The article proves a complex analytic inequality for stable Q-sheaves on Kähler varieties.
problem Proving a Bogomolov-Gieseker inequality for stable Q-sheaves on Kähler varieties.
method Complex analytic approach, including a new purely analytical proof and novel interpretation of orbifold Chern classes.
result Characterization of the equality case in the Bogomolov-Gieseker inequality and novel interpretation of the second orbifold Chern class.
Proves Kähler-Ricci shrinkers are complex analytic varieties.
problem Characterizing singular Kähler-Ricci shrinkers.
method Analyzes limits of Kähler-Ricci flows and applies algebraic geometry.
result Singular Kähler-Ricci shrinkers are locally algebraic complex-analytic varieties.
Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re(f) is Morse, we prove the existence of a stratified gradient-like vector field for Re(f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimensi…
By attaching a Lie algebra of germs of analytic vector fields to every point of a (real or complex) analytic variety V we construct the Nagano foliation of the variety. We prove that the Nagano foliation of V is a stratification. The treatment of the subject is totally coordinate free but relies on the Oka-Cartan-Serre…
We prove that a generic complex deformation of a generalized Kummer variety contains no complex analytic tori.
Explains complex analytic invariants of vector fields and foliations.
problem Integrating theories of singular varieties and foliations.
method Expository discussion of invariants.
result Introduces connections between complex analytic singular varieties and foliations.
For a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group) one has notions of the equivariant homological index and of the (reduced) equivariant radial index as elements of the ring of complex representations of th…
A geometric construction of Sullivan's Stiefel-Whitney homology classes of a real analytic variety X is given by means of the conormal cycle of an embedding of X in a smooth variety. We prove that the Stiefel-Whitney classes define additive natural transformations from certain constructible functions to homology. W…
For a lattice Γ of a simply connected solvable Lie group G, we describe the analytic germ in the variety of representations of Γ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by G. By this description, under…
A mapping bends Teichmüller spaces into character varieties, preserving symplectic structure.
problem Mapping Fricke-Teichmüller space to character variety of surface representations.
method Bending Fuchsian representations along a fixed measured lamination, proving equivariant symplectic embedding and properness.
result Continuous extension of bending map to Thurston boundary and geometric complexification.
Study non-existence of complex ball quotients in Torelli locus.
problem Non-existence of totally geodesic complex ball quotients in Torelli locus.
method Analytic techniques.
result Analytic techniques used to study non-existence.
We prove that refined analytic torsion on a manifold with boundary is an analytic section of the determinant line bundle over the representation variety. As a fundamental application we establish a gluing formula for refined analytic torsion on connected components of the complex representation space which contain a un…
We extend the holomorphic analytic torsion classes of Bismut and Köhler to arbitrary projective morphisms between smooth algebraic complex varieties. To this end, we propose an axiomatic definition and give a classification of the theories of generalized holomorphic analytic torsion classes for arbitrary projective mor…
Let X be a hyperkaehler manifold. Trianalytic subvarieties of X are subvarieties which are complex analytic with respect to all complex structures induced by the hyperkaehler structure. Given a 2-dimensional complex torus T, the Hilbert scheme T[n] classifying zero-dimensional subschemes of T admits a hype…
Proves properties of complex algebraic varieties and local systems.
problem Properties of complex algebraic varieties and local systems.
method Analytic Zariski open subsets and algebraic maps.
result Trivializing covering spaces of complex algebraic varieties.
The article examines twisted cohomologies on algebraic and analytic varieties.
problem Understanding and comparing twisted cohomologies on algebraic and analytic varieties.
method Comparison and definition of twisting parameters in both categories, algebraic and analytic.
result Reviewed isomorphisms of twisted cohomologies for cohomologous twisting parameters.
Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.
problem Understanding geometric structures of singular Kahler spaces.
method Proving RCD spaces homeomorphic to 3D projective varieties with bounded Nash entropy and Ricci curvature.
result Compact RCD spaces are equivalent to underlying projective varieties.
We show that a general n-dimensional polarized abelian variety (A,L) of a given polarization type and satisfying h0(A,L)≥28n⋅n!nn is projectively normal. In the process, we also obtain a sharp lower bound for the volume of a purely one-dimensional complex analytic subvariety i…
Study of Hermitian metrics on Lie algebroids over complex spaces.
problem Exploring Hermitian metrics on Lie algebroids over analytic spaces.
method Introduced Hermitian metrics on holomorphic Lie algebroids and examined associated characteristics foliations.
result Extended equivariant de Rham cohomology to Hermitian Lie algebroids.
This paper announces results on the behavior of some important algebraic and topological invariants --- Euler characteristic, arithmetic genus, and their intersection homology analogues; the signature, etc. --- and their associated characteristic classes, under morphisms of projective algebraic varieties. The formulas …
Lecture notes on using non-Archimedean geometry for complex variety degenerations.
problem Complex algebraic variety degenerations with non-Archimedean Berkovich spaces.
method Hybrid spaces and non-Archimedean pluripotential theory.
result Relation between convergence of psh metrics and Monge-Ampere measures in hybrid spaces.
Study pseudo-holomorphic disks in real analytic hypersurfaces using exterior differential systems.
problem Existence of pseudo-holomorphic disks in non-integrable real analytic hypersurfaces.
method Theory of exterior differential systems.
result Non-existence of certain equivalent structures in the non-integrable case.
We discuss replica analytic continuation using several simple models in order to prove mathematically the validity of replica analysis, which is used in a wide range of fields related to large scale complex systems. While replica analysis consists of two analytical techniques, the replica trick (or replica analytic con…
For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing 2-jets of the formal Segre varieti…
The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.
problem Identifying relationships between different geometric structures on complex varieties.
method Using perfect complexes and shifted symplectic geometries, the paper establishes a Lagrangian correspondence.
result A Lagrangian correspondence between shifted symplectic geometries of flat and Higgs perfect complexes.
Study on singularities of Chern-Ricci flow on complex manifolds.
problem Understanding finite-time singularities of the Chern-Ricci flow.
method Extending Guedj-Lu's approach to establish uniform a priori estimates for degenerate complex Monge-Ampère equations, applied to Chern-Ricci flows on complex log terminal varieties.
result Showed solutions starting from positive currents are smooth outside some analytic subset.
The paper proves a quadratic formality for Sasakian manifolds' representation varieties.
problem Analyzing the variety of representations of fundamental groups of Sasakian manifolds.
method Proving almost-formality of de Rham complex and vanishing cup product theorem.
result Quadratic formality of analytic germs of representation varieties.
We study the Gauss map and the dual variety of a real-analytic immersion of a connected compact real-analytic manifold into a sphere or into a hyperbolic space. The dual variety is defined to be the set of all normal directions of the immersion. First, we show that the image of the Gauss map characterizes the manifold.…
Let (X,h) be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of (X,h). In the fist part, assuming either dim(sing(X))=0 or dim(X)=2, we show that the rolled-up operator of the minimal L2-$\overline{\pa…
Study of twisted L2-torsion on 3-manifold character varieties.
problem Analyzing L2-torsion on character varieties of 3-manifolds. method Definition and study of twisted L2-torsion on character varieties, proving analyticity in hyperbolic cases. result Twisted L2-torsion is a real analytic function in a neighborhood of holonomy representations of hyperbolic 3-manifolds. Introduces stability conditions for polarized varieties, linking to K-stability.
problem Stability conditions for polarized varieties.
method Analogue of Bridgeland's stability for polarized varieties, Z-stability, Z-critical Kähler metrics.
result Polarized varieties with certain stability conditions admit Z-critical Kähler metrics.
Reconstructs fundamental groups from liquid local systems.
problem Reconstructing fundamental groups from category of liquid local systems.
method Theory of liquid vector spaces and liquid quasicoherent sheaves.
result Reconstructs topological fundamental group and twisted fundamental groupoids.
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 X⊂PN be a variety (respectively a patch of an analytic submanifold) and let x∈X be a general point. We show that if the projective second fundamental form of X at x is isomorphic to the second fundamental form of a point of a Segre Pn×Pm, n,m≥2, a Grassmaniann G(2,n+2), $n\geq 4…
New geometric conditions ensure compactness of ∂ˉ-Neumann problem.
problem Compactness of ∂ˉ-Neumann operator on specific domains. method Introduced new geometric conditions for a class of domains, proving compactness equivalence to boundary properties.
result Compactness of ∂ˉ-Neumann operator equivalent to boundary lack of analytic varieties. We study meromorphic actions of unipotent complex Lie groups on compact Kähler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable Kähler structures obtained by symplectic reduct…
The paper studies stability of CR structures on compact manifolds.
problem Stability of CR structures on compact manifolds.
method Smooth deformations and projective embeddings.
result Nearby structures still admit projective CR embeddings.
Abstract: Generalizes stability theories over toric varieties and Novikov type rings.
problem Stability theories over toric varieties and Novikov type base.
method Generalization of stability theories to families over toric varieties and their analytic analogues.
result Established properness of moduli of Calabi-Yau cones and Kahler-Ricci solitons.
Let X be a complex non-singular toric variety, E an equivariant and ample line bundle on X. We introduce a new functional V∞ on the set of smooth, positive and invariant metrics on E. We compare V∞ to some classical functionals. As an application, we study the variation of the holomorphic analyt…
Analyzes Kähler-Einstein metrics on families of Fano varieties.
problem Establishing Kähler-Einstein metrics on Fano varieties in families.
method Analytic method to show unique Kähler-Einstein metrics on neighboring fibers.
result Uniform a priori estimates and continuous variation of Kähler-Einstein potentials.
Characterizes metabelian distributions and geodesics in sub-Riemannian manifolds.
problem Characterizing metabelian distributions and geodesics in sub-Riemannian manifolds.
method Characterization of metabelian distributions in terms of principal bundle structures. Proof of geodesic properties for rank-2 distributions.
result For rank-2 metabelian distributions, geodesics are of class C1. Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension d admits, away from a closed analytic subset of positive codimension, …
The paper describes how Hodge loci are typically equidistributed in complex varieties.
problem Understanding the distribution of Hodge loci in complex varieties.
method Analyzing polarized variations of Hodge structures over smooth complex quasi-projective varieties.
result Hodge loci are either empty or equidistributed with respect to a pull-push form.
Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in areas like data mining, computer…
Holomorphic torsion invariant for log-Enriques surfaces derived from Borcherds products.
problem Holomorphic torsion invariant for log-Enriques surfaces.
method Introduced a holomorphic torsion invariant using Borcherds products.
result The invariant is given by the Petersson norm of an explicit Borcherds product.
Paper connects geometric and analytic aspects of Higgs bundles and pleated surfaces.
problem Understanding the asymptotic geometry of character varieties.
method Nonabelian Hodge correspondence and study of harmonic maps.
result Asymptotic correspondence between limiting configurations and geometric parameters.
Holomorphic map connects Hitchin components to character varieties.
problem Complex affine spheres and their properties.
method Mapping class group equivariant holomorphic map from Hitchin components to character varieties.
result Holomorphic map includes holonomies of SL(3,C)-opers.
To any completely integrable second-order system of real or complex partial differential equations in n > 1 independent variables and in one dependent variable, Mohsen Hachtroudi associated in 1937 a normal projective (Cartan) connection, and he computed its curvature. By means of a natural transfer of jet polynomials …