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.
New proof shows all conformal vector fields on complex hyperbolic space are Killing.
problem Proving all conformal vector fields on complex hyperbolic space are Killing.
method Local, analytic, and constructive approach using Lie group model and partial differential equations.
result Every conformal vector field on complex hyperbolic space is Killing.
We prove that a real-valued function (that is not assumed to be continuous) on a real analytic manifold is analytic whenever all its restrictions to analytic submanifolds homeomorphic to the 2-sphere are analytic. This is a real analog for the classical theorem of Hartogs that a function on a complex manifold is comple…
Paper derives analytical formulas for NLD-CEV moments with regime switching.
problem Analytical tractability of NLD-CEV models under stochastic regimes.
method Hybrid system approach using Feynman-Kac formula for solving interconnected PDEs.
result Exact closed-form expressions for fractional-order conditional moments.
We present the complex analytic and principal complex analytic realizability of a link in a 3-manifold M as a tool for understanding the complex structures on the cone C(M).
In a previous article the author extended the Witten deformation to singular spaces with cone-like singularities and to a class of Morse functions called admissible Morse functions. The method applies in particular to complex cones and stratified Morse functions in the sense of the theory developed by Goresky and MacPh…
This note provides a new proof of the real analyticity of the Liouville map.
problem Real analyticity of the Liouville map on Riemann surfaces.
method Complex analysis approach.
result Real analyticity of the Liouville map proved using complex analysis.
We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analyti…
In the case of a compact real analytic symplectic manifold M we describe an approach to the complexification of Hamiltonian flows [Se, Do1, Th1] and corresponding geodesics on the space of Kahler metrics. In this approach, motivated by recent work on quantization, the complexified Hamiltonian flows act, through the Gro…
Complex analytic sets' Lipschitz geometry at infinity characterized.
problem Characterize entire complex analytic sets based on their Lipschitz geometry at infinity.
method Proved a complex non-parametric version of Moser's Bernstein Theorem and characterized algebraicity.
result Entire complex analytic sets at infinity are affine linear subspaces if and only if they are bi-Lipschitz homeomorphic to algebraic sets.
Call a smooth knot (or smooth link) in the unit sphere in C2 analytic (respectively, smoothly analytic) if it bounds a complex curve (respectively, a smooth complex curve) in the complex ball. Let K be a smoothly analytic knot. For a small tubular neighbourhood of K we give a sharp lower bound for the 4…
Paper generalizes a theorem for real analytic singularities.
problem No specific problem stated; focuses on generalization.
method Generalization of a theorem for complex singularities.
result Generalized Join theorem for real analytic singularities.
It is a classical result that any complex analytic Lie supergroup G is split \cite{kosz}, that is its structure sheaf is isomorphic to the structure sheaf of a certain vector bundle. However, there do exist non-split complex analytic homogeneous supermanifolds. We study the question how to find out whether …
We deal with smooth real manifolds as well as complex analytic manifolds as well. It is well known that the concept of star product is powerful enough to produce all Poisson structures on real manifolds. According to [BdM] it is not known whether holomorphic star products exist on complex analytic manifolds. The main p…
In this paper, we study transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs, aiming to establish an analytic theory for log canonical singularities. As a result, in the case of purely log terminal pairs, we give an analytic proof of the injectivity theorem originally proved by the …
In this paper, we extend the Burghelea-Haller analytic torsion to the twisted de Rham complexes. We also compare it with the twisted refined analytic torsion defined by Huang.
Develops Bialynicki-Birula and Morse-Bott theory for complex analytic spaces.
problem Extending classical theories to complex analytic spaces with holomorphic C∗ actions. method Extends Bialynicki-Birula and Morse-Bott theories to non-compact complex manifolds and analytic spaces, proving existence and deriving geometric consequences.
result Existence of Bialynicki-Birula decompositions for C∗-invariant subspaces in complex manifolds. Examines properties of holomorphic fibrations in complex geometry.
problem Holomorphic fibrations in complex geometry.
method Analyzes properties of holomorphic fibrations.
result Provides insights into the structure of holomorphic fibrations.
In [1], we have explored the theoretical aspects of feature selection and evolutionary algorithms. In this chapter, we focus on optimization algorithms for enhancing data analytic process, i.e., we propose to explore applications of nature-inspired algorithms in data science. Feature selection optimization is a hybrid …
Recently, Cappell and Miller extended the classical construction of the analytic torsion for de Rham complexes to coupling with an arbitrary flat bundle and the holomorphic torsion for ∂ˉ-complexes to coupling with an arbitrary holomorphic bundle with compatible connection of type (1,1). Cappell and Mil…
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…
Unified framework for complex, split-complex, and dual numbers.
problem Analytic and geometric scope of real-analytic functions.
method Generalized Cauchy-Riemann structure and unified real algebra family.
result Milnor-Le type fibration theorem for nondegenerate algebras.
New trading model uses self-organization for real-world complexity.
problem Traditional trading models are too analytical for real-world conditions.
method Embraces self-organization, complex systems theory, and event-based time.
result Example Delta Engine algorithm demonstrates effectiveness.
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.
Let E be a flat complex vector bundle over a closed oriented odd dimensional manifold M endowed with a flat connection ∇. The refined analytic torsion for (M,E) was defined and studied by Braverman and Kappeler. Recently Mathai and Wu defined and studied the analytic torsion for the twisted de Rham complex…
The paper proves conditions for compact complex manifolds to be Kahler outside analytic subsets.
problem Conditions for compact complex manifolds to be Kahler outside an analytic subset.
method Analyzes balanced manifolds and uses Hironaka's examples to prove theorems.
result Compact complex manifolds that are Kahler outside an analytic subset are balanced.
Analytic realization of Thom-Smale complex for G-manifolds.
problem Realizing Thom-Smale complex for G-manifolds with Lie group action.
method Using G-invariant Witten instanton complex associated with a Morse-Bott function.
result Generalized Thom-Smale complex for G-manifolds including horizontal direction influence.
Improved neural network predicts spectral functions more accurately than traditional methods.
problem Reconstructing real-time spectral functions from imaginary-time Green's functions is ill-posed and challenging.
method Feature Learning Network (FL-net) for enhanced prediction accuracy.
result FL-net achieves at least 20% improvement over traditional methods like MEM.
Analytic torsion expansions for symmetric and complex homogeneous spaces.
problem Calculating the full asymptotic expansion of analytic torsion for various spaces.
method Explicit calculation and comparison with existing results.
result Explicit full asymptotic expansions for symmetric and complex homogeneous spaces.
Formula connects analytic torsion forms of fibration and its pieces.
problem Analytic torsion forms of a fibration and its pieces.
method Adiabatic limit and Witten deformation on flat vector bundle.
result Gluing formula relating analytic torsion forms.
We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification, existence of normal forms, pencil/fibration decomposition, infinitesimal symmetries. We…
We explicitly write down all eigenvalues of the Rumin Laplacian on the standard contact spheres, and express the analytic torsion functions associated with the {Rumin complex} in terms of the Riemann zeta function. In particular, we find that the functions vanish at the origin and determine the analytic torsions.
This paper deals with the question of analytic continuation of holonomy germs of holomorphic foliations. We prove that for a quasi-minimal Riccati foliation of the complex projective plane, any holonomy germ of the foliation between complex projective lines can be analytically continued along a generic Brownian path.
An important part of the classical theory of real or complex manifolds is the theory of (smooth, real analytic or complex analytic) vector bundles. With any vector bundle over a manifold (M,F) the sheaf of its (smooth, real analytic or complex analytic) sections is associated which is a locally free sheaf of F-modules,…
We prove that for every analytic curve in the complex plane, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Euc…
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 derive simple forms for saddle-node singular points of analytic foliations in the real or complex plane just by gluing foliated complex manifolds. We give the versal analytic deformation of the simplest model. We also derive a unique analytic form for those saddle-node having a central manifold. By this way, we reco…
In this paper we show that the Ray-Singer complex analytic torsion is trivial for even dimensional Calabi-Yau manifolds. Then we define the quaternionic analytic torsion for quaternionic manifolds and prove that they are metric independent. In dimension four, the quaternionic analytic torsion equals to the self-dual an…
Analytic torsion defined for rank 2 distributions on 5-manifolds.
problem Defining and analyzing analytic torsion for rank 2 distributions.
method Proposed an analytic torsion for Rumin complex associated with rank 2 distributions on 5-manifolds, established anomaly formulas, and showed coincidence with Ray-Singer torsion.
result The proposed torsion coincides with Ray-Singer torsion for certain nilmanifolds.
Analytic torsion matches Ray-Singer for specific nilmanifolds.
problem Matching analytic torsion with Ray-Singer in specific nilmanifolds.
method Examined rank two distributions on 5D nilmanifolds, proving torsion equality.
result Analytic torsion equals Ray-Singer torsion in these specific nilmanifolds.
Authors discuss complex and non-Archimedean geometry, proving a conjecture.
problem Proving a version of the Yau--Tian--Donaldson conjecture for Kähler metrics.
method Relation between complex, analytic, and non-Archimedean geometry.
result Sketch of proof for Yau--Tian--Donaldson conjecture.
We construct examples of C∞ smooth submanifolds in Cn and Rn of codimension 2 and 1, which intersect every complex, respectively real, analytic curve in a discrete set. The examples are realized either as compact tori or as properly imbedded Euclidean spaces, and are the graphs of quasianaly…
We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. As an application, we extend some known results concerning existen…
Paper constructs an invariant for a specific type of complex manifolds.
problem Analyzing an invariant for a specific class of complex manifolds.
method Using equivariant analytic torsion, the paper constructs an invariant for irreducible holomorphic symplectic manifolds with antisymplectic involution.
result A formula for the complex Hessian of the logarithm of the invariant is provided.
We extend the complex-valued analytic torsion, introduced by Burghelea and Haller on closed manifolds, to compact Riemannian bordisms. We do so by considering a flat complex vector bundle over a compact Riemannian manifold, endowed with a fiberwise nondegenerate symmetric bilinear form. The Riemmanian metric and the bi…
In this paper, we define a concept of a family of compact holomorphic Poisson manifolds on the basis of Kodaira-Spencer's deformation theory and deduce the integrability condition. We prove an analogue of their `Theorem of existence for complex analytic structures' under some analytic assumption, and establish an analo…
In the spirit of Ray and Singer we define a complex valued analytic torsion using non-selfadjoint Laplacians. We establish an anomaly formula which permits to turn this into a topological invariant. Conjecturally this analytically defined invariant computes the complex valued Reidemeister torsion, including its phase. …
Complex valued analytic torsion and dynamical zeta function studied on locally symmetric spaces.
problem Analyzing the Ruelle dynamical zeta function on locally symmetric spaces with flat vector bundles.
method Meromorphic extension and regularisation of the dynamical zeta function, relating it to the complex valued analytic torsion.
result The leading term of the dynamical zeta function at zero is related to the regularised determinant of the flat Laplacian.