonlinear interactions between the ITG turbulence and self-generated vortex flows and zonal flows, leading to an anisotropic structure of fluctuation and transport on the poloidal plane and in the toroidal direction. Magnetic islands greatly enhance turbulent transport of both particle and heat. The turbulent transport exhibits variations in the toroidal direction, with transport through the resonant layer near the island X-point being enhanced when the X-point is located at the outer mid-plane. A quantitative agreement is shown between simulations and KSTAR experiments in terms of time frequency and perpendicular wavevector spectrum.
New families of algebras and DG algebras with two simple modules are introduced and described. Using the twisted tensor product operation, we prove that such algebras have finite global dimension, and the resulting DG algebras are smooth. This description allows us to show that some of these DG algebras determine smooth proper noncommutative curves that provide smooth minimal noncommutative resolutions for singular rational curves.
In this paper, twisted tensor product of DG algebras is studied and sufficient conditions for smoothness of such a product are given. It is shown that in the case of finite-dimensional DG algebras, applying this operation offers great possibilities for constructing new examples of smooth DG algebras and algebras. In particular, examples are given of families of algebras of finite global dimension with two simple modules that have nontrivial moduli spaces.
This report is a summary of the mini-conference Workforce Development Through Research-Based, Plasma-Focused Science Education and Public Engagement held during the 2022 American Physical Society Division of Plasma Physics (APS DPP) annual meeting. The motivation for organizing this mini-conference originates from recent studies and community-based reports highlighting important issues with the current state of the plasma workforce. Here we summarize the main findings presented in the two speaker sessions of the mini-conference, the challenges and recommendations identified in the discussion sessions, and the results from a post-conference survey. We further provide information on initiatives and studies presented at the mini-conference, along with references to further resources.
We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with a full separable semi-exceptional collection. Moreover, we also show that it gives a characterization of such categories assuming that a subcategory is idempotent complete and has a classical generator.
We study the bounded derived categories of torus-equivariant coherent sheaves on smooth toric varieties and Deligne-Mumford stacks. We construct and describe full exceptional collections in these categories. We also observe that these categories depend only on the PL homeomorphism type of the corresponding simplicial complex.
The main purpose of this paper is to describe various phenomena and certain constructions arising in the process of studying derived noncommutative schemes. Derived noncommutative schemes are defined as differential graded categories of a special type. We review and discuss different properties of both noncommutative schemes and morphisms between them. In addition, the concept of geometric realization for derived noncommutative scheme is introduced and problems of existence and construction of such realizations are discussed. We also study the construction of gluing noncommutative schemes via morphisms and consider some new phenomena, such as phantoms, quasi-phantoms, and Krull-Schmidt partners, arising in the world of noncommutative schemes and allowing us to find new noncommutative schemes. In the last sections we consider noncommutative schemes that are related to basic finite dimensional algebras. It is proved that such noncommutative schemes have special geometric realizations under which the algebra goes to a vector bundle on a smooth projective scheme. Such realizations are constructed in two steps, one of which is the well-known construction of Auslander, while the second step is connected with a new concept of a well-formed quasi-hereditary algebra for which there are very particular geometric realizations sending standard modules to line bundles.
We discuss Krull--Schmidt partners for DG categories and produce bunches of them for smooth projective varieties.
In this paper we discuss different properties of noncommutative schemes over a field. We define a noncommutative scheme as a differential graded category of a special type. We study regularity, smoothness and properness for noncommutative schemes. Admissible subcategories of categories of perfect complexes on smooth projective schemes provide natural examples of smooth and proper noncommutative schemes that are called geometric noncommutative schemes. In this paper we show that the world of all geometric noncommutative schemes is closed under an operation of a gluing of differential graded categories via bimodules. As a consequence of the main theorem we obtain that for any finite dimensional algebra with separable semisimple part the category of perfect complexes over it is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme. Moreover, if the algebra has finite global dimension, then the full subcategory is admissible. We also provide a construction of a smooth projective scheme that admits a full exceptional collection and contains as a subcollection an exceptional collection given in advance. As another application of the main theorem we obtain, in characteristic 0, an existence of a full embedding for the category of perfect complexes on any proper scheme to the category of perfect complexes on a smooth projective scheme.
In this paper we construct strong exceptional collections of vector bundles on smooth projective varieties that have a prescribed endomorphism algebra. We prove the construction problem always have a solution. We consider some applications to noncommutative projective planes and to the quiver connected with the 3-point Ising function.
Explicit dyonic dilaton black holes of the four-dimensional Einstein-Maxwell-dilaton theory are known only for two particular values of the dilaton coupling constant $a =1,\sqrt{3}$, while for other $a$ numerical evidence was presented earlier about existence of extremal dyons in theories with the discrete sequence of dilaton couplings $a=\sqrt{n(n+1)/2}$ with integer $n$. Apart from the lower members $n=1,\,2$, this family of theories does not have motivation from supersymmetry or higher dimensions, and so far the above quantization rule has not been derived analytically. We fill this gap showing that this rule follows from analyticity of the dilaton at the $AdS_2\times S^2$ event horizon with $n$ being the leading dilaton power in the series expansion. We also present generalization for asymptotically anti-de Sitter dyonic black holes with spherical, plane and hyperbolic topology of the horizon.
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 conclude that cyclic covers on the symplectic side are mirror to orbifold quotients of the Landau-Ginzburg model.
In this paper we give a construction of phantom categories, i.e. admissible triangulated subcategories in bounded derived categories of coherent sheaves on smooth projective varieties that have trivial Hochschild homology and trivial Grothendieck group. We also prove that these phantom categories are phantoms in a stronger sense, namely, they have trivial K-motives and, hence, all their higher K-groups are trivial too.
We construct an exceptional collection $Υ$ of maximal possible length 6 on any of the Burniat surfaces with $K_X^2=6$, a 4-dimensional family of surfaces of general type with $p_g=q=0$. We also calculate the DG algebra of endomorphisms of this collection and show that the subcategory generated by this collection is the same for all Burniat surfaces. The semiorthogonal complement $\mathcal A$ of $Υ$ is an "almost phantom" category: it has trivial Hochschild homology, and $K_0(\mathcal A)=\bZ_2^6$.