By Bruno Harris
This topic has been of significant curiosity either to topologists and to quantity theorists. the 1st a part of this booklet describes the various paintings of Kuo-Tsai Chen on iterated integrals and the elemental crew of a manifold. the writer makes an attempt to make his exposition obtainable to starting graduate scholars. He then proceeds to use Chen’s buildings to algebraic geometry, exhibiting how this results in a few effects on algebraic cycles and the Abel–Jacobi homomorphism. ultimately, he offers a extra basic perspective touching on Chen’s integrals to a generalization of the idea that of linking numbers, and finally ends up with a brand new invariant of homology periods in a projective algebraic manifold. The booklet is predicated on a path given through the writer on the Nankai Institute of arithmetic within the fall of 2001.
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Additional info for Iterated Integrals and Cycles on Algebra
2 Consider triples of disjoint simple closed curves C1,C2, C3 as above and write P = A3(Xi)',C1 A C 2 A C 3 E P instead of C Y ~ A C Y ~ EA P . Such "decomposable elements" generate A3(3Ci)' = P . Proof: Let Al, B1, . , A g ,B, be standard simple closed curves in X , where we think of Ai as going around the i-th hole, and Bi going around the corresponding i-th handle with intersection number Ai . Bi = 1 and all other pairs being disjoint. Consider three types of elements Ci A cj A cr,: a) 1 5 i < j < k 5 g and each C1 = Al or Bl.
0 We come back to our program of constructing homomorphisms of Lie algebras, associative algebras, and Hopf algebras which we will show are isomorphisms. 6 Some work of Quillen Next, following Quillen and Lazard, for any group 7r we consider its descending central series and associated graded abelian group Grn, which is a Lie algebra (using group commutator in n). 20 Iterated Integrals and Cycles o n Algebraic Manifolds s, The function q : 7r 4 q(y) = y - 1 induces a homomorphism Q of a Lie algebra to an associative algebra Gr7r t GrR7r and so a homomorphism Q : U(Gr7r8 R) -+Gr(R7r) of Hopf algebras.
Equivalently, w(v, Jw)= g(v,w ) . Then, w(w,v) = -w(v, w) because W ( U , v) = -g(v, Jv) = -g(Jv, J 2 v ) = g(Jv,v) = g(v,J v ) = 0. Note that if v # 0, g(v,v) > 0 is equivalent to w(v, Jv) > 0: this says that w restricted to the 2-dimensional space with ordered basis v, J v is an orientation. In C" with Euclidean g as before, w(&, &)= 1 (and w = 0 on other pairs) says that w = d x l A d y l . * d x , A dy,. 4 A complex manifold ( X , J ) with Riemannian metric g such that J preserves g, is called a Kahler manifold (or has a "Kahler structure") if the 2-form w(v,w ) = -g(v, J w ) is a closed 2-form: dw = 0.
Iterated Integrals and Cycles on Algebra by Bruno Harris
Categories: Algebraic Geometry