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In the present paper, we outline and expound the fundamental and novel qualitative-cum-philosophical premises, principles, ideas, concepts, constructions and results that originate from our ongoing research project of applying the new conceptual panoply and the novel technical machinery of Abstract Differential Geometry (ADG) to various persistently outstanding issues in Quantum Gravity (QG) (Mallios and Raptis Int. J. Theor. Phys. 40, 1885, 2004, Int. J. Theor. Phys. 41, 1857, 2002, Int. J. Theor. Phys. 42, 1479, 2003; Raptis Int. J. Theor. Phys. 39, 1233, 2000, Int. J. Theor. Phys. 39, 1703, 2000, Int. J. Theor. Phys. 45, 79, 2006, Int. J. Theor. Phys. 46, 688, 2007, Int. J. Theor. Phys. 45, 1495, 2006, Int. J. Theor. Phys. 46, 1137, 2007, Int. J. Theor. Phys. 46, 3009, 2007; Raptis and Zapatrin Int. J. Theor. Phys. 39, 1, 2000, Class. Quant. Grav. 20, 4187, 2001). This paper may be regarded as a sequel to the paper (Raptis Int. J. Theor. Phys. 46, 3009, 2007) in the aftermath of the paper (Raptis 2024), which is currently in press. At the end of the paper, we discuss the potential philosophical repercussions of two possible future research routes that the main stream of our applications of ADG to QG may bifurcate towards in view of three independent, but overlapping, research papers that are currently under development (Raptis 2024a, 2024b, 2024c). [ABSTRACT FROM AUTHOR]

We study the coherent cohomology of generalized flag supervarieties. Our main observation is that these groups are closely related to the free resolutions of (certain generalizations of) determinantal ideals. In the case of super Grassmannians, we completely compute the cohomology of the structure sheaf: it is composed of the singular cohomology of a Grassmannian and the syzygies of a determinantal variety. The majority of the work involves studying the geometry of an analog of the Grothendieck-Springer resolution associated with the super Grassmannian; this takes place in the world of ordinary (non-super) algebraic geometry. Our work gives a conceptual explanation of the result of Pragacz-Weyman that the syzygies of determinantal ideals admit an action of the general linear supergroup. In a subsequent paper, we will treat other flag supervarieties in detail. [ABSTRACT FROM AUTHOR]

Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description of global dynamics. In this paper, a sheaf-theoretic approach to their continuation is developed. The algebraic structures are cast into a categorical framework to study their continuation systematically and simultaneously. Sheaves are built from this abstract formulation, which track the algebraic data as systems vary. Sheaf cohomology is computed for several classical bifurcations, demonstrating its ability to detect and classify bifurcations. [ABSTRACT FROM AUTHOR]

Topological states of matter can be classified only in terms of global topological invariants. These global topological invariants are encoded in terms of global observable topological phase factors in the state vectors of electrons. In condensed matter, the energy spectrum of the Hamiltonian operator has a band structure, meaning that it is piecewise continuous. The energy in each continuous piece depends on the quasi-momentum which varies in the Brillouin zone. Thus, the Brillouin zone of quasi-momentum variables constitutes the base localization space of the energy eigenstates of electrons. This is a continuous topological parameter space bearing the homotopy of a torus. Since the base localization space has the homotopy of a torus, if we vary the quasi-momentum in a direction, when the edge of the zone is reached, we obtain a closed path. Then, if we lift this loop from the base space to the sections of the sheaf-theoretic fibration induced by the localization of the energy eigenfunctions, we obtain a global topological phase factor which encodes the topological structure of the Brillouin zone. Because it is homotopically equivalent to a torus, the global phase factor turns out to be quantized, taking integer values. The experimental significance of this model stems from the recent discovery that there are observable global topological phase factors in fairly ordinary materials. In this communication, we show that it is the unitary representation theory of the discrete Heisenberg group in terms of commutative modular symplectic variables, giving rise to a joint commutative representation space endowed with an integral and Z 2 -invariant symplectic form that articulates the specific form of the topological conditions characterizing both the quantum Hall effect and the spin quantum Hall effect under a unified sheaf-theoretic cohomological framework. [ABSTRACT FROM AUTHOR]

The aim of this article is to study the geometry of Bott-Chern hypercohomology from the bimeromorphic point of view. We construct some new bimeromorphic invariants involving the cohomology for the sheaf of germs of pluriharmonic functions, the truncated holomorphic de Rham cohomology, and the de Rham cohomology. To define these invariants, by using a sheaf-theoretic approach, we establish a blow-up formula together with a canonical morphism for the Bott-Chern hypercohomology. In particular, we compute the invariants of some compact complex threefolds, such as Iwasawa manifolds and quintic threefolds.

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable ∞-category Cohb(ℝM) of complexes of sheaves with bounded coherent cohomology on a derived moduli stack ℝM. In the surface case, ℝM is a suitable derived enhancement of the moduli stack M of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case, we define three categorified Hall algebras associated with suitable derived enhancements of the moduli stack of Higgs sheaves on a curve X, the moduli stack of vector bundles with flat connections on X, and the moduli stack of finite-dimensional local systems on X, respectively. In the Higgs sheaves case we obtain a categorification of the K-theoretical and cohomological Hall algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in the other two cases our construction yields, by passing to K0, new K-theoretical Hall algebras, and by passing to H*BM, new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and the non-abelian Hodge correspondences can be lifted to the level of our categorified Hall algebras of a curve. [ABSTRACT FROM AUTHOR]

We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot–Chow morphism of tensor and wedge products of duals of tautological bundles. [ABSTRACT FROM AUTHOR]

A certain Grothendieck topology assigned to a metric space gives rise to a sheaf cohomology theory which sees the coarse structure of the space. Already constant coefficients produce interesting cohomology groups. In degree 0, they see the number of ends of the space. In this paper, a resolution of the constant sheaf via cochains is developed. It serves to be a valuable tool for computing cohomology. In addition, coarse homotopy invariance of coarse cohomology with constant coefficients is established. This property can be used to compute cohomology of Riemannian manifolds. The Higson corona of a proper metric space is shown to reflect sheaves and sheaf cohomology. Thus, we can use topological tools on compact Hausdorff spaces in our computations. In particular, if the asymptotic dimension of a proper metric space is finite, then higher cohomology groups vanish. We compute a few examples. As it turns out, finite abelian groups are best suited as coefficients on finitely generated groups.

We introduce a modular completion of the stack of maps from stable marked curves to the quotient stack [pt/C×], and use this stack to construct some gauge-theoretic analogues of the Gromov-Witten invariants. We also indicate the generalization of these invariants to the quotient stacks [X/C×], where X is a smooth proper complex algebraic variety.

We introduce a modular completion of the stack of maps from stable marked curves to the quotient stack [pt/C×], and use this stack to construct some gauge-theoretic analogues of the Gromov-Witten invariants. We also indicate the generalization of these invariants to the quotient stacks [X/C×], where X is a smooth proper complex algebraic variety.

Generalizing the Moret-Bailly pencil of supersingular abelian surfaces to higher dimensions, we construct for each field of characteristic p > 0 a smooth projective variety with trivial dualizing sheaf that does not lift to characteristic zero. Our approach heavily relies on local unipotent group schemes, the Beauville-Bogomolov decomposition for Kähler manifolds with c1 = 0, and equivariant deformation theory in mixed characteristics. [ABSTRACT FROM AUTHOR]

Computing the cohomology of the tensor product of two vector bundles is central in the study of their moduli spaces and in applications to representation theory, combinatorics and physics. These computations play a fundamental role in the construction of Brill–Noether loci, birational geometry and S-duality. Using recent advances in the Minimal Model Program for moduli spaces of sheaves on P 2 , we compute the cohomology of the tensor product of general semistable bundles on P 2 . This solves a natural higher rank generalization of the polynomial interpolation problem. More precisely, let v and w be two Chern characters of stable bundles on P 2 and assume that w is sufficiently divisible depending on v . Let V ∈ M (v) and W ∈ M (w) be two general stable bundles. We fully compute the cohomology of V ⊗ W . In particular, we show that if W is exceptional, then V ⊗ W has at most one nonzero cohomology group determined by the slope and the Euler characteristic, generalizing foundational results of Drézet, Göttsche and Hirschowitz. We characterize the invariants of effective Brill–Noether divisors on M (v) . We also characterize when V ⊗ W is globally generated. Our computation is canonical given the birational geometry of the moduli space, suggesting a roadmap for tackling analogous problems on other surfaces. [ABSTRACT FROM AUTHOR]

For a connected reductive algebraic group over an arbitrary number field, we consider a finite dimensional algebraic, irreducible representation of the group of its real points. Each adelic locally symmetric space corresponding to a level structure constructed using the group has an associated sheaf induced by this representation. The purpose of this note is to estimate the rate of growth of the total dimension of the pertinent cohomology with coefficients as either of the level structure or the associated sheaf varies. We obtain an upper bound on this total dimension. We also obtain a lower bound under certain topological conditions. Both the bounds are consistent with several classical dimension formulae as well as other known results. [ABSTRACT FROM AUTHOR]

To a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space. [ABSTRACT FROM AUTHOR]

Given a Heegaard splitting of a three-manifold Y, we consider the SL.2;C/character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an intersection a perverse sheaf of vanishing cycles. We prove that in our setting, the perverse sheaf is an invariant of Y, i.e., it is independent of the Heegaard splitting. The hypercohomology of the perverse sheaf can be viewed as a model for (the dual of) SL.2;C/instanton Floer homology. We also present a framed version of this construction, which takes into account reducible representations. We give explicit computations for lens spaces and Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. [ABSTRACT FROM AUTHOR]

This paper is an exposition on how Grothendieck's Quot scheme can be seen as a solution to the moduli problem of quotient sheaves. These schemes as corresponding moduli spaces play a significant role, not only in solving problems within geometry in general, but they are also applicable to problems of classification in algebra. These Quot schemes are introduced in context of the setting up of a solution to a moduli problem. This is then followed by its application to particular problems of classifying continuously varying families of quotient sheaves with a fixed Hilbert polynomial. The purpose of this investigation is to bridge the often abstract settings of moduli theory as pioneered by Grothendieck, and its concrete application through solving particular moduli problems in concrete projective geometric setting. In particular, this study will present how algebraic geometry through moduli theory helps reveal significant information about the family of objects which it parameterizes. [ABSTRACT FROM AUTHOR]

Abstract We prove that intermediate extensions of perverse constructible F p -sheaves commute with smooth pullbacks for embeddable schemes over a field of characteristic p. Along the way we also prove that the equivalence of categories of Cartier crystals with unit R [ F ] -modules commutes with f ! for a smooth morphism f : X → Y of embeddable schemes. [ABSTRACT FROM AUTHOR]

We generalize to the case of singular blocks the result in Bezrukavnikov and Lachowska [Quantum groups, Contemporary Mathematics 433 (American Mathematical Society, Providence, RI, 2007) 89–101] that describes the center of the principal block of a small quantum group in terms of sheaf cohomology over the Springer resolution. Then using the method developed in Lachowska and Qi [Int. Math. Res. Not., Preprint, 2017, arXiv:1604.07380], we present a linear algebro‐geometric approach to compute the dimensions of the singular blocks and of the entire center of the small quantum group associated with a complex semisimple Lie algebra. A conjectural formula for the dimension of the center of the small quantum group at an lth root of unity is formulated in type A. [ABSTRACT FROM AUTHOR]

We introduce and study a notion of Castelnuovo–Mumford regularity suitable for weighted projective spaces. [ABSTRACT FROM AUTHOR]

A complex contact structure γ is defined by a system of holomorphic local 1-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle Ker   γ of the tangent bundle and a line bundle L. In this paper, we prove that the sheaf of holomorphic k -vectors on a complex contact manifold splits into the sum of 𝒪 (∧ k Ker   γ) and 𝒪 (L ⊗ ∧ k − 1 Ker   γ) as sheaves of ℂ -module. The theorem induces the short exact sequence of cohomology of holomorphic k -vectors, and we obtain vanishing theorems for the cohomology of 𝒪 (∧ k Ker   γ). [ABSTRACT FROM AUTHOR]

We develop an elementary algebraic method to compute the center of the principal block of a small quantum group associated with a complex semisimple Lie algebra at a root of unity. The cases of |$\mathfrak{sl}_3$| and |$\mathfrak{sl}_4$| are computed explicitly. This allows us to formulate the conjecture that, as a bigraded vector space, the center of a regular block of the small quantum |$\mathfrak{sl}_m$| at a root of unity is isomorphic to Haiman's diagonal coinvariant algebra for the symmetric group |$S_{m}$|⁠. [ABSTRACT FROM AUTHOR]

The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable ∞-category C together with a collection of compact objects K ⊂ C we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring A and a suitable ideal in A , we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme X implies the local duality theorem of Alonso Tarrío, Jeremías López, and Lipman. As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts. [ABSTRACT FROM AUTHOR]

For arbitrary field coefficients K , we show that K -perverse sheaves on a complex affine torus satisfy the so-called propagation package , i.e., the generic vanishing property and the signed Euler characteristic property hold, and the corresponding cohomology jump loci satisfy the propagation property and codimension lower bound. The main ingredient used in the proof is Gabber–Loeser's Mellin transformation functor for K -constructible complexes on a complex affine torus, and the fact that it behaves well with respect to perverse sheaves. As a concrete topological application of our sheaf-theoretic results, we study homological duality properties of complex algebraic varieties, via abelian duality spaces . We provide new obstructions on abelian duality spaces by showing that their cohomology jump loci satisfy a propagation package. This is then used to prove that complex abelian varieties are the only complex projective manifolds which are abelian duality spaces. We also construct new examples of abelian duality spaces. For example, we show that if a smooth quasi-projective variety X satisfies a certain Hodge-theoretic condition and it admits a proper semi-small map (e.g., a closed embedding or a finite map) to a complex affine torus, then X is an abelian duality space. [ABSTRACT FROM AUTHOR]

We prove that wild ramification of a constructible sheaf on a surface is determined by that of the restrictions to all curves. We deduce from this result that the Euler–Poincaré characteristic of a constructible sheaf on a variety of arbitrary dimension over an algebraically closed field is determined by wild ramification of the restrictions to all curves. We similarly deduce from it that so is the alternating sum of the Swan conductors of the cohomology groups, for a constructible sheaf on a variety over a local field. [ABSTRACT FROM AUTHOR]

On the complement to a central plane line arrangement , a locally constant sheaf of complex vector spaces ℒa is associated to any multi-index a∈ℂn. Using the description of MacPherson and Vilonen of the category of perverse sheaves [7, 8], we obtain a criterion for the irreducibility and number of decomposition factors of the direct image as a perverse sheaf, where j:X→ℂ2 is the canonical inclusion. [ABSTRACT FROM AUTHOR]

In this paper, we define m -tail reflexive sheaves as reflexive sheaves on projective spaces with the simplest possible cohomology. We prove that the rank of any m -tail reflexive sheaf E on P n is greater or equal to n m − m . We completely describe m -tail reflexive sheaves on P n of minimal rank and we construct huge families of m -tail reflexive sheaves of higher rank. [ABSTRACT FROM AUTHOR]

This paper affirms a conjecture of MacPherson—that the derived category of cellular sheaves is equivalent to the derived category of cellular cosheaves. We give a self-contained treatment of cellular sheaves and cosheaves and note that certain classical dualities give rise to an exchange of sheaves with cosheaves. Following a result of Pitts that states that cosheaves are cocontinuous functors on the category of sheaves, we use the derived equivalence provided here to gain a novel description of compactly-supported sheaf cohomology. [ABSTRACT FROM AUTHOR]

Let X be a scheme such that the structural sheaf OX have zero higher cohomology group andMbe an OX-module on X. This paper investigate a sufficient condition for a quasi-coherent module M to have Hi(X,M) = 0 for all i > 0. [ABSTRACT FROM AUTHOR]

We determine an upper bound for the cohomological dimension of the complement of a closed subset in a projective variety which possesses an appropriate stratification. We apply the result to several particular cases, including the Bialynicki-Birula stratification; in this latter case, the bound is sharp. [ABSTRACT FROM AUTHOR]

Let the vector bundle E be a deformation of the tangent bundle over the Grassmannian G ( k , n ) . We compute the ring structure of sheaf cohomology valued in exterior powers of E , also known as the polymology. This is the first part of a project studying the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle, a generalization of ordinary quantum cohomology rings of Grassmannians. A companion physics paper [6] describes physical aspects of the theory, including a conjecture for the quantum sheaf cohomology ring, and numerous examples. [ABSTRACT FROM AUTHOR]

In this text, we generalize Čech cohomology to sheaves F with values in blue B -modules where B is a blueprint with −1. If X is an object of the underlying site, then the cohomology sets H l ( X , F ) turn out to be blue B -modules. For a locally free O X -module F on a monoidal scheme X , we prove that H l ( X , F ) + = H l ( X + , F + ) where X + is the scheme associated with X and F + is the locally free O X + -module associated with F . In an appendix, we show that the naive generalization of cohomology as a right derived functor is infinite-dimensional for the projective line over F 1 . [ABSTRACT FROM AUTHOR]

We present a robust categorical foundation for the duality theory introduced by Eisenbud and Schreyer to prove the Boij-Söderberg conjectures describing numerical invariants of syzygies. The new foundation allows us to extend the reach of the theory substantially. More explicitly, we construct a pairing between derived categories that simultaneously categorifies all the functionals used by Eisenbud and Schreyer. With this new tool, we describe the cone of Betti tables of finite, minimal free complexes having homology modules of specified dimensions over a polynomial ring, and we treat many examples beyond polynomial rings. We also construct an analogue of our pairing between derived categories on a toric variety, yielding toric/multigraded analogues of the Eisenbud-Schreyer functionals. [ABSTRACT FROM AUTHOR]

Motivated by the computation of the BPS-invariants on a local Calabi-Yau 3-fold suggested by S. Katz, we compute the Chow ring and the cohomology ring of the moduli space of stable sheaves of Hilbert polynomial 4m + 1 on the projective plane. As a byproduct, we obtain the total Chern class and Euler characteristics of all line bundles, which provide numerical data for the strange duality on the plane. [ABSTRACT FROM AUTHOR]

In this paper we study the quantum sheaf cohomology of Grassmannians with deformations of the tangent bundle. Quantum sheaf cohomology is a (0,2) deformation of the ordinary quantum cohomology ring, realized as the OPE ring in A/2-twisted theories. Quantum sheaf cohomology has previously been computed for abelian gauged linear sigma models (GLSMs); here, we study (0,2) deformations of nonabelian GLSMs, for which previous methods have been intractable. Combined with the classical result, the quantum ring structure is derived from the one-loop effective potential. We also utilize recent advances in supersymmetric localization to compute A/2 correlation functions and check the general result in examples. In this paper we focus on physics derivations and examples; in a companion paper, we will provide a mathematically rigorous derivation of the classical sheaf cohomology ring. [ABSTRACT FROM AUTHOR]

We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY's having reduced manifold equal to $$ {\mathrm{\mathbb{P}}}^1 $$ , namely the projective super space $$ {\mathrm{\mathbb{P}}}^{\left.1\right|2} $$ and the weighted projective super space $$ \mathbb{W}{\mathrm{\mathbb{P}}}_{(2)}^{\left.1\right|1} $$ . Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces $$ {\mathrm{\mathbb{P}}}^{\left.n\right|m} $$ . We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of $$ {\mathrm{\mathbb{P}}}^{\left.1\right|2} $$ , whose automorphism group turns out to be larger than the projective super group. By considering the cohomology of the super tangent sheaf, we compute the deformations of $$ {\mathrm{\mathbb{P}}}^{\left.1\right|m} $$ , discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that $$ {\mathrm{\mathbb{P}}}^{\left.1\right|2} $$ is self-mirror, whereas $$ \mathbb{W}{\mathrm{\mathbb{P}}}_{(2)}^{\left.1\right|1} $$ has a zero dimensional mirror. Also, the mirror map for $$ {\mathrm{\mathbb{P}}}^{\left.1\right|2} $$ naturally endows it with a structure of N = 2 super Riemann surface. [ABSTRACT FROM AUTHOR]

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 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 Poincaré 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. [ABSTRACT FROM AUTHOR]

We use supernatural bundles to build $\mathbf{GL}$-equivariant resolutions supported on the diagonal of $\mathbb{P}^{n}\times \mathbb{P}^{n}$, in a way that extends Beilinson’s resolution of the diagonal. We thus obtain results about supernatural bundles that largely parallel known results about exceptional collections. We apply this construction to Boij–Söderberg decompositions of cohomology tables of vector bundles, yielding a proof of concept for the idea that those positive rational decompositions should admit meaningful categorifications. [ABSTRACT FROM AUTHOR]