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An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of algebraic geometry, dynamical systems, geometry and topology lead to the study of such objects. It comes as a natural consequence that we can employ in our study algebraic, analytic, combinatorial and dynamical perspectives. These lecture notes aim to provide an expository introduction to this subject that will emphasize the aforementioned links between different areas of mathematics. We will associate to an Abelian differential a flat surface with conical singularities such that the underlying Riemann surface is obtained from a polygon by identifying edges with one another via translation. We will focus on studying these objects in families and describe some properties of the orbit as we vary the polygon by the action of $GL_2^{+}(mathbb{R})$ on the plane.
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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 normal crossing boundary. Locally, our compactification 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 our compactification is the orbifold corresponding to it. Moreover, we perform a real oriented blowup of the unprojectivized moduli space of multi-scale differentials such that the $mathrm{SL}_2(mathbb R)$-action in the interior of the moduli space extends continuously to the boundary.
A $k$-differential on a Riemann surface is a section of the $k$-th power of the canonical line bundle. Loci of $k$-differentials with prescribed number and multiplicities of zeros and poles form a natural stratification of the moduli space of $k$-differentials. In this paper we give a complete description for the compactification of the strata of $k$-differentials in terms of pointed stable $k$-differentials, for all $k$. The upshot is a global $k$-residue condition that can also be reformulated in terms of admissible covers of stable curves. Moreover, we study properties of $k$-differentials regarding their deformations, residues, and flat geometric structure.
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.
For a linear subvariety $M$ of a stratum of meromorphic differentials, we investigate its closure in the multi-scale compactification constructed by Bainbridge-Chen-Gendron-Grushevsky-Moller. We prove various restrictions on the type of defining linear equations in period coordinates for $M$ near its boundary, and prove that the closure is locally a toric variety. As applications, we give a fundamentally new proof of a generalization of the cylinder deformation theorem of Wright to the case of meromorphic strata, and construct a smooth compactification of the Hurwitz space of covers of the Riemann sphere.
We introduce a new class of $mathfrak{sl}_2$-triples in a complex simple Lie algebra $mathfrak{g}$, which we call magical. Such an $mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $mathfrak{g}$. Using this decomposition data, we explicitly parameterize special connected components of the moduli space of Higgs bundles on a compact Riemann surface $X$ for an associated real Lie group, hence also of the corresponding character variety of representations of $pi_1X$ in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or $mathrm{SO}_{p,q}$ with $1<pleq q$, and also constructs previously unknown components for the quaternionic real forms of $mathrm{E}_6$, $mathrm{E}_7$, $mathrm{E}_8$ and $mathrm{F}_4$. The classification of magical $mathfrak{sl}_2$-triples is shown to be in bijection with the set of $Theta$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmuller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $Theta$-positive Anosov representations, which indicate that this conjecture holds.
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