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.

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 Counting and Discrete Morse Theory

2019 ◽  
Vol 21 (1) ◽  
Author(s):  
Andrew Sack
Keyword(s):  
Morse Theory ◽  
Vector Fields ◽  
Discrete Morse Theory ◽  
Gradient Vector ◽  
Morse Functions

We examine enumerating discrete Morse functions on graphs up to equivalence by gradient vector fields and by restrictions on the codomain.  We give formulae for the number of discrete Morse functions on specific classes of graphs (line, cycle, and bouquet of circles).

scholarly journals Some examples of minimally degenerate Morse functions

1987 ◽  
Vol 30 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Frances Kirwan
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Morse Function ◽  
Compact Riemannian Manifold ◽  
Connected Components ◽  
Morse Inequalities ◽  
Poincaré Polynomial ◽  
Morse Functions ◽  
Image Position

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

Comparison between two complexes on a singular space

2017 ◽  
Vol 2017 (724) ◽  
pp. 1-52 ◽  
Author(s):  
Ursula Ludwig
Keyword(s):  
Stratified Spaces ◽  
Morse Functions ◽  
Singular Space ◽  
Witten Deformation

AbstractIn this article we generalise the Witten deformation for stratified spaces and a class of Morse functions which we call radial Morse functions. In the first part of the article we perform the Witten deformation on the complex of

scholarly journals The Pila–Wilkie theorem for subanalytic families: a complex analytic approach

2017 ◽  
Vol 153 (10) ◽  
pp. 2171-2194 ◽  
Author(s):  
Gal Binyamini ◽  
Dmitry Novikov
Keyword(s):  
Recent Result ◽  
Rational Points ◽  
Analytic Approach ◽  
Inductive Step ◽  
Elementary Functions ◽  
Analytic Set ◽  
Analytic Proof ◽  
Algebraic Hypersurfaces ◽  
Subanalytic Sets

We present a complex analytic proof of the Pila–Wilkie theorem for subanalytic sets. In particular, we replace the use of $C^{r}$-smooth parametrizations by a variant of Weierstrass division. As a consequence we are able to apply the Bombieri–Pila determinant method directly to analytic families without limiting the order of smoothness by a $C^{r}$ parametrization. This technique provides the key inductive step for our recent proof (in a closely related preprint) of the Wilkie conjecture for sets definable using restricted elementary functions. As an illustration of our approach we prove that the rational points of height $H$ in a compact piece of a complex-analytic set of dimension $k$ in $\mathbb{C}^{m}$ are contained in $O(1)$ complex-algebraic hypersurfaces of degree $(\log H)^{k/(m-k)}$. This is a complex-analytic analog of a recent result of Cluckers, Pila, and Wilkie for real subanalytic sets.

scholarly journals Discrete Morse Inequalities on Infinite Graphs

10.37236/127 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Rafael Ayala ◽  
Luis M. Fernández ◽  
José A. Vilches
Keyword(s):  
Morse Function ◽  
Special Kind ◽  
Infinite Graphs ◽  
Morse Inequalities ◽  
Morse Functions ◽  
Critical Elements ◽  
Finite Case ◽  
Discrete Morse Function

The goal of this paper is to extend to infinite graphs the known Morse inequalities for discrete Morse functions proved by R. Forman in the finite case. In order to get this result we shall use a special kind of infinite subgraphs on which a discrete Morse function is monotonous, namely, decreasing rays. In addition, we shall use this result to characterize infinite graphs by the number of critical elements of discrete Morse functions defined on them.

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