Defines relative $\dbar$-complex and studies its curvature properties.
problem Curvature properties of relative $\dbar$-complex and associated vector bundles.
method Definition and study of curvature properties of the relative $\dbar$-complex and associated vector bundles.
result Yamaguchi's theory on subharmonicity of the Green operator can be seen as a curvature property of the quotient bundle.
Analyzes log canonical pairs using harmonic integrals and dbar-equation theory.
problem Establishing an analytic proof of the injectivity theorem for log canonical pairs.
method Theory of harmonic integrals and L2-method for dbar-equation.
result Analytic proof of the injectivity theorem for purely log terminal pairs.
The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the h…
We prove the classical Nakano vanishing theorem with Hörmander L2-estimates on a compact Kähler manifold using Siu's so called $\partial\dbar$-Bochner-Kodaira method, thereby avoiding the Kähler identities completely. We then introduce singular hermitian metrics on holomorphic vector bundles, and proceed to prove a …
Proves a vanishing theorem for K"ahler manifolds.
problem Analyzing cohomology groups of K"ahler manifolds.
method Using higher direct image sheaves and harmonic integrals.
result Proves a vanishing theorem of Koll'ar-Ohsawa type.
We give a completely explicit formula for all harmonic maps of finite uniton number from a Riemann surface to the unitary group U(n) in any dimension, and so all harmonic maps from the 2-sphere, in terms of freely chosen meromorphic functions on the surface and their derivatives, using only combinations of projections …
We prove sharp inequalities for determinants of Toeplitz operators and twisted Laplace operators on the two-sphere, generalizing the Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures of Gillet-Soule and Fang motivated by Arakelov geometry is obtained; applications to SU(2)-invariant determ…
Survey on injectivity theorems using L^2-method and harmonic integrals.
problem Injectivity theorems and their applications in complex geometry.
method Combination of L^2-method for dbar-equation and harmonic integrals.
result Nadel type vanishing theorems and extension theorems for pluri-canonical sections of log pairs.
We reformulate Heegaard Floer homology in terms of holomorphic curves in the cylindrical manifold Sigma x [0,1] x R, where Sigma is the Heegaard surface, instead of Sym^g(Sigma). We then show that the entire invariance proof can be carried out in our setting. In the process, we derive a new formula for the index of the…
We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravity theories in four dimensions, dealing with N=0,...,8 supersymmetries, the cases where different parts of the R-symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group …
For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen's metric and interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula co…
Develops theory of relatively geometric actions on CAT(0) cube complexes.
problem Understand actions of relatively hyperbolic groups on CAT(0) cube complexes.
method Introduces and studies relatively geometric actions, proving key results.
result Proves full relatively quasi-convex subgroups are convex compact.
Study on opers over complex manifolds of dimension one.
problem Investigating opers over complex manifolds of dimension one.
method Introducing relative opers and differential operators, analyzing their equivalence.
result Bijective correspondence between relative opers and differential operators.
Outer automorphisms act loxodromically on a complex.
problem Understanding the dynamics of outer automorphisms on relative free factor complexes.
method Proving loxodromic action and north-south dynamics.
result Fully irreducible outer automorphisms act loxodromically on the relative free factor complex.
Combines relatively hyperbolic groups over a complex.
problem Combining relatively hyperbolic groups.
method Generalization of Martin's theorem for hyperbolic groups.
result Combination theorem for relatively hyperbolic groups.
Study on complex spaces and their properties.
problem Characterizing relative properties of complex spaces.
method Analyzing indefinite Euclidean and non-flat complex spaces.
result Indefinite Euclidean complex space is not a relative of an indefinite non-flat complex space form.
Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
problem Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
method Proving non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
result Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.
This paper categorifies Morse theory for manifolds with boundaries.
problem Categorifying Morse theory for manifolds with boundaries.
method Defining a relative Morse complex using handlebody decomposition and constructing an A∞-category structure. result The homology of the relative Morse complex is isomorphic to the relative singular homology.
Study of forms on inertia spaces using Grauert--Grothendieck complex.
problem Understanding basic relative forms on inertia spaces of Lie group actions.
method Use of Grauert--Grothendieck complex on differentiable spaces.
result Sheaf complex of basic relative forms is a fine resolution of Brylinski's sheaf.
We study two notions of relative differential cohomology, using the model of differential characters. The two notions arise from the two options to construct relative homology, either by cycles of a quotient complex or of a mapping cone complex. We discuss the relation of the two notions of relative differential cohomo…
Study complex hyperbolic lattices and their relation to strict hyperbolization.
problem Understanding the relationship between complex hyperbolic lattices and strict hyperbolization.
method Analyzing the fundamental groups of complex hyperbolic manifolds and spaces arising from strict hyperbolization.
result Uniform lattices in PU(n,1) cannot be fundamental groups of Charney-Davis strict hyperbolizations when n ≥ 2.
Clarifies boundary criterion for non-one-ended subgroups in cubulation theory.
problem Boundary criterion for relative cubulation in non-one-ended subgroups.
method Showed that if boundary criterion is satisfied for a relatively hyperbolic group, the group admits a relatively geometric action on a CAT(0) cube complex.
result The refinement of the boundary criterion is useful for constructing new relative cubulations.
Characterizes geometric actions on graphs with flexible stabilizers.
problem Understanding geometric actions on flexible stabilizers.
method Defining generalized fine actions and proving relative quasi-convexity criteria.
result Characterizes Bowditch boundary points in relatively geometric actions.
Defines relative Dolbeault homology and proves its equivalence with Čech-Dolbeault cohomology.
problem Comparing relative Dolbeault cohomology groups of complex manifolds.
method Uses Čech approach to define relative Dolbeault homology and proves equivalence with Čech-Dolbeault cohomology.
result Relative Dolbeault homology and Čech-Dolbeault cohomology are equivalent.
Proves small cancellation free products have geometric actions on CAT(0) cube complexes.
problem Proving small cancellation free products have geometric actions on CAT(0) cube complexes.
method Using a blown-up complex of groups and a boundary separation criterion, proving wall stabilizers form a rich family of subgroups.
result Proves $C'( rac16)$--small cancellation free products of residually finite groups are residually finite.
New groups with special properties found.
problem Finding new groups with specific geometric properties.
method Proved actions on CAT(0) cubical complexes under certain conditions.
result Many groups admit cocompact actions on CAT(0) cubical complexes.
New tests detect asphericity in complex pairs, simplifying previous proofs.
problem Detecting asphericity in complex pairs (L,K) where K is a subcomplex of L. method Developed relative weight tests for injective labeled oriented trees.
result Injective labeled oriented trees are aspherical, strengthening previous results.
We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) X is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) X is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) X is a higher rank symmetric space and $\JJ$ an equivariant collection of ma…
We prove a systolic inequality for the phi-relative 1-systole of a phi-essential 2-complex, where phi is a homomorphism from the fundamental group of the complex, to a finitely presented group G. Indeed we show that universally for any phi-essential Riemannian 2-complex, and any G, the area of X is bounded below by 1/8…
Introduces relative information gain for improving Gaussian process regression rates.
problem Improving the sample complexity of estimating or maximizing unknown functions.
method Introduces relative information gain, interpolates between effective dimension and information gain, and proves PAC-Bayesian bounds.
result Obtains minimax-optimal rates of convergence through the relative information gain.
We give a generalized and self-contained account of Haglund-Paulin's wallspaces and Sageev's construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to pro…
The paper studies cohomology of sheaf complexes and proves a relative de Rham theorem.
problem Cohomology of sheaf complexes and their representations.
method Intersection of Cech theory and derived categories, proving a relative de Rham theorem.
result Cohomology of sheaf complexes is canonically isomorphic to the relative cohomology of the sheaf.
We prove a relative isoperimetric inequalities for Lagrangian half disks in C2 with respect to a Lagrangian plane, or a complex plane, or a union of any two of Lagrangian or complex planes that intersect transversally at the origin.
Global theory of relative invariants and equivariant line bundles established.
problem Global theory of relative invariants and equivariant line bundles.
method Cohomological description of Pic_{\mathfrak{g}}(M) using Chevalley-Eilenberg complex and Čech complex.
result Characterization of polynomial divisors and multipliers of relative differential invariants.
Formula for Dolbeault cohomologies of complex manifolds via relative cohomology.
problem Calculating Dolbeault cohomologies of complex manifolds after blowing up.
method Introducing relative Dolbeault cohomology, proving blow-up formula, and using bimeromorphic invariance.
result Uniform proof of bimeromorphic invariance of Dolbeault cohomology numbers.
Extremal metrics linked to stability in complex geometry.
problem Existence and uniqueness of extremal metrics in complex geometry.
method Proving asymptotic relative Chow stability implies extremal metrics existence and uniqueness.
result Existence and uniqueness of extremal metrics in any polarization.
Formula for relative Chern character number on spin manifolds.
problem Computing relative Chern characteristic numbers on complex vector bundles.
method Formula derivation for homomorphisms between vector bundles over odd-dimensional spheres.
result Formula for index of twisted Dirac operators on spin manifolds.
Holomorphic symplectic structure on Lagrangian moduli space.
problem Understanding the structure of Lagrangian submanifolds in hyperKähler manifolds.
method Proving the existence of a natural holomorphic symplectic structure on the relative Albanese over the moduli space.
result The relative Albanese over the moduli space of complex Lagrangian submanifolds has a natural holomorphic symplectic structure.
The paper explores the geometry and dynamics of free splitting and free factor complexes for groups.
problem Understanding the large scale geometry and dynamics of free splitting and free factor complexes.
method Analyzing the actions of the relative outer automorphism group on these complexes and using tools like the Two Over All Theorem and filling paths.
result Hyperbolicity of the relative free splitting complex and relative free factor complex was proven.
This paper calculates hyperbolicity constants for pants and relative pants graphs.
problem Understanding how hyperbolicity constants change based on the surface.
method Study of hyperbolicity constants for pants and relative pants graphs for specific surfaces.
result Calculates hyperbolicity constants for the five-punctured sphere and twice punctured torus.
Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
problem Conditions for hyperbolic and relatively hyperbolic extensions of free groups.
method Using dynamics of outer automorphisms on the complex of free factors and investigating the geometry of the extension group.
result Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
Study finite-energy metrics over complex manifold degenerations.
problem Finite-energy metrics on complex manifolds with singularities.
method Investigate spaces of plurisubharmonic metrics with finite-energy conditions.
result Complete and geodesic metric structure on finite-energy metrics space.
New criteria for relative hyperbolicity in hierarchically hyperbolic spaces.
problem Characterizing relative hyperbolicity in hierarchically hyperbolic spaces.
method New formulation of relative hyperbolicity in terms of hierarchy structures, applied to graphs associated to surfaces.
result The separating curve graph of a surface is relatively hyperbolic when the surface has zero or two punctures.
Study of special elliptic isometries and their lengths in complex hyperbolic plane.
problem Classifying lengths of special elliptic isometries in complex hyperbolic plane.
method Classification and description of relative SU(2,1)-character varieties.
result Fully classified lengths of special elliptic isometries (2, 3, 4).
Defines de Rham relative cotangent complex in tangent categories.
problem Characterizing immersions, submersions, local diffeomorphisms, and unramified morphisms in tangent categories.
method Systematic study of morphisms and their interactions, using algebraic geometry, differential geometry, and Cartesian differential categories.
result Defines de Rham relative cotangent complex in an arbitrary tangent category.
There are two natural candidates for the group of relative Cheeger-Simons differential characters. The first directly extends the work of Cheeger and Simons and the second extends the description given by Hopkins and Singer of the Cheeger-Simons group as the homology of a certain cochain complex. We discuss both approa…
Given a complex 4-fold X with an (Calabi-Yau 3-fold) anti-canonical divisor Y, we study relative Donaldson-Thomas invariants for this pair, which are elements in the Donaldson-Thomas cohomologies of Y. We also discuss gluing formulas which relate relative invariants and DT4 invariants for Calabi-Yau 4-folds.
Efficiently accelerates attention calculation for Transformers with relative positional encoding.
problem Quadratic complexity of attention in long sequences.
method Kernelized attention with Fast Fourier Transform (FFT) for RPE.
result Achieves O(n log n) time complexity, mitigates training instability, and outperforms other models.