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Register a new userDimensions of the Ascending and Descending Sets in Complex Stratified Morse Theory
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Authors
Mikhail Grinberg
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We present a new construction of gradient-like vector fields in the setting of Morse theory on a complex analytic stratification. We prove that the ascending and descending sets for these vector fields possess cell decompositions satisfying the dimension bounds conjectured by M. Goresky and R. MacPherson. Similar results by C.-H. Cho and G. Marelli have recently appeared in arXiv:0908.1862.
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We develop the idea of self-indexing and the technology of gradient-like vector fields in the setting of Morse theory on a complex algebraic stratification. Our main result is the local existence, near a Morse critical point, of gradient-like vector fields satisfying certain ``stratified dimension bounds up to fuzz for the ascending and descending sets. As a global consequence of this, we derive the existence of self-indexing Morse functions.
We analyze the axiomatic strength of the following theorem due to Rival and Sands in the style of reverse mathematics. Every infinite partial order $P$ of finite width contains an infinite chain $C$ such that every element of $P$ is either comparable with no element of $C$ or with infinitely many elements of $C$. Our main results are the following. The Rival-Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $mathsf{I}Sigma^0_2 + mathsf{ADS}$ over $mathsf{RCA}_0$. For each fixed $k geq 3$, the Rival-Sands theorem for infinite partial orders of width $leq! k$ is equivalent to $mathsf{ADS}$ over $mathsf{RCA}_0$. The Rival-Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $mathsf{SADS}$ over $mathsf{RCA}_0$. Here $mathsf{RCA}_0$ denotes the recursive comprehension axiomatic system, $mathsf{I}Sigma^0_2$ denotes the $Sigma^0_2$ induction scheme, $mathsf{ADS}$ denotes the ascending/descending sequence principle, and $mathsf{SADS}$ denotes the stable ascending/descending sequence principle. To our knowledge, the
We establish pointwise and distributional fractal tube formulas for a large class of relative fractal drums in Euclidean spaces of arbitrary dimensions. A relative fractal drum (or RFD, in short) is an ordered pair $(A,Omega)$ of subsets of the Euclidean space (under some mild assumptions) which generalizes the notion of a (compact) subset and that of a fractal string. By a fractal tube formula for an RFD $(A,Omega)$, we mean an explicit expression for the volume of the $t$-neighborhood of $A$ intersected by $Omega$ as a sum of residues of a suitable meromorphic function (here, a fractal zeta function) over the complex dimensions of the RFD $(A,Omega)$. The complex dimensions of an RFD are defined as the poles of its meromorphically continued fractal zeta function (namely, the distance or the tube zeta function), which generalizes the well-known geometric zeta function for fractal strings. These fractal tube formulas generalize in a significant way to higher dimensions the corresponding ones previously obtained for fractal strings by the first author and van Frankenhuijsen and later on, by the first author, Pearse and Winter in the case of fractal sprays. They are illustrated by several interesting examples. These examples include fractal strings, the Sierpinski gasket and the 3-dimensional carpet, fractal nests and geometric chirps, as well as self-similar fractal sprays. We also propose a new definition of fractality according to which a bounded set (or RFD) is considered to be fractal if it possesses at least one nonreal complex dimension or if its fractal zeta function possesses a natural boundary. This definition, which extends to RFDs and arbitrary bounded subsets of $mathbb{R}^N$ the previous one introduced in the context of fractal strings, is illustrated by the Cantor graph (or devils staircase) RFD, which is shown to be `subcritically fractal.
In 2009, the first author introduced a new class of zeta functions, called `distance zeta functions, associated with arbitrary compact fractal subsets of Euclidean spaces of arbitrary dimension. It represents a natural, but nontrivial extension of the theory of `geometric zeta functions of bounded fractal strings. In this memoir, we introduce the class of `relative fractal drums (or RFDs), which contains the classes of bounded fractal strings and of compact fractal subsets of Euclidean spaces as special cases. Furthermore, the associated (relative) distance zeta functions of RFDs, extend (in a suitable sense) the aforementioned classes of fractal zeta functions. This notion is very general and flexible, enabling us to view practically all of the previously studied aspects of the theory of fractal zeta functions from a unified perspective as well as to go well beyond the previous theory. The abscissa of (absolute) convergence of any relative fractal drum is equal to the relative box dimension of the RFD. We pay particular attention to the question of constructing meromorphic extensions of the distance zeta functions of RFDs, as well as to the construction of transcendentally $infty$-quasiperiodic RFDs (i.e., roughly, RFDs with infinitely many quasiperiods, all of which are algebraically independent). We also describe a class of RFDs (and, in particular, a new class of bounded sets), called {em maximal hyperfractals}, such that the critical line of (absolute) convergence consists solely of nonremovable singularities of the associated relative distance zeta functions. Finally, we also describe a class of Minkowski measurable RFDs which possess an infinite sequence of complex dimensions of arbitrary multiplicity $mge1$, and even an infinite sequence of essential singularities along the critical line.
A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere $S^2$ of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold $M$ and a generic complex structure $Lin S^2$, the complex manifold $(M,L)$ has algebraic dimension 0. A stronger result is proven when the hypercomplex nilmanifold is abelian. Consider the Lie algebra of left-invariant vector fields of Hodge type (1,0) on the corresponding nilpotent Lie group with respect to some complex structure $Iin S^2$. A hypercomplex nilmanifold is called abelian when this Lie algebra is abelian. We prove that all complex subvarieties of $(M,L)$ for generic $Lin S^2$ on a hypercomplex abelian nilmanifold are also hypercomplex nilmanifolds.
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