Quasi-modular forms attached to elliptic curves, I

In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the Ramanujan differential equation between Eisenstein series. We also explain the notion of period map constructed from elliptic integrals. This turns out to be the bridge between the algebraic notion of a quasi-modular form and the one as a holomorphic function on the upper half plane. In this way we also get another interpretation, essentially due to Halphen, of the Ramanujan differential equation in terms of hypergeometric functions. The interpretation of quasi-modular forms as sections of jet bundles and some related enumerative problems are also presented.Comment: 51 page

The Gauss-Manin connection on the Hodge structures

Pour tout sch\'ema simplicial complexe $X_{\bullet}$ il existe une application canonique $\nabla:H^{\ast}(X_{\bullet})\longrightarrow \Omega^1_{{\mathbb C}/{\mathbb Q}}\otimes H^{\ast}(X_{\bullet})$, appel\'ee la connexion de Gau\ss-Manin. Nous montrons qu'il existe une unique connexion fonctorielle sur toute structure de Hodge-Tate mixte ayant certaines propri\'et\'es de la connexion de Gau\ss-Manin. Cette connexion n'est pas int\'egrable en g\'en\'eral, et alors son int\'egrabilit\'e est une condition non triviale pour qu'une structure de Hodge soit g\'eom\'etrique. Dans des cas particuliers, je donne des formules explicites pour la connexion de Gau\ss-Manin sur la cohomologie singuli\`ere des vari\'et\'es alg\'ebriques sur ${\mathbb C}$ dans les termes de la structure de Hodge

Hodge metrics and positivity of direct images

Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we show that the direct image of an adjoint semi-ample line bundle by a projective submersion has a continuous metric with Griffiths semi-positive curvature. This shows that for every holomorphic semi-ample vector bundle $E$ on a complex manifold, and every positive integer $k$, the vector bundle $S^kE\otimes\det E$ has a continuous metric with Griffiths semi-positive curvature. If $E$ is ample on a projective manifold, the metric can be made smooth and Griffiths positive.Comment: revised and expanded version of "A positivity property of ample vector bundles