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(Co)-Homology theories of manifolds come equipped with Poincare duality and intersection theory, which allows us to intersect closed submanifolds ( after moving and stretching) to get lower dimensional submanifolds. We will review Poincare duality and algebraic intersection theories that one gets from algebraic submanifolds of an algebraic manifold. Then we will see that for enumerative geometry purpose why we have to develop a good intersection theory of Moduli spaces. Classical Riemann Roch and Grothendieck Riemann Roch will be reviewed, which relates algebraic vector bundles on a algebraic variety to its algebraic intersection theory. Then we will generalize Grothendieck Riemann Roch to the category of Moduli spaces. |