The principle of topological economy in algebraic geometry

Abstract

This part of the lecture is not related to the first part, so it can be understood independently. We start with an example. Example 1. In ℂP n , consider two algebraic varieties X and Y of complementary dimensions. In general position, they intersect in finitely many points. Let [ X ] and [ Y ] be the homology classes realized by the varieties X and Y , and let [ X ] o [ Y ] be the intersection index of these classes (which is an integer). It is equal to the number of “positive” intersection points of X with Y minus the number of “negative” intersection points. Thus, the number #( X ∩ Y ) of all intersection points is not smaller than the intersection index [ X ] o [ Y ] (and has the same parity). The Bezout Theorem asserts that #( X ∩ Y ) is equal to the number [ X ] o [ Y ], i.e., there is no inequality! The point is that the orientation of complex manifolds is such that each intersection makes a contribution of +1, not –1, in the total intersection index. Negative intersections are “expensive,” they increase the number of intersection points of X with Y in comparison with the “topologically necessary” number. A propos , the same considerations imply that a polynomial of degree n has precisely n roots, not more. This (well-known) and the following (newer) examples lead to a “principle of economy,” which, in its turn, can be used to state further conjectures. These conjectures can be verified in particular cases; sometimes, they can be proved and become theorems. But in most cases, they remain conjectures, i.e., assertions which we may try to disprove, for a long time.