F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. of Math. 135 (1992), 567-600.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in L 1 loc (X) and OE( Omega 0 ) is an L 1 loc (X) form canonically defined in terms of a singular connection naturally associated to . This result holds for C 1 meromorphic sections which are atomic. The notion of an atomic section of a vector bundle was first introduced and studied in [HS]. The formulae obtained include a generalization of the Poincar e Lelong formula to C 1 meromorphic sections of a bundle of arbitrary rank. Introduction. The main result of this paper is a generalization of the following version of the argument principle for smooth maps from a manifold into ....

....that bundle. In this paper we generalize their work by computing the currents associated to C 1 meromorphic sections of a vector bundle. The analytic underpinning for [HL] and for the results described here, is given in detail in Harvey and Semmes paper Zero divisors of atomic functions , [HS]. Let F X be a rank n complex vector bundle over an oriented manifold, and let OE be an Ad invariant polynomial on gl(n; C ) For each choice of connection on F the Chern Weil theory constructs a d closed differential form OE( Omega F ) on X, called the OE Chern form, which represents ....

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F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. Math. 135 (1992), 567--600.

....ff : E F over X is the pull back ff = ff (ffffff) of ff ff ff. For these reasons ff ff ff is called the universal mapping. There are analogous universal mappings of direct sum and tensor product type. Historical Comment: Notions of Atomicity. The first notion of atomicity, introduced in [HS] applied only to sections of a vector bundle. It consisted of analytic conditions on the section under which its vanishing determines a well defined, integrally flat current dual to the Euler class. In [HL 2 ] the notion of k atomicity was introduced for bundle maps ff : E F (for 0 k rank ....

.... F is geometrically atomic. This is proved below. 6 Note that sections of a vector bundle F are in natural one to one correspondence with bundle mappings ff : K F where K is the trivial (real or complex) line bundle. Example 1.4. Any section of a vector bundle which is atomic in the sense of [HS] is geometrically atomic. See [HL 6 ] Example 1.5. Any normal bundle map is geometrically atomic. Such maps are open and dense in the C topology. See x3 for details. Example 1.6. Suppose that ff : E F is a geometrically atomic bundle map, and suppose there exists a map ae : Hom(E;F ) ....

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F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. of Math. 135 (1992), 567-600.

....F is geometrically atomic. This is proved below. Note that sections of a vector bundle F are in natural one to one correspondence with bundle mappings ff : K F where K is the trivialized (real or complex) line bundle. Example 1.4. Any section of a vector bundle which is atomic in the sense of [HS] is geometrically atomic. See [HL 6 ] Example 1.5. Any normal bundle map (See x3) is geometrically atomic. Example 1.6. Suppose that ff : E F is a geometrically atomic bundle map, and suppose there exists a map ae : Hom(E;F ) Gamma Hom(E 0 ; F 0 ) with the properties that for each x ....

.... ) v 10 2 K Omega K 0 oe Phi ae (v 01 ; 1 t L j Omega a 0 j (v 01 ) v 01 2 K Omega K 0 oe Phi ae (v 11 ; 1 t a j Omega a 0 j (v 11 ) v 11 2 K Omega K 0 oe The support of the current P kk 0 will lie the set of limit points of such sequences of graphs of 1 t j ff j Omega ff 0 j where t j 0. Setting ff j = t j L; a) ff j = t j L 0 ; a 0 ) and sending t j 0 gives planes of the form (5.6) We shall show that all other limit points lie in a subanalytic set B with the property that (5.9) dim(B) dimfHom(K; I ) Theta Hom(K ....

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F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. of Math. 135 (1992), 567-600.

....a formula: Phi( Omega 1 ) Gamma Phi( Omega 2 ) dT , where Omega i is the curvature of the i th connection and T is a canonically defined smooth form. The gaugeinvariant forms T are important in the study of the space of connections and they lead to well known secondary invariants [CS] ChS] The aim here is to combine these results and derive MacPherson type formulas locally on the manifold. Assume bundles E and F are equipped with metrics and connections, and let ff : E F be a smooth bundle map. We shall derive formulas which explicitly express each Chern Weil form ....

....It holds for normal bundle maps and for all real analytic bundle maps. Furthermore, it cuts robustly across the cases mentioned above. Within each special case the geometrically atomic maps are generic. The concept of geometric atomicity strictly generalizes the notion of atomicity introduced in [HS] that is, any section ff : R F which is atomic is geometrically atomic. Furthermore, there is an analytic criterion analogous to that in [HS] which implies geometric atomicity. This will be discussed in part II. A basic feature of geometric atomicity is that it enables the construction of ....

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F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. of Math. 135 (1992), 567-600.

....the behaviour of the section in a neighbourhood of its zero and pole sets to the topology of the bundle. Our results are analogous to those obtained in the study of meromorphic sections of complex vector bundles (see [Z] The results build on the work of Harvey, Lawson and Semmes (see [HL] [HS]) who studied smooth sections of vector bundles (i.e. no pole set) The formulae obtained can be applied to study the differential topology of maps into spheres and real projective spaces. For instance we will prove the following result about maps, such as the Gauss map, from an n dimensional ....

....is the inverse image of the zero section of V and the pole set P is the inverse image of spatial infinity, P(V ) Note that generically Z has codimension n and P has codimension 1 in X. Harvey and Semmes defined the zero divisor, Div 0 ( of an atomic section of the vector bundle V X (see [HS] or Definition 1.4 below) The zero divisor is a codimension n current on X which is supported on the zero set Z of and encodes the multiplicity of vanishing of on Z. If 0 is a regular value of then is atomic and Div 0 ( Z] is the current of integration over the (suitably oriented) ....

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F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. Math. 135 (1992), 567--600.

....integrally flat currents and their cohomological properties. This simple approach is via sheaf theory and is quite distinct from the form of the theory given in the literature. The theory of dependency currents relies heavily on the theory of zero divisor currents which was originally developed in [HS] for atomic sections of an oriented vector bundle over an oriented manifold. The notion of an atomic section provides a generalization of the notion of a section being transverse to zero, one which is both useful and vastly more general. The zero divisor is a d closed locally integrally flat ....

....but disagrees with that in [Z1,2] A. Divisors in the nonorientable case. In this subsection we define and study the divisor of a section of V X. The divisor is defined to be an O V twisted current. Note that, if V and X are oriented, the definition of divisor given below agrees with that of [HS]. The solid angle kernel, is the L 1 loc form on R n obtained by pulling back the normalized volume form on the unit sphere to R n f0g by the radial projection map. The current equation d = 0] on R n , where [0] denotes the point mass at the origin, motivates the definition of ....

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F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. Math. 135 (1992), 567--600.

....the behaviour of the section in a neighbourhood of its zero and pole sets to the topology of the bundle. Our results are analogous to those obtained in the study of meromorphic sections of complex vector bundles (see [Z] The results build on the work of Harvey, Lawson and Semmes (see [HL] [HS]) who studied smooth sections of vector bundles (i.e. no pole set) The formulae obtained can be applied to study the differential topology of maps into spheres and real projective spaces. For instance we will prove the following theorem. Theorem. Let X be a compact oriented n dimensional ....

....g S: 4) Let , j be O(V ) twisted forms on X. Then p ( j) e e j: 5) Let be an O(V ) twisted form on X and S an odd current on e X. Then p (e S) e p (S) 2. Divisors of atomic sections of a nonorientable bundle. In this section we apply the work of Harvey and Semmes [HS] to define the notion of an atomic section of the projective compactification of a bundle V X and define the zero and pole divisors of such a section. To do this we are forced to make sense of the divisor of a section of a nonorientable vector bundle. In [HS] orientability assumptions were ....

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F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. Math. 135 (1992), 567--600.

....AND EULER SPARKS OF ATOMIC SECTIONS F. Reese Harvey and John Zweck The purpose of this paper is twofold. The first purpose is to give a simple treatment of the basic theory of atomic sections and their zero currents or divisors. This theory (see [HS,HL1]) places the classical theory of characteristic classes on a new and more general analytic footing based on de Rham currents and geometric measure theory. The second purpose is to discuss an instance of the Cheeger Simons theory of differential characters from this analytic perspective. ....

....crucial ingredients in the development of this theory was the concept of the divisor, or zero current, denoted by Div(ff) of a section ff of a vector bundle. Sections which have divisors are called atomic sections. The precise definitions of atomic section and divisor current were introduced in [HS]. An abundance of examples of this Chern Weil current theory were presented as geometric residue theorems in [HL2] These examples all reduced to the Euler case (i.e. to the case of zero currents of sections of vector bundles) using methods from intersection theory (see [Fu] but in a C 1 ....

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F.R. Harvey and S. Semmes, Zero divisors of atomic functions, Ann. Math. 135 (1992), 567--600.

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