The main concrete result of this paper is enumeration of genus-two curves with complex structure fixed in ℙ ² and ℙ ³. Along the way, rational curves with certain simple singularities are counted as well. While the methods described can be used to count positive-genus curves in some other cases, the most powerful direct applications of the machinery developed are to enumeration of rational curves with a very large class of singularities in projective spaces.

In this paper we present a way of computing a lower bound for the genus of any smooth representative of a homology class of positive self-intersection in a smooth four-manifold $X$ with second positive Betti number $b^{+}_{2}(X) = 1$. We study the solutions of the Seiberg-Witten equations on the cylindrical end manifold which is the complement of the surface representing the class. The result can be formulated as a form of generalized adjunction inequality. The bounds obtained depend only on the rational homology type of the manifold, and include the Thom conjecture as a special case. We generalize this approach to derive lower bounds on the number of intersection points of $n$ algebraically disjoint surfaces of positive self-intersection in manifolds with $b^{+}_{2}(X) = n$.

An anti-holomorphic multiplication by the integers $O_d$ of a quadratic imaginary number field, on a principally polarized complex abelian variety $A_{\mathbb{C}}$ is an action of $O_d$ on $A_{\mathbb{C}}$ such that the purely imaginary elements act in an anti-holomorphic manner. The coarse moduli space $X_{\mathbb{R}}$ of such $A$ (with appropriate level structure) is shown to consist of finitely many isomorphic connected components, each of which is an arithmetic quotient of the quaternionic Siegel space, that is, the symmetric space for the complex sym- plectic group. The moduli space $X_{\mathbb{R}}$ is also identified as the fixed point set of a certain anti-holomorphic involution $\tau$ on the complex points $X_{\mathbb{C}}$ of the Siegel moduli space of all principally polarized abelian varieties (with appropriate level structure). The Siegel moduli space $X_{\mathbb{C}}$ admits a certain rational structure for which the involution $\tau$ is rationally defined. So the space $X_{\mathbb{R}}$ admits the structure of a rationally defined, real algebraic variety.