Analytic sets with unique infinite tangent cone are algebraic.
problem Characterizing analytic sets with unique infinite tangent cones.
method Analytic and algebraic set properties, degree of complex algebraic sets.
result Degree of Lipschitz normally embedded sets equals their infinite tangent cone degree.
Proves algebraic cones for LCK manifolds with potential.
problem Characterizing algebraic cones for LCK manifolds.
method Analyzes LCK manifolds as complex submanifolds of Hopf manifolds and covers, proving algebraicity of the resulting cones.
result Affine algebraic structure on cones is independent of manifold choice.
Holomorphic tensors on algebraic cones are invariant under certain group actions.
problem Holomorphic tensors on products of algebraic cones
method Using algebraic structures and embeddings
result Holomorphic tensors are invariant under group actions
Study left-invariant pseudo-Riemannian metrics on Lie groups focusing on null cone Lie algebras.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups in the null cone.
method Use bracket flow on Lie algebra to study metrics on Lie groups.
result Classify all cases of null cone Lie algebras in signatures (1,q) and (2,q).
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.
New method classifies special Vinberg cones of rank 4.
problem Classifying special Vinberg cones of rank 4.
method Using Clifford Nil-algebras and directed acyclic graphs.
result Explicit classification of rank 4 special Vinberg cones.
Kähler-Ricci flows' tangent cones are algebraic varieties.
problem Understanding the structure of Kähler-Ricci flows' tangent cones.
method Analyzing tangent cones as normal affine algebraic varieties and using Hörmander's L2 estimate. result The regular set of tangent cones coincides with the algebraic regular set.
We investigate subgroups of SL (n,Z) which preserve an open nondegenerate convex cone in real n-space and admit in that cone as fundamental domain a polyhedral cone of which some faces are allowed to lie on the boundary. Examples are arithmetic groups acting on selfdual cones, Weyl groups of certain Kac-Moody algebras …
Study of g-vector cones in cluster algebras from weighted orbifolds.
problem Determine the closure of g-vector cones in cluster algebras. method Analyzing g-vector cones in a cluster algebra defined from a weighted orbifold. result Closure of the union of g-vector cones is Rn except for specific weighted orbifolds. Study left-invariant pseudo-Riemannian metrics on Lie groups using moving bracket approach.
problem Characterize left-invariant pseudo-Riemannian metrics on Lie groups with vanishing scalar curvature invariants.
method Using the moving bracket approach, analyze Lie algebras of dimensions ≤ 6 and semi-simple Lie algebras.
result All Lie algebras of dimension ≤ 6 except 3-dimensional solvable Lie algebra are in the null cone, leading to VSI metrics.
By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is dec…
Paper proves unique tangent maps for complex maps into algebraic varieties.
problem Proving uniqueness of tangent maps for complex maps into algebraic varieties.
method Developed techniques to prove uniqueness of tangent maps for weakly holomorphic and locally approximable maps.
result Established unique tangent cone property for weakly holomorphic maps into projective algebraic varieties.
Study of tangent cones at infinity for algebraic sets.
problem Characterizing algebraic sets based on their tangent cones at infinity.
method Definition and analysis of tangent cones C4,∞(X) and C5,∞(X), proving properties and relations. result Affine linear subspace characterization based on C5,∞(X)'s dimension. Constructs moduli spaces for Calabi-Yau cones and Sasaki-Einstein manifolds.
problem Proper moduli spaces for K-polystable Q-Fano cones and their links.
method Algebraic construction using local normalized volume and higher Θ-stable reduction.
result Alternative algebraic proof of proper moduli spaces for Q-Fano varieties.
We construct two infinite families of algebraic minimal cones in Rn. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byp…
Bounded symmetric domains are biholomorphic to tube domains over Finsler symmetric cones.
problem Characterizing biholomorphic mappings between tube domains and bounded symmetric domains.
method Analyzing properties of Finsler symmetric cones and unital JB-algebras.
result Tube domains over Finsler symmetric cones are biholomorphic to bounded symmetric domains.
Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.
problem Approximation and interpolation for regular immersions directed by algebraically elliptic cones.
method Uses homotopy-theoretic necessary and sufficient conditions for approximation and interpolation.
result Homotopy-theoretic conditions for approximation and interpolation are satisfied in many cases of interest.
The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreduci…
Cone structures in quantum field theory linked to information geometry.
problem Understanding geometric structures in quantum field theory.
method Analyzing invariant cones under modular automorphism groups and their relation to Wishart laws.
result Explicit connection between CAH cones and Wishart laws.
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.
By applying the symplectic cutting operation to cotangent bundles, one can construct a large number of interesting symplectic cones. In this paper we show how to attach algebras of pseudodifferential operators to such cones and describe the symbolic properties of the algebras.
The paper explores cone structures and their connections to parabolic geometries in complex manifolds.
problem Understanding cone structures and their properties in complex manifolds.
method Analyzes cone structures induced by parabolic geometries and VMRT structures, focusing on local invariants.
result Establishes a local differential-geometric version of a global algebraic-geometric recognition theorem.
The study describes special real manifolds and invariant admissible cubics in Vinberg cones.
problem Understanding special real manifolds and invariant admissible cubics in Vinberg cones.
method Simplified Vinberg theory using Nil-algebras to describe invariant functions and polynomials.
result Examples of continuous families of non-homogeneous special real manifolds.
In symmetric cones, a non-empty locus satisfies the WDVV equation, generalizing previous results.
problem Finding a non-empty locus in symmetric cones where the WDVV equation holds.
method Combining algebraic/geometric and analytic approaches, including Calabi's work on Monge-Ampère equations.
result A non-empty locus in symmetric cones satisfies the WDVV equation, generalizing previous results.
New algorithm for online optimization over symmetric cones, unifying previous methods.
problem Online convex optimization over symmetric cones.
method Symmetric-Cone Multiplicative Weights Update (SCMWU) algorithm.
result SCMWU is a no-regret algorithm.
This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this …
Exact formulas for volumes of specific knot cone-manifolds.
problem Finding exact volumes of cone-manifolds with two-bridge knots.
method Provided exact integral formulas using Chebyshev polynomials and algebraic equations.
result Exact formulas for hyperbolic and spherical volumes of cone-manifolds.
We study constrained generalized Killing spinors over the metric cone and cylinder of a (pseudo-)Riemannian manifold, developing a toolkit which can be used to investigate certain problems arising in supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually…
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…
The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a `projective cone', a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying…
A new algebraic method extracts symmetry anomalies from 5D SCFTs.
problem Extracting global symmetry anomalies from 5D superconformal field theories.
method Path algebra of branes probing Calabi-Yau cones provides a complementary approach.
result Combinatorial approach to symmetry anomalies in 5D SCFTs.
Observing a linear superposition principle, a family of new minimal hypersurfaces in Euclidean space is found, as well as that linear combinations of generalized helicoids induce new algebraic minimal cones of arbitrarily high degree.
Let C be a cone in the space of algebraic curvature tensors. Moreover, let (M,g) be a compact Einstein manifold with the property that the curvature tensor of (M,g) lies in the cone C at each point on M. We show that (M,g) has constant sectional curvature if the cone C satisfies certain structure conditions.
Study on Lie groups with negative Ricci curvature, including open questions and a new cone.
problem Understanding Lie groups with negative Ricci curvature metrics.
method Overview and introduction of a new cone C(n) for solvable Lie algebras.
result Introduction of a new open cone C(n) that parametrizes solvable Lie algebras with negative Ricci curvature metrics.
The paper studies horofunction compactifications of symmetric cones under Finsler distances.
problem Understanding horofunction compactifications of symmetric cones under Finsler distances.
method Establishing a correspondence between horofunction compactifications of symmetric cones and normed spaces, using Thompson and Hilbert distances.
result Explicit extensions of the exponential map and characterizations of horofunctions for Thompson and Hilbert distances.
We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to…
We give a vertex algebra proof of the Berglund-Hübsch duality of nondegenerate invertible potentials. We suggest a way to unify it with the Batyrev-Borisov duality of reflexive Gorenstein cones.
The study connects curvature operators' positivity to manifold topology.
problem Positivity of curvature operators and their geometric implications.
method Analysis of Garding cones and positivity properties of curvature operators.
result Shifted cone conditions on curvature operators constrain manifold topology.
This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2π. We show that these are in a natrual way projective algebraic var…
Proves openness of balanced HKT cone and studies hyperholomorphic vector fields.
problem Understanding balanced HKT structures on compact hypercomplex manifolds.
method Analyzes Lie algebra of hyperholomorphic vector fields and proves harmonicity properties.
result Proves openness of balanced HKT cone and non-existence of certain fields.
In this paper we continue to study Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry. Our main focus is on the algebro-geometric meaning of Riemannian tangent cones and rescaled limits.
We determine the moduli space of metric 2-step nilpotent Lie algebras of dimension up to 6. This space is homeomorphic to a cone over a 4-dimensional contractible simplicial complex.
New tools for studying Hsiang algebras discovered, linking them to known algebraic structures.
problem Classifying Hsiang algebras and understanding their properties.
method Introducing quasicomposition and tripling constructions to study Hsiang algebras.
result The triple of a quasicomposition algebra is an exceptional Hsiang algebra.
Extends isoparametric foliations and area-minimizing cones in product manifolds.
problem Generalizing isoparametric foliations and area-minimizing cones in SnimesSn. method Analyzes isoparametric foliations and area-minimizing cones, extending known results.
result Extends known area-minimizing cones to codimension-two cases, yielding infinitely many families of area-minimizing subcones.
I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures of K-con…
Promotes Poisson deformations to hyperkähler structures.
problem Deforming hyperkähler cone metrics.
method Twistor methods for universal Poisson deformations.
result Produces incomplete hyperkähler metrics with applications.
Study extends JB-algebra structure group results to infinite dimensions.
problem Extend results for real Jordan algebras to infinite dimensional JB-algebras.
method Prove structure groups, cone preserving groups, and automorphism groups are embedded Banach-Lie groups; describe components via cones, isotopes, and central projections; apply to special JB-algebra of self-adjoint operators.
result Full description of components of structure group and automorphism group, including their Banach-Lie algebras and connected components.
We characterize H-like Lie algebras in terms of subspaces of cones over conjugacy classes in so(Rq), translating the classification problem for H-like Lie algebras to an equivalent problem in linear algebra. We study properties of H-like Lie algebras, present new methods for constructing them, in…