We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof. We furthermore study an involution which acts on the quasi modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their

In this note we show that if a projective manifold admits a Khler metric with negative holomorphic sectional curvature then the canonical bundle of the manifold is ample. This confirms a conjecture of the second author.

This note answers some questions on holomorphic curves and their distribution in an algebraic surface of positive index. More specifically, we exploit the existence of natural negatively curved "pseudo-Finsler" metrics on a surface S of general type whose Chern numbers satisfy c(2)1>2c2 to show that a holomorphic map of a Riemann surface to S whose image is not in any rational or elliptic curve must satisfy a distance decreasing property with respect to these metrics. We show as a consequence that such a map extends over isolated punctures. So assuming that the Riemann surface is obtained from a compact one of genus q by removing a finite number of points, then the map is actually algebraic and defines a compact holomorphic curve in S. Furthermore, the degree of the curve with respect to a fixed polarization is shown to be bounded above by a multiple of q - 1 irrespective of the map.

In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three dimensional manifold M has a boundary with strongly positive mean curvature, the effect of this mean curvature can influence the internal geometry of M. For example, if the scalar curvature of M is greater than certain constant related to this boundary effect, no incompressible surface of higher genus can exist.

We derive bounds for eigenvalues of the Laplacian of graphs using discrete versions of the Sobolev inequalities and heat kernel estimates.