Generalized Nakano positivity for certain singular cases.
problem Nakano positivity of direct image sheaves for singular cases.
method Generalization of Berndtsson's result to singular cases.
result Nakano positivity for direct image sheaves in special singular cases.
Paper proves direct image sheaf positivity for certain Kähler fibrations.
problem Proving positivity of direct image sheaves for Kähler fibrations.
method Uses singular Hermitian line bundles and Narasimhan-Simha metric.
result Direct image sheaf is singular Nakano semi-positive.
The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a rela…
The paper studies curvature properties of sheaves of twisted holomorphic forms on families of compact Kähler manifolds.
problem Investigating curvature properties of sheaves of twisted holomorphic forms on families of compact Kähler manifolds.
method Derives a general curvature formula and explores special cases.
result Provides insight into geometric and analytical properties of curvature in various contexts.
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 slopes of direct images in complex manifolds, proving a Mehta-Ramanathan type theorem.
problem Distribution of Harder-Narasimhan slopes in direct image sheaves.
method Analyzing asymptotic distributions of slopes under base changes of families of complex projective manifolds.
result Asymptotic distribution of slopes can be recovered from base changes over generic curves.
We shall show that q-semipositivity of the vector bundle E over a Kähler total space X implies the Griffiths-semipositivity of the q-th direct image of O(KX/B⊗E). As an application, we shall give a negative-curvature criterion for the generalized Weil-Petersson metric on t…
In this paper I give new elementary proofs of basic results of Gelfand, Kapranov and Zelevinskywhich express discriminants and resultants in terms of determinants of direct images of Cayley-Koszul complexes of sheaves.
Paper proves structure of compact Kähler 3-folds with specific bundles.
problem Characterizing compact Kähler 3-folds with nef anti-canonical bundles.
method Minimal Model Program, positivity of direct image sheaves, Q-conic bundles, orbifold vector bundles.
result Compact Kähler 3-folds with nef anti-canonical bundles are essentially one of three types.
Study on algebraic fiber spaces and their anti-canonical divisors.
problem Understanding positivity conditions and base loci of algebraic fiber spaces.
method Algebraic and analytic methods for positivity of direct image sheaves.
result Algebraic fiber spaces with semi-ample relative anti-canonical divisor have a product structure.
The paper establishes lower bounds on Yang-Mills functionals for fibrations.
problem Analyzing the stability and nefness of direct image sheaves in fibrations.
method Generalizing mean curvature and Harder-Narasimhan filtrations to arbitrary polarized fibrations.
result Optimal lower bounds on fibered Yang-Mills functionals in terms of direct image sheaves.
For proper surjective holomorphic maps from K"ahler manifolds to analytic spaces, we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by Nakano semi-positive vector bundles by means of the higher direct image sheaves, by using the theory of harmonic integrals developed by Takegoshi…
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…
In this paper, we investigate the geometry of the moduli space of curves by using the curvature properties of direct image sheaves of vector bundles. We show that the moduli space (Mg,ωWP) of curves with genus g>1 has dual-Nakano negative and semi-Nakano-negative curvature, and in particular, it has non-positi…
Characterizes curvature positivity for Riemannian metrics on flat vector bundles.
problem Characterizing Nakano positivity of Riemannian flat vector bundles.
method Using solvability of the d equation with specific L2 estimates and inspired by recent works on Hermitian holomorphic vector bundles. result Alternative proof of matrix-valued Prekopa's theorem.
A new method for clearing liability networks using sheaves on directed hypergraphs.
problem Clearing in liability networks using a novel mathematical approach.
method Associate a liability sheaf on a directed hypergraph to a liability network, identifying clearing configurations as global sections of this sheaf.
result Clearing configurations are precisely the global sections of the sheaf, and the sheaf construction is functorial under change of coefficient category.
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.
The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) G-module, where G-modules are struc…
The paper extends a conjecture about Euler characteristics of perverse sheaves on certain Kähler manifolds.
problem The conjecture about Euler characteristics of perverse sheaves on compact aspherical Kähler manifolds.
method The method involves expressing the Euler characteristic as an intersection number involving the characteristic cycle, and using curvature conditions to deduce non-negativity. For the second result, the local system is shown to underlie a complex variation of Hodge structures, leading to the desired inequality from curvature properties of the period map.
result The conjecture holds for compact aspherical Kähler manifolds with non-positive holomorphic bisectional curvature and for projective manifolds with a faithful semi-simple rigid local system.
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.
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.
Let (M,I,J,K) be a hyperkaehler manifold, dimRM=4n. We study positive, Dolbeault-closed (2p,0)-forms on (M,I). These forms are quaternionic analogues of the positive (p,p)-forms. We construct an injective homomorphism mapping Dolbeault-closed (2p,0)-forms to closed (n+p,n+p)-forms, and positive $(2p,…
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.
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 …
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.
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.
New geometric model for knot homology using monodromic Hecke category.
problem Geometric description of Khovanov-Rozansky knot homology.
method Monodromic model based on Soergel bimodules and Hecke category.
result Geometric description of Khovanov-Rozansky knot homology.
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.
Study first-order locally convex Lie algebroids in Bastiani calculus.
problem Define and study first-order locally convex Lie algebroids.
method Define sheaves of Lie algebroid forms and morphisms, prove category structure, study representations and cohomology.
result First-order locally convex Lie algebroids form a category and have applications in Lie II theorems.
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…
We establish a Kobayashi-Hitchin correspondence for the stable Higgs sheaves on a compact Kaehler manifold. Using it, we also obtain a Kobayashi-Hitchin correspondence for the stable Higgs G-sheaves, where G is any complex reductive linear algebraic group.
Develops a new method to study algebraic tangent cones of sheaves using valuations.
problem Analyzing tangent cones of torsion-free sheaves on algebraic varieties.
method Introduces a slope stability theory and uses it to define a canonical tangent cone for quasi-regular valuations.
result Shows the existence of a canonical tangent cone for torsion-free sheaves, up to equivalence.
Unified framework for Morita invariant cohomology of Lie groupoids.
problem Proving Morita invariance of cohomology theories for Lie groupoids.
method Viewing cohomology as sheaves of modules on the nerve of the groupoid and establishing criteria for Morita invariance.
result Established criteria for Morita invariant cohomology theories.
For (X,L) a polarized toric variety and G a torus of automorphisms of (X,L), denote by Y the GIT quotient X/G. We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on Y to the category of torus equivariant reflexive sheaves on X. We show, under a genericity assumption o…
We extend Nadel's results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal sheaves on toric del Pezzo surfaces which do not admit Kähler-Einstein metrics. We also show t…
Sheaves on graphs link to noncommutative geometry.
problem Exploring noncommutative geometry concepts on graphs.
method Sheaf theory and simplicial sets.
result Enhanced understanding of discrete noncommutative geometry.
Geometrically computes sheaves linking HOMFLY-PT homology to Hilbert schemes.
problem Linking HOMFLY-PT homology to geometric structures on Hilbert schemes.
method Geometric sheaf theory, Hochschild homology formality, Hilbert schemes of points.
result Established formalism connecting HOMFLY-PT homology to coherent sheaves on Hilbert schemes.
Study homotopy sheaves on categories and their presheaves, proving descent properties.
problem Homotopy sheaves on categories and their presheaves.
method Homotopy right Kan extension, pretopologies, Yoneda embedding.
result Preserves homotopy sheaves and induces equivalence between sheaves and colimit-preserving sheaves.
This is a large audience version of our previous work (see math.AG/0301146) in which we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of ∂ˉ-coherent sheaves. We also include here the complete proof of our main Theorem.
Proves Verdier duality for sheaves on stratified spaces.
problem Verdier duality for constructible sheaves on stratified spaces.
method Uses conically smooth stratified spaces and Lurie's Verdier duality.
result Shows equivalence between constructible sheaves and cosheaves.
Extends six operations to sheaves in any symmetric monoidal category.
problem Extending six operations to a broader class of sheaves.
method Develops formalism for sheaves in any closed symmetric monoidal ∞-category, proving properties of locally contractible geometric morphisms and relating pullbacks and colimits.
result Establishes the six functor formalism for a wider range of sheaves, including those with values in any closed symmetric monoidal ∞-category.
In this expository article we first give an overview on multiplier ideal sheaves and geometric problems in Kählerian and Sasakian geometries. Then we review our recent results on the relationship between the support of the subschemes cut out by multiplier ideal sheaves and the invariant whose non-vanishing obstructs th…
In this note we construct Nadel multiplier ideal sheaves using the Ricci flow on Fano manifolds. This extends a result of Phong, Sesum and Sturm. These sheaves, like their counterparts constructed by Nadel for the continuity method, can be used to obtain an existence criterion for Kahler-Einstein metrics.
Extends h-principle to stratified spaces using sheaf and jet theories.
problem Applying h-principle to stratified spaces.
method Developed new sheaf and bundle theories for stratified spaces, and proved the h-principle.
result Stratified continuous sheaves and homotopy fiber sheaves lead to the parametric h-principle.