We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.

Abstract. In this paper we continue the program pioneered by D'Hoker and Phong, and recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the chiral superstring measure by constructing modular forms satisfying certain factorization constraints. We give new expressions for their proposed ansätze in genera 2 and 3, respectively, which admit a straightforward generalization. We then propose an ansatz in genus 4 and verify that it satisfies the factorization constraints and gives a vanishing cosmological constant. We further conjecture a possible formula for the superstring amplitudes in any genus, subject to the condition that certain modular forms admit holomorphic roots.

We construct a compactification PΞMg,n(µ) 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 normal crossing boundary. Locally, PΞMg,n(µ) can be described as the normalization of an explicit blowup of the incidence variety compactification, which was defined in [BCGGM18] as the closure of the stratum of abelian differentials in the closure of the Hodge bundle. We also define families of projectivized multi-scale differentials, which gives a proper Deligne-Mumford stack, and PΞMg,n(µ) is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized space ΞMg,n(µ) such that the SL2(R)-action in the interior of the moduli space extends continuously to the boundary.A multi-scale differential on a pointed stable curve is the data of an enhanced level structure on the dual graph, prescribing the orders of poles and zeroes at the nodes, together with a collection of meromorphic differentials on the irreducible components satisfying certain conditions. Additionally, the multi-scale differential encodes the data of a prong-matching at the nodes, matching the incoming and outgoing horizontal trajectories in the flat structure. The construction of PΞMg,n(µ) furthermore requires defining families of multi-scale differentials, where the underlying curve can degenerate, and understanding the notion of equivalence of multi-scale differentials under various rescalings.Our construction of the compactification proceeds via first constructing an augmented Teichmüller space of flat surfaces, and then taking its suitable quotient. Along the way, we give a complete proof of the fact that the conformal and quasiconformal topologies on the (usual) augmented Teichmüller space agree.

We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new proof of the Diaz' bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods and their degenerations.conjecture: the Jacobians of curves are exactly those indecomposable principally polarized abelian varieties (ppav) whose theta-functions provide explicit solutions of the Kadomtsev-Petviashvili (KP) equation, was the first evidence of the now well-accepted usefulness of combining the techniques of integrable systems and algebraic geometry to obtain new results in both fields.Novikov's conjecture was proved by Shiota in [37], and until relatively recently had remained the most effective solution of the Riemann-Schottky problem, the problem of characterizing Jacobians among all ppavs. A much stronger characterization of Jacobians was suggested by Welters, who, inspired by Novikov's conjecture and Gunning's theorem [15], conjectured in [38] that a ppav is a Jacobian if and only if its Kummer variety has at least one trisecant (and then it follows that in fact it has a four-dimensional family of trisecants).Recall that for a ppav X with principal polarization Θ the Kummer variety K(X) is the image of the complete linear system |2Θ|. This is to say that the coordinates for the embedding K : X/ ± 1 ֒→ CP 2 g −1 are given by a basis of the sections of |2Θ|, consisting of theta functions of the second orderfor all ε ∈ (Z/2Z) 2g , where τ is the period matrix of X. A projective (m − 2)-dimensional plane CP m−2 ⊂ CP 2 g −1 intersecting K(X) in at least m points is called an m-secant of the Kummer variety.

Abstract. In this paper we prove a conjecture of Hershel Farkas [8] that if a 4-dimensional principally polarized abelian variety has a vanishing theta-null, and the hessian of the theta function at the corresponding point of order two is degenerate, the abelian variety is a Jacobian.We also discuss possible generalizations to higher genera, and an interpretation of this condition as an infinitesimal version of Andreotti and Mayer's local characterization of Jacobians by the dimension of the singular locus of the theta divisor.