By Roger A. Johnson

This vintage textual content explores the geometry of the triangle and the circle, focusing on extensions of Euclidean thought, and interpreting intimately many really fresh theorems. numerous hundred theorems and corollaries are formulated and proved thoroughly; various others stay unproved, for use via scholars as routines. 1929 variation.

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1. Chern classes of vector bundles. X, and Chern classes of vector bundles on X should lie in A· X. Although such theories exist, at this time there is not yet a simple geometric construction of such a cohomology theory. Indeed, it would be extremely useful to have such a theory, perhaps analogous to Go~'s realization of ordinary cohomology via "geometric coqcles" [26]. () At any rate, any such theory should have "c~~ts" A'X ® AkX .... Ak_,X, and Chern classes c,(£) E A'X. In particular, a bundle £ on X should determine homomorphisms - - - - c,(E)()- ~ Ak_iX, -> by a ....

E. regular sequences locally defining X in Y remain regular sequences on V, we recover again the formula X· V = [W]. There is a simple but important refinement of these constructions and formulas. The morphism g: V ..... Y can be an arbitrary morphism; it need not be a closed imbedding. Defining W to be the inverse image scheme g-I( X), h: W ..... X the induced morphism, one still has the normal cone C = CwV imbedded in N = h*NxY, and X· V E A*W can be constructed by intersecting [C] with the zero section in N.

Grassmannians and flag varieties. In general the computation of the ring A* X, for a nonsingular projective variety X, is a very difficult problem. One has AD X = Z. and AI X = Pic( X),. but for p ;;. 2, there is little general knowledge of AP X. Mumford [45) showed that, for general surfaces, it is impossible to give A2 X any natural, finite dimensional, algebraic geometric structure. Collino [12) has calculated A* X for X a symmetric product of a curve, and Bloch and Murre [8) have done the same for certain Fano threefolds.