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F. Reese Harvey

Publications and source records attributed to F. Reese Harvey.

47 records · Page 3Linked to original sources
arXiv2007

Boundaries of Positive Holomorphic Chains and the Relative Hodge Question

We characterize the boundaries of positive holomorphic chains (with both compact and non-compact support) in an arbitrary complex manifold. We then consider a compact oriented real submanifold of dimension 2p-1 in a compact Kahler manifold X and address the question of which relative homology classes in H_{2p}(X,M;Z) are represented by positive holomorphic chains. Specifically, we define what it means for a class u in H_{2p}(X,M;Z) to be of type (p,p) and positive. It is then shown that u has these properties if and only if u = [T+S] where T is a positive holomorphic chain with dT = du and S is a positive (p,p)-current with dS = 0.

math.CV↗
arXiv2007

D-bar Sparks, I

A d-bar-analogue of differential characters for complex manifolds is introduced and studied using a new theory of homological spark complexes. Many essentially different spark complexes are shown to have isomorphic groups of spark classes. This has many consequences: It leads to an analytic representation of O*-gerbes with connection, it yields a soft resolution of the sheaf O* by currents on the manifold, and more generally it gives a Dolbeault-Federer representation of Deligne cohomology as the cohomology of certain complexes of currents. It is shown that the d-bar-spark classes H^*(X) carry a functorial ring structure. Holomorphic bundles have Chern classes in this theory which refine the integral classes and satisfy Whitney duality. A version of Bott vanishing for holomorphic foliations is proved in this context.

math.DG↗
arXiv2006

Remarks on the Alexander-Wermer Theorem for Curves

We give a new proof of the Alexander-Wermer Theorem that characterizes the oriented curves in C^n which bound positive holomorphic chains, in terms of the linking numbers of the curve with algebraic cycles in the complement. In fact, we establish a slightly stronger version which applies to a wider class of boundary 1-cycles. Arguments here are based on the Hahn-Banach Theorem and some geometric measure theory. Several ingredients in the original proof have been eliminated.

math.CV↗
arXiv2006

Projective Linking and Boundaries of Positive Holomorphic Chains in Projective Manifolds, Part II

In Part I of this paper we introduced the notion of the projective linking number Link(M,Z) of a compact oriented real submanifold M of dimension 2p-1 in complex projective n-space P^n with an algebraic subvariety Z of codimension p in P^n - M. It is shown here that a basic conjecture concerning the projective hull of real curves in P^2 implies the following result: M is the boundary of a positive holomorphic p-chain T of mass M(T) < K in P^n if and only if Link(M,Z) > -K p!deg(Z) for all algebraic subvarieties Z of codimension p in P^n - M. An analogous result is implied in any projective manifold X. The above theorem implies a duality between relative and absolute homology classes which are represented by positive holomorphic chains.

math.CV↗
arXiv2006

Plurisubharmonic Functions in Calibrated Geometries

In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. This paper investigates, in depth, questions of: pseudo-convexity and cores, positive phi-currents, Duval-Sibony Duality, and boundaries of phi-submanifolds, all in the context of a general calibrated manifold (X,phi). Analogues of totally real submanifolds are used to construct enormous families of strictly phi-convex spaces with every topological type allowed by Morse Theory. Specific calibrations are used as examples throughout. Analogues of the Hodge Conjecture in calibrated geometry are considered.

math.CV↗
arXiv2005

Lefschetz-Pontrjagin Duality for Differential Characters

A theory of differential characters is developed for manifolds with boundary. This is done from both the Cheeger-Simons and the deRham-Federer viewpoints. The central result of the paper is the formulation and proof of a Lefschetz-Pontrjagin Duality Theorem, which asserts that the pairing: Ch^k(X,dX) x Ch^{n-k-1}(X) --> S^1, given by (a,b) l-->(a*b)[X], induces isomorphisms: D : Ch^k(X,dX) --> Hom^{smooth}(Ch^{n-k-1}(X), S^1) D': Ch^{n-k-1}(X) --> Hom^{smooth}(Ch^k(X, dX), S^1) onto the smooth Pontrjagin duals. In particular, D and D' are injective with dense range in the group of all continuous homomorphisms into the circle. A coboundary map is introduced which yields a long sequence for the character groups associated to the pair (X,dX). The relation of the sequence to the duality mappings is analyzed.

math.DG↗
arXiv2005

Boundaries of Varieties in Projective Manifolds

We give a characterization of the boundaries of holomorphic chains in complex projective space in terms of certain non-linear moment conditions. This extends previous work of the authors and complements results of Dolbeault and Henkin.

math.CV↗
arXiv2005

The de Rham-Federer Theory of Differential Characters and Character Duality

The theory of differential characters is developed completely from a de Rham - Federer viewpoint. Characters are defined as equivalence classes of special currents, called sparks, which appear naturally in the theory of singular connections. There are many different spaces of currents which yield the character groups. A multiplication of de Rham-Federer characters is defined using transversality results for flat and rectifiable currents. There is a natural equivalence of ring functors from deRham - Federer characters to Cheeger-Simons characters. Differential characters have a topology and natural smooth Pontrjagin duals (introduced here). A principal result is the formulation and proof of duality for characters on oriented manifolds. It is shown that the pairing (a,b) --> a*b([X]) given by multiplication and evaluation on the fundamental cycle, gives an isomorphism of the group of differential characters of degree k with the dual to characters in degree n-k-1 where n = dim(X). It is also shown that there are natural Thom homomorphisms for differential characters. Gysin maps are also defined for differential characters. Many examples including Morse sparks, Hodge sparks and characteristic sparks are examined in detail.

math.DG↗
arXiv2005

Projective Hulls and the Projective Gelfand Transformation

We introduce the notion of a ``projective hull'' for subsets of complex projective varieties, parallel to the idea of the polynomial hull in affine varieties. With this concept, a generalization of J. Wermer's classical theorem on the hull of a curve in $C^n$ is established in the projective setting. The projective hull is shown to have interesting properties and is related to various extremal functions and capacities in pluripotential theory. A main analytic result asserts that for any point x in the projective hull K^ of a compact set K in $P^n$, there exists a positive current T of bidimension (1,1) with support in the closure of K^, and a probability measure $μ$ on K with $d d^cT=μ- [x]$. This result generalizes to any Kahler manifold and has strong consequences for the structure of K^. We also introduce the notion of a projective spectrum for Banach graded algebras, parallel to the Gelfand spectrum of a Banach algebra. This projective spectrum has universal properties exactly like those in the Gelfand case. Moreover, the projective hull is shown to play a role (for graded algebras) completely analogous to that played by the polynomial hull in the study of finitely generated Banach algebras. This paper gives foundations for generalizing many of the results on boundaries of varieties in $C^n$ to general algebraic manifolds.

math.CV↗
arXiv2001

Finite volume flows and Morse theory

In this paper we present a new approach to Morse theory based on the de Rham-Federer theory of currents. The full classical theory is derived in a transparent way. The methods carry over uniformly to the equivariant and the holomorphic settings. Moreover, the methods are substantially stronger than the classical ones and have interesting applications to geometry. They lead, for example, to formulae relating characteristic forms and singularities of bundle maps.

math.DG↗