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In [Ven16] , Venkatesh conjectures a relation between the action of derived Hecke operators and an action by motivic cohomology groups. In [HV18] , Harris and Venkatesh reformulate this conjecture for the case of cohomology groups of coherent sheaves associated to modular forms of weight one. We refer to the latter as the HV conjecture. Recently, the work of Darmon, Harris, Rotger and Venkatesh in [DHRV] proves that this conjecture holds for dihedral weight one forms. The following article focuses therefore on the case of exotic forms, describes methods to compute explicitly all key ingredients appearing in the HV conjecture and provides further numerical evidence for it.

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

This paper aims at studying the homotopy category of cotorsion flat left modules $${{\mathbb {K}}({\mathrm{CotF}}\text {-}R)}$$ over a ring R. We prove that if R is right coherent, then the homotopy category $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ of dg-cotorsion complexes of flat R-modules is compactly generated. This uses firstly the existence of cotorsion flat preenvelopes over such rings and, secondly, the existence of a complete cotorsion pair $$({{\mathbb {K}}_{\mathrm{p}}({\mathrm{Flat}}\text {-}R)}, {\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R))$$ in the homotopy category $${{\mathbb {K}}({\mathrm{Flat}}\text {-}R)}$$ of complexes of flat R-modules, for arbitrary R. In the setting of quasi coherent sheaves over a Noetherian scheme, this cotorsion pair was discovered in the literature. However, we use a more elementary argument that gives this cotorsion pair for arbitrary R. Next we deal with cotorsion flat resolutions of complexes and define and study the notion of cotorsion flat dimension for complexes of flat R-modules. We also obtain an equivalence $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)\approx {{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ of triangulated categories where $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ is the homotopy category of projective R-modules. Combined with the aforementioned result, this recovers a result from Neeman, asserting the compact generation of $${{\mathbb {K}}({\mathrm{Proj}}\text {-}R)}$$ over right coherent R. Also we get the unbounded derived category $${\mathbb {D}} (R)$$ of R as a Verdier quotient of $${\mathbb {K}}(\mathrm{dg}\text {-}\mathrm{CotF}\text {-}R)$$ .

We construct a minimal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian IGr(3, 7). Moreover, we show that IGr(3, 7) admits a full exceptional collection consisting of equivariant vector bundles.

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

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.

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

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 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!

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

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

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 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.

We study triangulated categories and torsion theories in them, and compare two definitions of torsion theories in this work. The most important types of torsion theories—weight structures and t-structures (and admissible triangulated subcategories) are also been considered. One of the aims of this paper is to show that a number of basic definitions and properties of weight and t-structures naturally extend to arbitrary torsion theories (in particular, we define smashing and cosmashing torsion theories). This can optimize some of the proofs. Similarly, the definitions of orthogonal and adjacent weight and t-structures are generalized. We relate the adjacency of torsion theories with Brown-Comenetz duality and Serre functors. These results may be applied to the study of t-structures in compactly generated triangulated categories and in derived categories of coherent sheaves. The relationship between torsion theories and projective classes is described.

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 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 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

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.

Alexander L. Rosenberg has constructed a spectrum for abelian categories which is able to reconstruct a quasi-separated scheme from its category of quasi-coherent sheaves. In this note we present a detailed proof of this result which is due to Ofer Gabber. Moreover, we determine the automorphism class group of the category of quasi-coherent sheaves.

Let L be a cyclic L ∞ -algebra of dimension 3 with finite dimensional cohomology only in dimension one and two. By transfer theorem there exists a cyclic L ∞ -algebra structure on the cohomology H ⁎ ( L ) . The inner product plus the higher products of the cyclic L ∞ -algebra defines a superpotential function f on H 1 ( L ) . We associate with an analytic Milnor fiber for the formal function f and define the Euler characteristic of L is to be the Euler characteristic of the etale cohomology of the analytic Milnor fiber. In this paper we prove a Thom–Sebastiani type formula for the Euler characteristic of cyclic L ∞ -algebras. As applications we prove the Joyce–Song formulas about the Behrend function identities for semi-Schur objects in the derived category of coherent sheaves over Calabi–Yau threefolds. A motivic Thom–Sebastiani type formula and a conjectural motivic Joyce–Song formula for the motivic Milnor fiber of cyclic L ∞ -algebras are also discussed.

We define and study infinite root stacks of fine and saturated logarithmic schemes, a limit version of the root stacks introduced by Niels Borne and the second author. We show in particular that the infinite root stack determines the logarithmic structure, and recovers the Kummer-flat topos of the logarithmic scheme. We also extend the correspondence between parabolic sheaves and quasi-coherent sheaves on root stacks to this new setting.

We construct a moduli space of stable pairs over a smooth projective variety, parametrizing morphisms from a fixed coherent sheaf to a varying sheaf of fixed topological type, subject to a stability condition. This generalizes the notion used by Pandharipande and Thomas, following Le Potier, where the fixed sheaf is the structure sheaf of the variety. We then describe the relevant deformation and obstruction theories. We also show the existence of the virtual fundamental class in special cases., Comment: A new reference added and the email address corrected. Comments are welcome!

Let E ⁎ be a finite complex of locally free sheaves on a complex manifold X. We prove that to every connection of type ( 1 , 0 ) on E ⁎ it is canonically associated an L ∞ morphism g : A X 0 , ⁎ ( H o m O X ⁎ ( E ⁎ , E ⁎ ) ) ⇝ A X ⁎ , ⁎ A X ≥ 2 , ⁎ [ 2 ] that lifts the 1-component of the Buchweitz-Flenner semiregularity map. An application to deformations of coherent sheaves on projective manifolds is given.

Let $\mathcal{H}$ be a connected hereditary abelian category with tilting objects. It is proved that the cluster-tilting graph associated with $\mathcal{H}$ is always connected. As a consequence, we establish the connectedness of the tilting graph for the category $\operatorname{coh}\mathbb{X}$ of coherent sheaves over a weighted projective line $\mathbb{X}$ of wild type. The connectedness of tilting graphs for such categories was conjectured by Happel-Unger, which has immediately applications in cluster algebras. For instance, we deduce that there is a bijection between the set of indecomposable rigid objects of the cluster category $\mathcal{C}_{\mathbb{X}}$ of $\operatorname{coh}\mathbb{X}$ and the set of cluster variables of the cluster algebra $\mathcal{A}_{\mathbb{X}}$ associated with $\operatorname{coh}\mathbb{X}$., Comment: 22 pages, minor changes

We present a Langlands dual realization of the putative category of affine character sheaves. Namely, we calculate the categorical center and trace (also known as the Drinfeld center and trace, or categorical Hochschild cohomology and homology) of the affine Hecke category starting from its spectral presentation. The resulting categories comprise coherent sheaves on the commuting stack of local systems on the two-torus satisfying prescribed support conditions, in particular singular support conditions as appear in recent advances in the Geometric Langlands program. The key technical tools in our arguments are: a new descent theory for coherent sheaves or D-modules with prescribed singular support; and the theory of integral transforms for coherent sheaves developed in the companion paper [BNP]., Comment: 28 pages

We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module-theoretic variant of Strassman's theorem that we prove in the appendix.

We discuss Calabi-Yau and fractional Calabi-Yau semiorthogonal components of derived categories of coherent sheaves on smooth projective varieties. The main result is a general construction of a fractional Calabi-Yau category from a rectangular Lefschetz decomposition and a spherical functor. We give many examples of application of this construction and discuss some general properties of Calabi-Yau categories., Comment: 24 pages, v2: minor corrections

It has recently been noticed that the degeneracies of the Poisson bracket of linearized gravity on constant curvature Lorentzian manifold can be described in terms of the cohomologies of a certain complex of differential operators. This complex was first introduced by Calabi and its cohomology is known to be isomorphic to that of the (locally constant) sheaf of Killing vectors. We review the structure of the Calabi complex in a novel way, with explicit calculations based on representation theory of GL(n), and also some tools for studying its cohomology in terms of of locally constant sheaves. We also conjecture how these tools would adapt to linearized gravity on other backgrounds and to other gauge theories. The presentation includes explicit formulas for the differential operators in the Calabi complex, arguments for its local exactness, discussion of generalized Poincar\'e duality, methods of computing the cohomology of locally constant sheaves, and example calculations of Killing sheaf cohomologies of some black hole and cosmological Lorentzian manifolds., Comment: tikz-cd diagrams, 69 pages

On the Grassmannian of 2-dimensional subspaces in a finite dimensional linear space we construct a Karoubian complete strongly exceptional PO set of coherent sheaves, parametrized by the cosets of the Weyl group of the general linear group of the linear space modulo the Weyl group of the parabolic subgroup stabilizing the subspace, from subquotients of the Frobenius direct image of the structure sheaf of the Grassmannian defined over a field of large positive characteristic. Our collection diverges from the one discovered by Kapranov. We also show in the general setting over any field of positive characteristic that the sheaf corresponding to the longest element of the cosets is a direct summand of the Frobenius direct image of the structure sheaf of the homogeneous space.

In algebraic geometry, a number of invariants for classifying algebraic varieties are obtained from the cohomology groups of coherent sheaves. Some typical algorithms to compute the dimensions of the cohomology groups have been proposed by Decker and Eisenbud, and their algorithms have been implemented over compute algebra systems such as Macaulay2 and Magma. On the other hand, Maruyama showed an alternative method to compute the dimensions in his textbook; we call the method Maruyama’s method in this paper. However, Maruyama’s method was not described in an algorithmic format, and it has not been implemented yet. In this paper, we give an explicit algorithm of his method to compute the dimensions and bases of the cohomology groups of coherent sheaves. We also analyze the complexity of our algorithm, and implemented it over Magma. By our implementation, we examine the computational practicality of our algorithm. Moreover, we give some possible applications of our algorithm in algebraic geometry over fields of positive characteristics.

In this paper we study the derived categories of coherent sheaves on Grassmannians Gr ( k , n ) , defined over the ring of integers. We prove that the category D b ( Gr ( k , n ) ) has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of G L k . This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection [13] , which was constructed over a field of characteristic zero. We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of G L n − k . The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors. We also construct a tilting vector bundle on Gr ( k , n ) . We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over Z , and we identify the objects of D b ( Gr ( k , n ) ) , which correspond to the standard and costandard modules in both structures. All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change). Similar results over fields of arbitrary characteristic were obtained independently in [7] , by different methods.

For a given Fourier–Mukai equivalence of bounded derived categories of coherent sheaves on smooth quasi-projective varieties, we construct Fourier–Mukai equivalences of derived factorization categories of gauged Landau–Ginzburg (LG) models. As an application, we obtain some equivalences of derived factorization categories of K-equivalent gauged LG models. This result is an equivariant version of the result of Baranovsky and Pecharich, and it also gives a partial answer to Segal's conjecture. As another application, we prove that if the kernel of the Fourier–Mukai equivalence is linearizable with respect to a reductive affine algebraic group action, then the derived categories of equivariant coherent sheaves on the varieties are equivalent. This result is shown by Ploog for finite groups case.

We define a deformation of our earlier link homologies for fundamental representations of sl_m. The deformed homology of a link is isomorphic to the deformed homology of the disjoint union of its components. Moreover, there exists a spectral sequence starting with the old homology and converging to this deformed homology., Comment: 19 pages