Related papers: Modular forms from Noether-Lefschetz theory
We study the moduli space of solutions to the Seiberg-Witten equations with $N$ spinors on a compact Riemann surface. These moduli spaces arise in a program to define a new enumerative invariant of 3-manifolds. They are also of independent…
Moduli spaces of algebraic curves and closely related to them Hurwitz spaces, that is, spaces of meromorphic functions on the curves, arise naturally in numerous problems of algebraic geometry and mathematical physics, especially in…
Working over imperfect fields, we give a comprehensive classification of genus-one curves that are regular but not geometrically regular, extending the known case of geometrically reduced curves. The description is given intrinsically, in…
We connect two notions of tautological ring: one for the moduli space of curves (after Mumford, Faber, etc.), and the other for the Jacobian of a curve (after Beauville, Polishchuk, etc.). The motivic Lefschetz decomposition on the Jacobian…
We prove the existence of a smoothing for a toroidal crossing space under mild assumptions. By linking log structures with infinitesimal deformations, the result receives a very compact form for normal crossing spaces. The main approach is…
We describe a supersymmetric generalization of the construction of Kontsevich and Arbarello, De Concini, Kac, and Procesi, which utilizes a relation between the moduli space of curves with the infinite-dimensional Sato Grassmannian. Our…
We investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact Kaehler manifold X. These solutions are known to be related to polystable triples via a Kobayashi-Hitchin type…
The modularity of an elliptic curve $E/\mathbb Q$ can be expressed either as an analytic statement that the $L$-function is the Mellin transform of a modular form, or as a geometric statement that $E$ is a quotient of a modular curve…
The space of elliptic modular forms of fixed weight and level can be identfied with a space of intertwining operators, from a holomorphic discrete series representation of SL2(R) to a space of automorphic forms. Moreover, multiplying…
We calculate the homomorphism of the cohomology induced by the Krichever map of moduli spaces of curves into infinite-dimensional Grassmannian. This calculation can be used to compute the homology classes of cycles on moduli spaces of…
We contribute to the foundations of tropical geometry with a view towards formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show…
We establish several formulas relating periods of modular forms on quaternion algebras over number fields to special values of L-functions. Our main inputs are the cohomological techniques for working with periods introduced in [Mol21],…
We consider a moduli space of lattice polarized K3 surfaces with the additional information of a frame of the trascendental cohomology with respect to the lattice polarization. This moduli space is proved to be quasi-affine, and the…
Theorems of Barth-Lefschetz type describe restrictions on the topology of varieties of small codimension. R. Schoen and J. Wolfson, using Morse theory on a path space, have described a technique to prove theorems of this kind for complex…
Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the number of such curves that pass through g generic points and we…
It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain…
We analyze cohomological properties of the Krichever map and use the results to study Weierstrass cycles in moduli spaces and the tautological ring.
We construct a finite dimensional Kaehler manifold with a holomorphic, symplectic circle action whose symplectic reduced spaces may be identified with the tau-vortex moduli spaces (or tau-stable pairs). The Morse theory of the circle action…
We construct real polarizable Hodge structures on the reduced leafwise cohomology of K\"ahler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's K\"ahlerian…
We study the moduli space of meromorphic 1-forms on complex algebraic curves having at most simple poles with fixed nonzero residues. We interpret the Bergman tau function on this moduli space as a section of a line bundle and study its…