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In this paper, we prove a decomposition result for the Chow groups of projectivizations of coherent sheaves of homological dimension $\le 1$. In this process, we establish the decomposition of Chow groups for the cases of Cayley's trick and standard flips. Moreover, we apply these results to study the Chow groups of symmetric powers of curves, nested Hilbert schemes of surfaces, and the varieties resolving Voisin maps for cubic fourfolds., Comment: v2. Many grammatical problems fixed. References updated

We show a new neutral-fermionic presentation of Ikeda-Naruse's $K$-theoretic $Q$-functions, which represent a Schubert class in the $K$-theory of coherent sheaves on the Lagrangian Grassmannian. Our presentation provides a simple description and yields straightforward proof of two types of Pfaffian formulas for them. We present a dual space of $G\Gamma$, the vector space generated by all $K$-theoretic $Q$-functions, by constructing a non-degenerate bilinear form that is compatible with the neutral fermionic presentation. We give a new family of dual $K$-theoretic $Q$-functions, their neutral-fermionic presentations, and Pfaffian formulas., Comment: 27 pages, 1 figure v4: Major revision. The title has been changed. (The old title was "Pfaffian formulas in K-theory and the Boson-Fermion Correspondence")

We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of Bondal-Van den Bergh. This extends a recent result of Neeman and is new even in the affine case. The main ingredient includes Gabber's weak local uniformization theorem and the notions of boundedness and descendability of a morphism of schemes., 5 pages

Let Λ be a basic finite dimensional algebra over an algebraically closed field $\Bbbk $ , and let $\widehat {\Lambda }$ be the repetitive algebra of Λ. In this article, we prove that if $\widehat {V}$ is a left $\widehat {\Lambda }$ -module with finite dimension over $\Bbbk $ , then $\widehat {V}$ has a well-defined versal deformation ring $R(\widehat {\Lambda },\widehat {V})$ , which is a local complete Noetherian commutative $\Bbbk $ -algebra whose residue field is also isomorphic to $\Bbbk $ . We also prove that $R(\widehat {\Lambda }, \widehat {V})$ is universal provided that $\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk $ and that in this situation, $R(\widehat {\Lambda }, \widehat {V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $\mathbb {P}^{1}_{\Bbbk }$ .

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of Toen and Vaquie. In this paper we show that when $X$ is quasi-compact and semi-separated, any quasi-coherent sheaf on $X$ may be expressed as a directed colimit of its finitely generated quasi-coherent submodules. Thereafter, we introduce a notion of "field objects" in $(\mathcal C,\otimes,1)$ that satisfy several properties similar to those of fields in usual commutative algebra. Finally we show that the points of a Noetherian, quasi-compact and semi-separated scheme $X$ over such a field object $K$ in $(\mathcal C,\otimes,1)$ can be recovered from certain kinds of functors between categories of quasi-coherent sheaves. The latter is a partial generalization of some recent results of Brandenburg and Chirvasitu.

We develop a version of Hodge theory for a large class of smooth formally proper quotient stacks $X/G$ analogous to Hodge theory for smooth projective schemes. We show that the noncommutative Hodge-de Rham sequence for the category of equivariant coherent sheaves degenerates. This spectral sequence converges to the periodic cyclic homology, which we canonically identify with the topological equivariant K-theory of $X$ with respect to a maximal compact subgroup of $G$, equipping the latter with a canonical pure Hodge structure. We also establish Hodge-de Rham degeneration for categories of matrix factorizations for a large class of equivariant Landau-Ginzburg models., Comment: 47 pages, updated to match the published version, to avoid confusion. Following referee's suggestion, we reorganized the paper so that all matrix factorization material appears in its own separate section

Let $f$ and $g$ be two cuspidal modular forms and let $\mathcal{F}$ be a Coleman family passing through $f$, defined over an open affinoid subdomain $V$ of weight space $\mathcal{W}$. Using ideas of Pottharst, under certain hypotheses on $f$ and $g$ we construct a coherent sheaf over $V \times \mathcal{W}$ which interpolates the Bloch-Kato Selmer group of the Rankin-Selberg convolution of two modular forms in the critical range (i.e. the range where the $p$-adic $L$-function $L_p$ interpolates critical values of the global $L$-function). We show that the support of this sheaf is contained in the vanishing locus of $L_p$., Comment: Final version. To appear in Canadian Jour. Math

In this paper, we define the $K$-theoretic Hall algebra for dimension $0$ coherent sheaves on a smooth projective surface, prove that the algebra is associative, and construct a homomorphism to a shuffle algebra introduced by Negut [ 10].

Suppose $$F:{\mathcal {D}}(X)\rightarrow {\mathcal {T}}$$ is an exact functor from the bounded derived category of coherent sheaves on a smooth projective variety X to a triangulated category $${\mathcal {T}}$$ . If F possesses left and right adjoints, then the Bondal–Orlov criterion gives a simple way of determining if F is fully faithful. We prove a natural extension of this theorem to the case when X is a smooth and proper DM stack with projective coarse moduli space.

We show fullness of the exceptional collections of maximal length constructed by A. Kuznetsov and A. Polishchuk in the bounded derived categories of coherent sheaves on Lagrangian Grassmannians., 26 pages

Given a certain kind of linear representation of a reductive group, referred to as a quasi-symmetric representation in recent work of \v{S}penko and Van den Bergh, we construct equivalences between the derived categories of coherent sheaves of its various geometric invariant theory (GIT) quotients for suitably generic stability parameters. These variations of GIT quotient are examples of more complicated wall crossings than the balanced wall crossings studied in recent work on derived categories and variation of GIT quotients. Our construction is algorithmic and quite explicit, allowing us to: 1) describe a tilting vector bundle which generates the derived category of such a GIT quotient, 2) provide a combinatorial basis for the K-theory of the GIT quotient in terms of the representation theory of G, and 3) show that our derived equivalences satisfy certain relations, leading to a representation of the fundamental groupoid of a "K\"ahler moduli space" on the derived category of such a GIT quotient. Finally, we use graded categories of singularities to construct derived equivalences between all Deligne-Mumford hyperk\"ahler quotients of a symplectic linear representation of a reductive group (at the zero fiber of the algebraic moment map and subject to a certain genericity hypothesis on the representation), and we likewise construct actions of the fundamental groupoid of the corresponding K\"ahler moduli space., Comment: 37 pages, v2: added Sections 5.1 and 6; v3: added Remark 3.10, Remark 6.9, Example 6.10 elaborating on action of fundamental groupoid on K-theory

We remark that the combination of the works of Ben-Bassat-Brav-Bussi-Joyce and Alper-Hall-Rydh imply the conjectured local description of the moduli stacks of semi-Schur objects in the derived category of coherent sheaves on projective Calabi-Yau 3-folds. This result was assumed in the author's previous papers to apply wall-crossing formulas of DT type invariants in the derived category, e.g. DT/PT correspondence, rationality, etc. We also show that the above result is applied to prove the higher rank version of DT/PT correspondence and rationality.

This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilbn (S) and Hilbn (Sτ) are birational.

We introduce an invariant, associated to a coherent sheaf over a projective morphism of schemes, which controls when sheaf cohomology can be passed through the given morphism. We then use this invariant to estimate the stability indexes of the regularity and a*-invariant of powers of homogeneous ideals., Comment: 16 pages

We prove an Atiyah-Segal isomorphism for the higher $K$-theory of coherent sheaves on quotient Deligne-Mumford stacks over $\C$. As an application, we prove the Grothendieck-Riemann-Roch theorem for such stacks. This theorem establishes an isomorphism between the higher $K$-theory of coherent sheaves on a Deligne-Mumford stack and the higher Chow groups of its inertia stack. Furthermore, this isomorphism is covariant for proper maps between Deligne-Mumford stacks., Final version (50 pages), to appear in J. Algebraic Geom. (JAG)

Given a smooth variety $X$ with an action of a finite group $G$, and a semiorthogonal decomposition of the derived category, $\mathcal{D}([X/G])$, of $G$-equivariant coherent sheaves on $X$ into subcategories equivalent to derived categories of smooth varieties, we construct a similar semiorthogonal decomposition for a smooth $G$-invariant divisor in $X$ (under certain technical assumptions). Combining this procedure with the semiorthogonal decompositions constructed in [PV15], we construct semiorthogonal decompositions of some equivariant derived categories of smooth projective varieties., 23 pages. Minor errors and typos have been fixed. Some references have been added

We use recollement and HRS-tilt to describe bounded t-structures on the bounded derived category $\mathcal{D}^b(\mathbb{X})$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of virtual genus $\leq 1$. We will see from our description that the combinatorics in classification of bounded t-structures on $\mathcal{D}^b(\mathbb{X})$ can be reduced to that in classification of bounded t-structures on bounded derived categories of finite dimensional right modules over representation-finite finite dimensional hereditary algebras., Revised version

We show that local-global compatibility (at split primes) away from $p$ holds at all points of the $p$-adic eigenvariety of a definite $n$-variable unitary group. The novelty is we allow non-classical points, possibly non-\'{e}tale over weight space. More precisely we interpolate the local Langlands correspondence for GL(n) across the eigenvariety by considering the fibers of its defining coherent sheaf. We employ techniques of Scholze from his new approach to the local Langlands conjecture., Comment: 28 pages, comments are welcome!

The objective of this work is to present the reader with the study of some mathematical tools used in current problems of algebraic geometry, assuming only some knowledge in algebra and topology. We treat basic concepts and results in the theory of sheaves and schemes that we later use to understand the correspondence between local cohomology and sheaf cohomology of quasi-coherent sheaves over projective schemes. Then, with this background we are able to state some open problems that are related to the Hilbert polynomial and to the Castelonuovo-Mumford regularity of a coherent sheaf. O objetivo deste trabalho é apresentar ao leitor o estudo de algumas ferramentas matemáticas utilizadas nos problemas atuais da geometria algébrica, pressupondo apenas alguns conhecimentos em álgebra e topologia. Expõe conceitos e resultados básicos na teoria de feixes e esquemas, que logo são usados para entender a correspondência que existe entre a cohomologia local e a cohomologia de feixes, no caso de feixes quasi-coherentes sobre esquemas projetivos. Finalmente enunciamos alguns problemas em aberto relacionados com o polinômio de Hilbert e a regularidade de Castelonuovo-Mumford de um feixe coherente.

We review the recent proof of the N.Takahashi's conjecture on genus $0$ Gromov-Witten invariants of $(\mathbb{P}^2, E)$, where $E$ is a smooth cubic curve in the complex projective plane $\mathbb{P}^2$. The main idea is the use of the algebraic notion of scattering diagram as a bridge between the world of Gromov-Witten invariants of $(\mathbb{P}^2, E)$ and the world of moduli spaces of coherent sheaves on $\mathbb{P}^2$. Using this bridge, the N.Takahashi's conjecture can be translated into a manageable question about moduli spaces of coherent sheaves on $\mathbb{P}^2$. This survey is based on a three hours lecture series given as part of the Beijing-Zurich moduli workshop in Beijing, 9-12 September 2019., Comment: Expository paper. 18 pages, 1 figure

In this paper we study derived equivalences for Symplectic reflection algebras. We establish a version of the derived localization theorem between categories of modules over Symplectic reflection algebras and categories of coherent sheaves over quantizations of Q-factorial terminalizations of the symplectic quotient singularities. To do this we construct a Procesi sheaf on the terminalization and show that the quantizations of the terminalization are simple sheaves of algebras. We will also sketch some applications: to the generalized Bernstein inequality and to perversity of wall crossing functors., Comment: 17 pages, v2 acknowledgements added, v3 19 pages, the exposition is made more detailed; v4 accepted version, significant modifications

We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian $\text{G}(k,n)$ we conjecture that the residual category associated with Fonarev's Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for $k = p$, a prime number, modulo completeness of Fonarev's collection (and for $p = 3$ we check this completeness)., Final version. To appear in PLMS

We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the number of points at which the cyclic moduli space intersects a Hitchin fibre; and we describe explicitly certain $\mathbb{C}^\times$-flows into the nilpotent cone. We also extend this description to moduli of certain twisted cyclic quivers whose rank vector has components larger than 1. We show that, for certain choices of underlying bundle, such moduli spaces decompose as a product of cyclic quiver varieties in which each node is a line bundle., 18 pages, 1 figure

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., updated version, 34 pages

Let X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos of quasi-coherent sheaves over X. As it turns out, every topos point of is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if and are equivalent.The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ∞, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.

We describe a strategy for the construction of finitely generated $G$-equivariant $\mathbb{Z}$-graded modules $M$ over the exterior algebra for a finite group $G$. By an equivariant version of the BGG correspondence, $M$ defines an object $\mathcal{F}$ in the bounded derived category of $G$-equivariant coherent sheaves on projective space. We develop a necessary condition for $\mathcal{F}$ being isomorphic to a vector bundle that can be simply read off from the Hilbert series of $M$. Combining this necessary condition with the computation of finite excerpts of the cohomology table of $\mathcal{F}$ makes it possible to enlist a class of equivariant vector bundles on $\mathbb{P}^4$ that we call strongly determined in the case where $G$ is the alternating group on $5$ points.

We realize explicit symmetries of Bridgeland stability conditions on any abelian threefold given by Fourier-Mukai transforms. In particular, we extend the previous joint work with Maciocia to study the slope and tilt stabilities of sheaves and complexes under the Fourier-Mukai transforms, and then to show that certain Fourier-Mukai transforms give equivalences of the stability condition hearts of bounded t-structures which are double tilts of coherent sheaves. Consequently, we show that the conjectural construction proposed by Bayer, Macri and Toda gives rise to Bridgeland stability conditions on any abelian threefold by proving that tilt stable objects satisfy the Bogomolov-Gieseker type inequality. Our proof of the Bogomolov-Gieseker type inequality conjecture for any abelian threefold is a generalization of the previous joint work with Maciocia for a principally polarized abelian threefold with Picard rank one case, and also this gives an alternative proof of the same result in full generality due to Bayer, Macri and Stellari. Moreover, we realize the induced cohomological Fourier-Mukai transform explicitly in anti-diagonal form, and consequently, we describe a polarization on the derived equivalent abelian variety by using Fourier-Mukai theory.

Let X be a compact Calabi–Yau 3-fold, and write M , M ‾ for the moduli stacks of objects in coh ( X ) , D b coh ( X ) . There are natural line bundles K M → M , K M ‾ → M ‾ , analogues of canonical bundles. Orientation data on M , M ‾ is an isomorphism class of square root line bundles K M 1 / 2 , K M ‾ 1 / 2 , satisfying a compatibility condition on the stack of short exact sequences. It was introduced by Kontsevich and Soibelman [35, §5] in their theory of motivic Donaldson–Thomas invariants, and is also important in categorifying Donaldson–Thomas theory using perverse sheaves. We show that natural orientation data can be constructed for all compact Calabi–Yau 3-folds X, and also for compactly-supported coherent sheaves and perfect complexes on noncompact Calabi–Yau 3-folds X that admit a spin smooth projective compactification X ↪ Y . This proves a long-standing conjecture in Donaldson–Thomas theory. These are special cases of a more general result. Let X be a spin smooth projective 3-fold. Using the spin structure we construct line bundles K M → M , K M ‾ → M ‾ . We define spin structures on M , M ‾ to be isomorphism classes of square roots K M 1 / 2 , K M ‾ 1 / 2 . We prove that natural spin structures exist on M , M ‾ . They are equivalent to orientation data when X is a Calabi–Yau 3-fold with the trivial spin structure. We prove this using our previous paper [33] , which constructs ‘spin structures’ (square roots of a certain complex line bundle K P E • → B P ) on differential-geometric moduli stacks B P of connections on a principal U ( m ) -bundle P → X over a compact spin 6-manifold X.

In this paper, we construct recollements and ladders for exceptional curves by using reduction/insertion functors due to $p$-cycle construction. As applications to weighted projective lines, we classify recollements for the category of coherent sheaves over a weighted projective line, and give an explicit description of ladders in two different levels: the bounded derived category of coherent sheaves and the stable category of vector bundles., Comment: 21 pages

In this paper, we give complex geometric descriptions of the notions of algebraic geometric positivity of vector bundles and torsion-free coherent sheaves, such as nef, big, pseudo-effective and weakly positive, by using singular Hermitian metrics. As an applications, we obtain a generalization of Mori's result. We also give a characterization of the augmented base locus by using singular Hermitian metrics on vector bundles and the Lelong numbers., 18pages v4: completely revised version, to appear in Michigan Mathematical Journal

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

Extending work of Meinhardt and Partsch, we prove that two varieties are isomorphic away from a subset of a given dimension if and only if certain quotients of their categories of coherent sheaves are equivalent. This result interpolates between Gabriel’s reconstruction theorem and the fact that two varieties are birational if and only if they have the same function field.

We determine versal non-commutative deformations of some simple collections in the categories of perverse coherent sheaves arising from tilting generators for projective morphisms., 20 pages

We define an equivalence relation among coherent sheaves on a projective variety called biliaison. We prove the existence of sheaves that are minimal in a biliaison class in a suitable sense, and show that all sheaves in the same class can be obtained from a minimal one using certain deformations and other basic moves. Our results generalize the main theorems of liaison theory of subvarieties to sheaves, and provide a framework to study sheaves and subvarieties simultaneously., 21 pages, added a reference by Burragina that treats a special case of our theory

Suppose $(X,\Omega,g)$ is a compact Spin(7)-manifold, e.g. a Riemannian 8-manifold with holonomy Spin(7), or a Calabi-Yau 4-fold. Let $G$ be U$(m)$ or SU$(m)$, and $P\to X$ be a principal $G$-bundle. We show that the infinite-dimensional moduli space ${\mathcal B}_P$ of all connections on $P$ modulo gauge is orientable, in a certain sense. We deduce that the moduli space ${\mathcal M}_P^{Spin(7)}\subset{\mathcal B}_P$ of irreducible Spin(7)-instanton connections on $P$ modulo gauge, as a manifold or derived manifold, is orientable. This improves theorems of Cao and Leung arXiv:1502.01141 and Mu\~noz and Shahbazi arXiv:1707.02998. If $X$ is a Calabi-Yau 4-fold, the derived moduli stack $\boldsymbol{\mathscr M}$ of (complexes of) coherent sheaves on $X$ is a $-2$-shifted symplectic derived stack $(\boldsymbol{\mathcal M},\omega)$ by Pantev-To\"en-Vaqui\'e-Vezzosi arXiv:1111.3209, and so has a notion of orientation by Borisov-Joyce arXiv:1504.00690. We prove that $(\boldsymbol{\mathscr M},\omega)$ is orientable, by relating algebro-geometric orientations on $(\boldsymbol{\mathscr M},\omega)$ to differential-geometric orientations on ${\mathcal B}_P$ for U$(m)$-bundles $P\to X$, and using orientability of ${\mathcal B}_P$. This has applications to the programme of defining Donaldson-Thomas type invariants counting moduli spaces of (semi)stable coherent sheaves on a Calabi-Yau 4-fold, as in Donaldson and Thomas 1998, Cao and Leung arXiv:1407.7659, and Borisov and Joyce arXiv:1504.00690. This is the third in a series arXiv:1811.01096, arXiv:1811.02405 on orientations of gauge-theoretic moduli spaces., Comment: 57 pages. (v2) Major rewrite: new title, added author, new material on Calabi-Yau manifolds

Let $X_N$ be the second infinitesimal neighborhood of a closed point in $N$-dimensional affine space. In this note we study $D^b(coh\, X_N)$, the bounded derived category of coherent sheaves on $X_N$. We show that for $N\geq 2$ the lattice of triangulated subcategories in $D^b(coh\, X_N)$ has a rich structure (which is probably wild), in contrast to the case of zero-dimensional complete intersections. We also establish a relation between triangulated subcategories in $D^b(coh\, X_N)$ and universal localizations of a free graded associative algebra in $N$ variables. Our homological methods produce some applications to the structure of such universal localizations., 32 pages, comments are welcome. v2: organization of the paper is changed, minor changes to the text, some reference added

Given a reduced effective divisor D on a smooth variety X, we describe the generating function for the classes of the Hodge ideals of D in the Grothendieck group of coherent sheaves on X in terms of the motivic Chern class of the complement of the support of D. As an application, we compute the generating function for the Hilbert series of Hodge ideals of a hyperplane arrangement in terms of the Poincare polynomial of the arrangement., Comment: 19 pages; v.2: details added regarding the equivariant motivic Chern class, following suggestions of J. Schuermann; v.3: reference added to a result of Xia Liao, final version, to appear in Journal of Singularities

We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules on spaces such as (possibly singular) complex quasi-projective varieties. These formulae generalize our previous results for symmetric and alternating powers of such coefficients, and apply also to other Schur functors. The proofs of these results are reduced via an equivariant K\"{u}nneth formula to a more general generating series identity for abstract characters of tensor powers $\mathcal{V}^{\otimes n}$ of an element $\mathcal{V}$ in a suitable symmetric monoidal category $A$. This abstract approach applies directly also in the equivariant context for spaces with additional symmetries (e.g., finite group actions, finite order automorphisms, resp., endomorphisms), as well as for introducing an abstract plethysm calculus for symmetric sequences of objects in $A$., Comment: completely rewritten, title changed, a new section on abstract plethysm included, new references added

We calculate a semi-orthogonal decomposition of the bounded derived category of coherent sheaves on P(1,1,1,3) using a tilting bundle., Comment: 9 pages, a section for preliminaries is added

Generalizing the continuous rank function of Barja-Pardini-Stoppino, in this paper we consider cohomological rank functions of $\mathbb Q$-twisted (complexes of) coherent sheaves on abelian varieties. They satisfy a natural transformation formula with respect to the Fourier-Mukai-Poincar\'e transform, which has several consequences. In many concrete geometric contexts these functions provide useful invariants. We illustrate this with two different applications, the first one to GV-subschemes and the second one to multiplication maps of global sections of ample line bundles on abelian varieties., Comment: 28 pages, minor changes. Final version to appear on Annales Scient. ENS

The standard Lie bialgebra structure on an affine Kac-Moody algebra induces a Lie bialgebra structure on the underlying loop algebra and its parabolic subalgebras. In this paper we classify all classical twists of the induced Lie bialgebra structures in terms of Belavin-Drinfeld quadruples up to a natural notion of equivalence. To obtain this classification we first show that the induced bialgebra structures are defined by certain solutions of the classical Yang-Baxter equation (CYBE) with two parameters. Then, using the algebro-geometric theory of CYBE, based on torsion free coherent sheaves, we reduce the problem to the well-known classification of trigonometric solutions given by Belavin and Drinfeld. The classification of twists in the case of parabolic subalgebras allows us to answer recently posed open questions regarding the so-called quasi-trigonometric solutions of CYBE., Comment: 30 pages, 2 figures

In the present paper, we give a definition of the quantum group $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$$ of the circle $$S^1:={\mathbb {R}}/{\mathbb {Z}}$$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$ associated to the rational circle $$S^1_{\mathbb {Q}}:={\mathbb {Q}}/{\mathbb {Z}}$$ as a direct limit of $$\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(n))$$’s, where the order is given by divisibility of positive integers. The quantum group $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $$X_\infty $$ over a fixed smooth projective curve X defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus $$g_X$$, of $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$. Moreover, we show that $$\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(+\infty ))$$ and $$\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(\infty ))$$ are subalgebras of $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$. As proved by T. Kuwagaki in the appendix, the quantum group $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$$ naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle $$S^1$$.

We describe new irreducible components of the Gieseker–Maruyama moduli scheme M ( 3 ) of semistable rank 2 coherent sheaves with Chern classes c 1 = 0 , c 2 = 3 , c 3 = 0 on P 3 , general points of which correspond to sheaves whose singular loci contain components of dimensions both 0 and 1. These sheaves are produced by elementary transformations of stable reflexive rank 2 sheaves with c 1 = 0 , c 2 = 2 along a disjoint union of a projective line and a collection of points in P 3 . The constructed families of sheaves provide first examples of irreducible components of the Gieseker–Maruyama moduli scheme such that their general sheaves have singularities of mixed dimension.

This work deals with an algebro–geometric theory of solutions of the classical Yang–Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstras cubic curves.

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj ( A ) analogous to Proj ( R ) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves on MProj ( A ) , and we prove several basic results regarding these. We show that: 1. every quasicoherent sheaf F on MProj ( A ) can be constructed from a graded A-set in analogy with the construction of quasicoherent sheaves on Proj ( R ) from graded R-modules 2. if F is coherent on MProj ( A ) , then F ( n ) is globally generated for large enough n, and consequently, that F is a quotient of a finite direct sum of invertible sheaves 3. if F is coherent on MProj ( A ) , then Γ ( MProj ( A ) , F ) is finitely generated over A 0 (and hence a finite set if A 0 = { 0 , 1 } ). The last part of the paper is devoted to classifying coherent sheaves on P 1 in terms of certain directed graphs and gluing data. The classification of these over F 1 is shown to be much richer and combinatorially interesting than in the case of ordinary P 1 , and several new phenomena emerge.