Related papers: Higher Ramanujan equations I: moduli stacks of abe…

We describe higher dimensional generalizations of Ramanujan's classical differential relations satisfied by the Eisenstein series $E_2$, $E_4$, $E_6$. Such "higher Ramanujan equations" are given geometrically in terms of vector fields…

In the first part of this work, we have considered a moduli space $B_g$ classifying principally polarized abelian varieties of dimension $g$ endowed with a symplectic-Hodge basis, and we have constructed the higher Ramanujan vector fields…

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…

In this article we define the algebra of differential modular forms and we prove that it is generated by Eisenstein series of weight $2,4$ and 6. We define Hecke operators on them, find some analytic relations between these Eisenstein…

For every genus g, we construct a smooth, complete, rational polarized algebraic variety DM_g together with a normal crossing divisor D = sum D_i, such that for every moduli space M_C(2,0) of semistable topologically trivial vector bundles…

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack $\mathcal K^{\text{abs}}_g$ of abstract Kummer varieties and the second one is the stack $\mathcal K^{\text{em}}_g$ of…

We prove that smooth, separated Deligne--Mumford stacks in mixed characteristic with quasi-projective coarse moduli space are global quotient stacks and satisfy the resolution property. This builds on work of Kresch and Vistoli and of…

In this article we prove that for all primes $p\not=2,3$, the Ramanujan vector field has an invariant algebraic curve and then we give a moduli space interpretation of this curve in terms of Cartier operator acting on the de Rham cohomology…

Abstract. Let G be a complex reductive group and A be an Abelian variety of dimension d over $\mathbb{C}$. We determine the Poincar\'e polynomials and also the mixed Hodge polynomials of the moduli space $\mathcal{M}_{A}^{H}(G)$ of G-Higgs…

In the first part of the paper we characterize certain systems of first order nonlinear differential equations whose space of solutions is an $\mathfrak{sl}_2(\mathbb{C})$-module. We prove that such systems, called Ramanujan systems of…

This is the second part of a series of papers devoted to develop Homotopical Algebraic Geometry. We start by defining and studying generalizations of standard notions of linear and commutative algebra in an abstract monoidal model category,…

In this article, we derive estimates of Teichm\"uller modular forms, and associated invariants. Let $\mathcal{M}_{g}$ denote the moduli space of compact hyperbolic Riemann surfaces of genus $g\geq 2$, and let $\overline{M}_{g}$ be the…

We construct a compactification of the moduli spaces of abelian differentials on Riemann surfaces with prescribed zeroes and poles. This compactification, called the moduli space of multi-scale differentials, is a complex orbifold with…

These are expanded notes from some talks given during the fall 2002, about ``homotopical algebraic geometry'' (HAG) with special emphasis on its applications to ``derived algebraic geometry'' (DAG) and ``derived deformation theory''. We use…

In "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld introduced the "very good" property for a smooth complex equidimensional stack. They prove that for a semisimple complex group G, the moduli…

For $k \in \mathbb{Z}_{>0}$, let $\mathcal{H}^{(k)}_{g,n}$ denote the vector bundle over $\mathfrak{M}_{g,n}$ whose every fiber consists of meromorphic $k$-differentials with poles of order at most $k-1$ on a fixed Riemman surface of genus…

We compute some Hodge and Betti numbers of the moduli space of stable rank $r$ degree $d$ vector bundles on a smooth projective curve. We do not assume $r$ and $d$ are coprime. In the process we equip the cohomology of an arbitrary…

This paper describes the derivation of the level 5 versions of Ramanujan's system of ordinary differential equations satisfied by the Eisenstein series, $E_2(q),E_4(q)$, and $E_6(q).$

Jet manifolds and vector bundles allow one to employ tools of differential geometry to study differential equations, for example those arising as equations of motions in physics. They are necessary for a geometrical formulation of…

We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented by an infinite…