scholarly journals A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

2011 ◽  
Vol 61 (5) ◽  
pp. 1749-1777 ◽  
Author(s):  
Ursula Ludwig
Keyword(s):  
Algebraic Curves ◽  
Morse Inequalities ◽  
Witten Deformation

scholarly journals WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

2021 ◽  
pp. 1-38
Author(s):  
Xianzhe Dai ◽  
Junrong Yan
Keyword(s):  
Essential Role ◽  
Morse Function ◽  
Noncompact Manifold ◽  
Morse Inequalities ◽  
Bounded Geometry ◽  
Noncompact Manifolds ◽  
Witten Laplacian ◽  
Relative Cohomology ◽  
Witten Deformation ◽  
Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

scholarly journals AN ANALYTIC APPROACH TO THE STRATIFIED MORSE INEQUALITIES FOR COMPLEX CONES

2013 ◽  
Vol 24 (12) ◽  
pp. 1350100
Author(s):  
URSULA LUDWIG
Keyword(s):  
Morse Theory ◽  
Singular Points ◽  
Analytic Approach ◽  
Morse Inequalities ◽  
Singular Spaces ◽  
Analytic Proof ◽  
Morse Functions ◽  
Complex Cone ◽  
Witten Deformation

In a previous paper 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 MacPherson. It is well-known from stratified Morse theory that the singular points of the complex cone contribute to the stratified Morse inequalities in middle degree only. In this paper, an analytic proof of this fact is given.

scholarly journals Novikov type inequalities for differential forms with non-isolated zeros

1997 ◽  
Vol 122 (2) ◽  
pp. 357-375 ◽  
Author(s):  
MAXIM BRAVERMAN ◽  
MICHAEL FARBER
Keyword(s):  
Critical Points ◽  
Differential Forms ◽  
Explicit Estimate ◽  
Morse Inequalities ◽  
De Rham Complex ◽  
Analytic Proof ◽  
Rham Complex ◽  
De Rham ◽  
Flat Bundle ◽  
Witten Deformation

We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R. Bott) and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. The proof uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian.In particular, we obtain a new analytic proof of the degenerate Morse inequalities of Bott.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

scholarly journals Real plane algebraic curves with prescribed singularities

Topology ◽  
1993 ◽  
Vol 32 (4) ◽  
pp. 845-856 ◽  
Author(s):  
Eugenii Shustin
Keyword(s):  
Algebraic Curves ◽  
Real Plane
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