In this paper we construct various non-trivial and non-tautological cohomology classes on compactified and uncompactified strata of curves with a differential, by using the geometry of the boundary stratification of the moduli space of multi-scale differentials.

We show that when the number of zeros or poles is at least four, every connected component of the strata of differentials in genus one with prescribed zero and pole orders is not an orbifold K(π,1). For quadratic differentials, this provides infinitely many counterexamples to a conjecture attributed to Kontsevich, as well as to a folklore conjecture concerning the contractibility of spaces of Bridgeland stability conditions.

Multiscale differentials arise as limits of holomorphic differentials with prescribed orders of zeros on nodal curves. In this paper, we address the conjecture concerning Gorenstein contractions of multiscale differentials, initially proposed by Ranganathan and Wise and further elaborated upon by Battistella and Bozlee. Specifically, we show that multiscale differentials can be contracted into Gorenstein singularities level by level from the top down. At each level, these differentials can descend to become generators of the dualizing bundle at the singularities. Additionally, the global residue condition, which governs the smoothability of multiscale differentials, is a special case of the residue condition for descent differentials.

The tautological rings of strata of differentials are known to be generated by divisor classes. In this paper, we give lower bounds on the degrees of relations among them, depending on the genus g and the number of simple zeros. For strata with more than 4g/3 simple zeros, our results show that there are no relations in degrees less than ⌊ g/3 ⌋ + 1. Moreover, we conjecture that, outside of a few exceptions, there is always a non-trivial relation in degree ⌊ g/3 ⌋ + 1, and prove the conjecture for all strata of holomorphic abelian differentials with g ≤ 30. We also prove that the cohomology rings of strata of holomorphic differentials with sufficiently many simple zeros stabilize to the free algebra on the tautological divisor class. Finally, we show that for a large class of holomorphic abelian strata, containing hyperelliptic differentials, the tautological ring is non-trivial for sufficiently large g.

Here we completely determine the spin parity of k-differentials with prescribed zero and pole orders on Riemann surfaces of genus zero and one. This result was previously obtained conditionally by the first author and Quentin Gendron assuming the truth of a number-theoretic hypothesis Conjecture A.10. We prove this hypothesis by reformulating it in terms of Jacobi symbols, reducing the proof to a combinatorial identity and standard facts about Jacobi symbols. The proof was obtained by AxiomProver and the system formalized the proof of the combinatorial identity in Lean/Mathlib (see the Appendix).

Differentials on Riemann surfaces correspond to translation surfaces with conical singularities, and affine transformations acting on them preserve the orders of these singularities. This viewpoint allows the moduli spaces of differentials to appear in various guises across many areas, including algebraic geometry, dynamical systems, combinatorial enumeration, and mathematical physics. Over the past few decades, remarkable progress has been made in computing invariants of these moduli spaces, classifying linear subvarieties, understanding degenerations and compactifications, and developing intersection theory on these spaces. Despite these advances, our understanding of the topology of moduli spaces of differentials remains limited, and many fundamental questions are still open. In this survey, we aim to present, from an algebro-geometric perspective, the known results and open problems concerning the topology of moduli spaces of differentials, as well as their connections to other aspects of the field, with the hope of inspiring further developments in the coming decade.

Weil-Petersson and Masur-Veech volumes measure the sizes of moduli spaces of Riemann surfaces equipped with hyperbolic and flat metrics, respectively. Over the past several decades, the computation of these volumes has inspired remarkable developments in combinatorial enumeration, intersection theory, and recursion relations. In this survey, we review key results, methods, open problems, as well as interesting parallels that emerge in the approaches to computing both types of volumes.

Given a partition of 2, the stratum (1) parametrizes meromorphic differen-tial one-forms on the Riemann sphere CP & sup1; with zeros and p poles of orders prescribed by p. The isoresidual fibration is defined by assigning to cach differential in (4) its configuration of residues at the poles. In the case of differentials with n = 2 zeros, generic isoresidual fibers are complex curves endowed with a canonical translation structure, which we describe exten-sively in this paper. Quantitative characteristics of the translation structure on isoresidual fiber curves provide rich discrete invariants for these fibers. We determine the Euler characteristic of generic isoresidual fiber curves from intersection-theoretic computations, we describe a wall and chamber structure for the Euler characteristic of generic isoresidual fiber curves in terms of the partition a, and we classify the connected components of generic inoresidual fibers for strata in genus zero with an arbitrary number of zeros.

Given a holomorphic differential on a smooth curve, we associate to it a Gorenstein singularity with 𝔾_m-action via a test configuration. This construction decomposes the strata of holomorphic differentials into negatively graded versal deformation spaces of such singularities, refining Pinkham's correspondence between monomial singularities and Weierstrass semigroups to the case of Gorenstein singularities with multiple branches in the framework of Looijenga's deformations with good 𝔾_m-action. Additionally, this construction provides a natural description for the singular curves that appear in the boundary of the versal deformation spaces, generalizing various special cases from symmetric semigroups and local complete intersections to arbitrary Gorenstein curves that admit canonical divisors with prescribed orders of zeros. Our construction provides a uniform approach to describe the resulting singularities and their invariants, such as weights and characters, initially studied by Alper–Fedorchuk–Smyth. As an application, we classify the unique Gorenstein singularity with 𝔾_m-action for each nonvarying stratum of holomorphic differentials in the work of Chen–Möller and Yu–Zuo, identify each nonvarying stratum with the locus of smooth deformations of the corresponding singularity, and study when these nonvarying strata can be compactified by weighted projective spaces. Additionally, we classify such singularities with bounded α-invariants in the Hassett–Keel log minimal model program for ℳ_g. We also study the slopes of these singularities and utilize them to bound the slopes of effective divisors in ℳ_g. Finally, we show that the loci of subcanonical points with fixed semigroups have trivial tautological rings and provide a criterion to determine whether they are affine.

Multi-scale differentials were constructed by M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. M & ouml;ller, from the viewpoint of flat and complex geometry, for the purpose of compactifying moduli spaces of curves together with a differential with prescribed orders of zeros and poles. Logarithmic differentials were constructed by S. Marcus and J. Wise, as a generalization of stable rubber maps from Gromov-Witten theory. Modulo the global residue condition that isolates the main components of the compactification, we show that these two kinds of differentials are equivalent, and establish an isomorphism of their (coarse) moduli stacks. Moreover, we describe the rubber and multi-scale spaces as an explicit blowup of the moduli space of stable pointed rational curves in the case of genus zero, and as a global blowup of the incidence variety compactification for arbitrary genera, which implies their projectivity. We also propose a refined double ramification cycle formula in the twisted Hodge bundle which interacts with the universal line bundle class.

We provide a complete description of realizable relative period representations for holomorphic differentials on Riemann surfaces with prescribed orders of zeros and additional invariants given by the hyperelliptic structure and spin parity. This answers a question posed by Simion Filip.

We investigate the count of meromorphic differentials on the Riemann sphere possessing a single zero, multiple poles with prescribed orders and fixed residues at each pole. Gendron and Tahar previously examined this problem with respect to general residues using flat geometry, while Sugiyama approached it from the perspective of fixed point multipliers of polynomial maps in the case of simple poles. In our study, we employ intersection theory on compactified moduli spaces of differentials, enabling us to handle arbitrary residues and pole orders, which provides a complete solution to this problem. We also determine interesting combinatorial properties for the solution formula as well as related intersection numbers.

Let $\mathcal{P}(\mu)^{\circ}$ be a connected component of the projectivized stratum of differentials on smooth complex curves, where the zero and pole orders of the differentials are specified by $\mu$. When the complex dimension of $\mathcal{P}(\mu)^{\circ}$ is at least two, Dozier--Grushevsky--Lee, through explicit degeneration techniques, showed that the boundary of $\mathcal{P}(\mu)^{\circ}$ is connected in the multi-scale compactification constructed by Bainbridge--Chen--Gendron--Grushevsky--M\"oller. A natural question is whether the connectedness of the boundary of $\mathcal{P}(\mu)^{\circ}$ is determined by its intrinsic properties. In the case of meromorphic differentials, we provide a concise explanation that the boundary of $\mathcal{P}(\mu)^{\circ}$ is always connected in any complete algebraic compactification, based on the fact that the strata of meromorphic differentials are affine varieties. We also observe that the same result holds for linear subvarieties of meromorphic differentials, as well as for the strata of $k$-differentials with a pole of order at least $k$. In the case of holomorphic differentials, using properties of Teichm\"uller curves, we provide an alternative argument showing that the horizontal boundary of $\mathcal{P}(\mu)^{\circ}$ and every irreducible component of its vertical boundary intersect non-trivially in the multi-scale compactification.

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^+(ℝ) on the plane.

Here we give an explicit construction of a globally defined strictly plurisubharmonic function on projectivized strata of strictly meromorphic differentials with prescribed orders of zeros and poles. In particular, this yields a flat-geometric proof that these strata do not contain positive-dimensional complete subvarieties.

We describe the principal boundary of an arbitrary affine invariant submanifold of REL zero in terms of level graphs of the multi-scale compactification of strata of Abelian differentials with prescribed orders of zeros. We show that the area Siegel–Veech constant of the affine invariant submanifold can be obtained by using volumes of the principal boundary strata. As an application, we prove the conjectural formula in [CMS23a] that computes the area Siegel–Veech constant via intersection theory in the case of REL zero. In particular, the formula holds for strata of quadratic differentials with odd orders of zeros and for the gothic locus. We also explicitly describe the principal boundary components of the gothic locus and their individual contributions to the area Siegel–Veech constant

We describe the Harder--Narasimhan filtration of the Hodge bundle for Teichm\"uller curves in the non-varying strata of quadratic differentials appearing in [CM2]. Moreover, we show that the Hodge bundle on the non-varying strata away from the irregular components can split as a direct sum of line bundles. As applications, we determine all individual Lyapunov exponents of algebraically primitive Teichm\"uller curves in the non-varying strata and derive new results regarding the asymptotic behavior of Lyapunov exponents.

Consider the strata of primitive $k$-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by $k$. The map that assigns to each $k$-differential the $k$-residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the $k$-factorial function, for the cardinality of each fiber. We prove this formula using intersection calculations on the multi-scale compactification of the strata of $k$-differentials. In special cases, we also give alternative proofs using flat geometry. Finally, we present an application to cone spherical metrics with dihedral monodromy.

We provide a complete description of realizable period representations for meromorphic differentials on Riemann surfaces with prescribed orders of zeros and poles, hyperelliptic structure and spin parity.

This paper lays the foundation for determining the Kodaira dimension of the projectivized strata of Abelian differentials with prescribed zero and pole orders in large genus. We work with the moduli space of multi-scale differentials constructed in [BCGGM2] which provides an orbifold compactification of these strata. We establish the projectivity of the moduli space of multi-scale differentials, describe the locus of canonical singularities, and compute a series of effective divisor classes. Moreover, we exhibit a perturbation of the canonical class which allows the corresponding pluri-canonical forms to extend over the locus of non-canonical singularities. As applications, we certify general type for strata with few zeros as well as for strata with equidistributed zero orders when g is sufficiently large. In particular, we show general type for the odd spin components of the minimal strata for g > 12.