Characterizes tangent cones for specific connections on reflexive sheaves.
problem Analyzing tangent cones of admissible Hermitian-Yang-Mills connections over reflexive sheaves.
method Algebro-geometric characterization of analytic tangent cones.
result Complete characterization of tangent cones for admissible Hermitian-Yang-Mills connections over reflexive sheaves.
The paper studies equivariant sheaves on toric varieties and their quotients.
problem Understanding stability of sheaves on toric GIT quotients.
method Defining equivariant sheaves and showing stability preservation under certain conditions.
result Stability of sheaves on toric GIT quotients is related to combinatorial criteria.
We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geom…
In this paper, we study semistable Higgs sheaves over compact Kähler manifolds, we prove that there is an approximate admissible Hermitian-Einstein structure on a semi-stable reflexive Higgs sheaf and consequently, the Bogomolove type inequality holds on a semi-stable reflexive Higgs sheaf.
Existence of metrics on non-Kähler varieties, generalizing previous work.
problem Existence of metrics on non-Kähler varieties.
method Definition of slope stability and existence of singular Hermite-Einstein metrics.
result Existence and uniqueness of singular Hermite-Einstein metrics for slope-stable sheaves.
Establishes Hermite-Einstein metrics on complex spaces with singularities.
problem Existence of Hermite-Einstein metrics on complex spaces with singularities.
method Established existence of estimable Hermite-Einstein metrics for stable reflexive coherent sheaves on compact normal Kähler spaces with klt singularities.
result Obtained precise results for varieties with klt singularities.
In this paper, we study the asymptotic behavior of the Hermitian-Yang-Mills flow on a reflexive sheaf. We prove that the limiting reflexive sheaf is isomorphic to the double dual of the graded sheaf associated to the Harder-Narasimhan-Seshadri filtration, this answers a question by Bando and Siu.
Characterizes stable sheaves for equality in orbifold BG inequality.
problem Stability of sheaves on compact Kähler varieties with klt singularities.
method Characterization of stable reflexive sheaves for BG equality.
result Characterizes stable reflexive sheaves for equality in BG inequality.
The paper proves a theorem and characterizes connections over normal varieties.
problem The study addresses the stability and connections over normal varieties.
method The authors prove a complete version of the Donaldson-Uhlenbeck-Yau theorem and use it to show the polystability of reflexive sheaves.
result An admissible Hermitian-Yang-Mills connection defines a polystable reflexive sheaf and gives a lower bound for discriminants.
Study Kobayashi-Hitchin correspondence for special sheaves on Kähler manifolds.
problem Understanding the Kobayashi-Hitchin correspondence for specific sheaves.
method Using Hermitian-Yang-Mills flow on Kähler manifolds with simple normal crossing divisors.
result Established the correspondence for saturated reflexive parabolic sheaves.
We introduce the notion of T-stability for torsion-free Higgs sheaves as a natural generalization of the notion of T-stability for torsion-free coherent sheaves over compact complex manifolds. We prove similar properties to the classical ones for Higgs sheaves. In particular, we show that only saturated flags of to…
The paper studies foliations on smooth projective varieties and their properties.
problem Characterizing and understanding foliations on smooth projective varieties.
method Develops a structure theorem for smooth projective varieties with almost nef regular foliations, using a smooth morphism and MRC fibration.
result An almost nef regular foliation on a smooth projective variety can be decomposed into a numerically flat regular foliation and a smooth morphism.
It is known that given a stable holomorphic pair (E,φ), where E is a holomorphic vector bundle on a compact Kähler manifold X and φ is a holomorphic section of E, the vector bundle E admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(\E ,φ)$, where $\E$ is a reflexive shea…
The paper studies coherent sheaves on subvarieties of Hopf manifolds.
problem Understanding coherent sheaves on subvarieties of Hopf manifolds.
method Proves a version of GAGA theorem, shows natural algebraic structure, and uses quotient and embedding properties.
result Any reflexive coherent sheaf on M is filtrable. Let M be a hyperkaehler manifold, and F a torsion-free and reflexive coherent sheaf on M. Assume that F (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then F is stable and its singulariti…
Study local third Chern class for point singularities on threefolds.
problem Understanding gauge theory singularity contributions on threefolds.
method Local algebraic data and deformation invariance, K-theoretic interpretation.
result Local third Chern class can be computed from family data and is deformation invariant.
We give a simple direct proof of uniqueness of tangent cones for singular projectively Hermitian Yang-Mills connections on reflexive sheaves at isolated singularities modelled on μ-polystable holomorphic bundles over Pn−1.
We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical Kähler manifolds: If E is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a μ-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Ya…
The paper proves a key inequality for a specific type of complex spaces.
problem Establishing a mathematical inequality for a class of complex spaces.
method Analytical approach involving Higgs sheaves and orbifolds.
result Proves the Miyaoka-Yau inequality for minimal Kähler klt spaces.
Paper studies convergence of Yang-Mills-Higgs flow on Kähler manifolds.
problem Analyzing convergence of Yang-Mills-Higgs flow for twisted Higgs pairs.
method Proves convergence to a reflexive twisted Higgs sheaf outside a closed subset.
result Limiting twisted Higgs sheaf is isomorphic to the double dual of graded twisted Higgs sheaves.
Criterion for projectivisation on klt spaces, characterizing quotients and stability.
problem Characterizing finite quotients of projective spaces and Abelian varieties.
method Criterion based on reflexive sheaves and stability conditions.
result Characterization of finite quotients using Q-Chern class inequalities and stability condition. Establishes Kobayashi-Hitchin correspondence for nef and big classes.
problem Analyzing stability and positivity in algebraic geometry.
method Introducing adapted currents and metrics to establish correspondence.
result Equality cases of Bogomolov-Gieseker and Miyaoka-Yau inequalities.
A simple quantitative example of a reflexive feedback process and the resulting price dynamics after an exogenous price shock to a financial network is presented. Furthermore, an outline of a theory that connects financial reflexivity, which stems from cross-ownership and delayed or incomplete information, and no-arbit…
Proves stability in Weyl polytopes using optimal transport.
problem Stability of Weyl polytopes under optimal transport.
method Optimal transport stability for reflexive Weyl polytopes.
result Weak metric SYZ conjecture holds for Delzant reflexive Weyl polytopes.
Study shows awareness of reflexivity improves LLMs' financial forecasting accuracy.
problem Improving LLMs' ability to forecast financial markets during boom-bust cycles.
method Evaluated three LLMs under four conditions of reflexivity awareness in two market episodes.
result Reflexivity awareness improves forecasting accuracy differently across models and contexts.
The mathematical model proposed by George Soros for his theory of reflexivity is analyzed under the framework of discrete dynamical systems. We show the importance of the notion of fixed points for explaining the behavior of a reflexive system governed by its cognitive and manipulative functions. The interrelationship …
This article generalizes the work of Ballmann and Światkowski to the case of Reflexive Banach spaces and uniformly convex Busemann spaces, thus giving a new fixed point criterion for groups acting on simplicial complexes.
The paper solves Monge-Ampère equations on reflexive polytopes, linking solvability to SYZ conjecture and tropical geometry.
problem Solvability of Monge-Ampère equations on reflexive polytopes.
method Analyzes reflexive polytopes with height functions, proving conditions for Monge-Ampère solvability and linking to SYZ conjecture.
result Conditions for Monge-Ampère solvability are necessary and sufficient, and solvability implies the SYZ conjecture for Calabi-Yau hypersurfaces.
Study proves existence of precotangent bundles for Grassmannians.
problem Existence of precotangent bundles for Grassmannians.
method Proof for Grassmannians of reflexive Banach spaces and p-restricted Grassmannians of polarized Hilbert space. result Existence of bundle predual to tangent bundle (precotangent bundle).
New examples of Cappell-Shaneson knot pairs with same Alexander polynomial found.
problem Determine dimensions for non-reflexive knot pairs.
method Constructing new examples of Cappell-Shaneson knot pairs.
result Found examples of Cappell-Shaneson knot pairs with same Alexander polynomial but inequivalent.
We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension n≥5. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also…
The paper proves a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.
problem Proving a Donaldson-Uhlenbeck-Yau theorem for normal varieties and semistable bundles.
method Analyzing degenerating families of projective normal varieties and studying the limiting behavior of semistable bundles.
result Improves several previously known algebro-geometric results on normalized tautological classes and proves a new version of the singular Donaldson-Uhlenbeck-Yau theorem.
Study co-Higgs sheaves on toric varieties, finding explicit examples.
problem Characterizing and understanding co-Higgs sheaves on toric varieties.
method Characterization and explicit computation of examples.
result Explicit examples of co-Higgs sheaves on toric varieties computed.
We review the notion of Gieseker stability for torsion-free Higgs sheaves. This notion is a natural generalization of the classical notion of Gieseker stability for torsion-free coherent sheaves. We prove some basic properties that are similar to the classical ones for torsion-free coherent sheaves over projective alge…
Study sheaves of Lie-Rinehart algebras and their morphisms, generalizing Lie algebroid concepts.
problem Understanding sheaves of Lie-Rinehart algebras and their morphisms.
method Introduced morphisms and comorphisms, proved factorization theorems, and defined higher homotopy groups and groupoids.
result Sheaves of Lie-Rinehart algebras over smooth manifolds induce partitions into orbits of the fundamental groupoid.
Bayesian reflex models AI learning like the autonomic nervous system.
problem Online learning in dynamic AI environments.
method Bayesian online algorithms with belief maintenance, sequential updating, and uncertainty-driven action balancing.
result Unified framework for adaptive AI learning.
Constructs coordinate systems from spectral curve sheaves.
problem Creating coordinate systems from spectral curve sheaves.
method Finite-gap integration methods for orthogonal curvilinear coordinates.
result Constructs coordinate systems over reducible spectral curves.
We consider a general regularised interpolation problem for learning a parameter vector from data. The well known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is at the core of kernel methods in machine learning as it makes…
New potentials found for sheaves on Calabi-Yau 4-folds.
problem Understanding sheaves on Calabi-Yau 4-folds.
method Derived Quot-stacks and Lagrangian distributions.
result Globally defined −1-shifted potentials on sheaves. Paper constructs Chern character for coherent sheaves.
problem Chern character for coherent sheaves with values in Bott-Chern cohomology.
method Based on Block's fundamental construction, constructs Chern character.
result Proves Riemann-Roch-Grothendieck formula for coherent sheaves.
This note is mostly an exposition of an unpublished result of Deligne, which introduces an analogue of perverse t-structure on the derived category of coherent sheaves on a Noetherian scheme with a dualizing complex. Construction extends to the category of coherent sheaves equivariant under an action of an algebraic …
Augmentations and sheaves linked for Legendrian graphs.
problem Understanding categorical Legendrian isotopy invariants.
method Equivalence between augmentation category and DG category of sheaves.
result Proved 'augmentations are sheaves' for Legendrian graphs.
Develops equivariant Chern characters for coherent sheaves with group actions.
problem Computing Chern characters for coherent sheaves on manifolds with group actions.
method Introduces equivariant Chern characters and proves Riemann-Roch-Grothendieck theorem in Bott-Chern cohomology.
result Establishes a Riemann-Roch-Grothendieck theorem for coherent sheaves with finite group actions.
Bipartite data is common in data engineering and brings unique challenges, particularly when it comes to clustering tasks that impose on strong structural assumptions. This work presents an unsupervised method for assessing similarity in bipartite data. Similar to some co-clustering methods, the method is based on regu…
We classify the simple sheaves microsupported along the conormal bundle of a knot. We also establish a correspondence between simple sheaves up to local systems and augmentations, explaining the underlying reason why knot contact homology representations detect augmentations.
The paper connects connections on sheaves to an L∞ morphism lifting semiregularity maps.
problem Understanding connections on sheaves and their relationship to semiregularity maps.
method Proves a canonical association of a connection of type (1,0) on a sheaf to an L∞ morphism. result Establishes a connection between connections on sheaves and an L∞ morphism lifting semiregularity maps. Defines sheaves and Čech cohomology for diffeological spaces and classifies principal bundles.
problem Defining sheaves and Čech cohomology for diffeological spaces.
method Defines sheaves for diffeological spaces and constructs Čech cohomology. Uses Čech cohomology to classify principal bundles.
result First degree Čech cohomology classes classify diffeological principal G-bundles. Given a smooth projective toric variety XΣ of complex dimension n, Fang-Liu-Treumann-Zaslow \cite{FLTZ} showed that there is a quasi-embedding of the differential graded (dg) derived category of coherent sheaves Coh(XΣ) into the dg derived category of constructible sheaves on a torus Sh(Tn,ΛΣ). Recently, K…