We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

Zhou et al. [‘On weakly non-decreasable quasiconformal mappings’, J. Math. Anal. Appl.386 (2012), 842–847] proved that, in a Teichmüller equivalence class, there exists an extremal quasiconformal mapping with a weakly nondecreasable dilatation. They asked whether a weakly nondecreasable dilatation is a nondecreasable dilatation. The aim of this paper is to give a negative answer to their problem. We also construct a Teichmüller class such that it contains an infinite number of weakly nondecreasable extremal representatives, only one of which is nondecreasable.

We outline an algorithm to compute $\unicode[STIX]{x1D703}(z,\unicode[STIX]{x1D70F})$ in genus two in quasi-linear time, borrowing ideas from the algorithm for theta constants and the one for $\unicode[STIX]{x1D703}(z,\unicode[STIX]{x1D70F})$ in genus one. Our implementation shows a large speed-up for precisions as low as a few thousand decimal digits. We also lay out a strategy to generalize this algorithm to genus $g$.

We compute equations for real multiplication on the divisor classes of genus-2 curves via algebraic correspondences. We do so by implementing van Wamelen’s method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant $5$ and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.

We classify the horizontal $\text{SL}(2)$s and $\mathbb{R}$-split polarized mixed Hodge structures on a Mumford–Tate domain.

To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

We consider the system F4 (a, b, c) of differential equations annihilating Appell's hypergeometric series F4(a,b,c;x). We find the integral representations for four linearly independent solutions expressed by the hypergeometric series F4 . By using the intersection forms of twisted (co)homology groups associated with them, we provide the monodromy representation of F4(a, b, c) and the twisted period relations for the fundamental systems of solutions of F 4.

Thurston introduced shear deformations (cataclysms) on geodesic laminations–deformations including left and right displacements along geodesics. For hyperbolic surfaces with cusps, we consider shear deformations on disjoint unions of ideal geodesics. The length of a balanced weighted sum of ideal geodesics is defined and the Weil–Petersson (WP) duality of shears and the defined length is established. The Poisson bracket of a pair of balanced weight systems on a set of disjoint ideal geodesics is given in terms of an elementary $2$-form. The symplectic geometry of balanced weight systems on ideal geodesics is developed. Equality of the Fock shear coordinate algebra and the WP Poisson algebra is established. The formula for the WP Riemannian pairing of shears is also presented.

We endow certain GKZ-hypergeometric systems with a natural structure of a mixed Hodge module, which is compatible with the mixed Hodge module structure on the Gauß–Manin system of an associated family of Laurent polynomials. As an application we show that the underlying perverse sheaf of a GKZ-system with rational parameter has quasi-unipotent local monodromy.

We show that there is no stable Siegel modular form that vanishes on every moduli space of curves.

We show that the zero locus of an admissible normal function on a smooth complex algebraic variety is algebraic. In Part II of the paper, which is an appendix, we compute the Tannakian Galois group of the category of one-variable admissible real nilpotent orbits with split limit. We then use the answer to recover an unpublished theorem of Deligne, which characterizes the ${\mathrm{sl} }_{2} $-splitting of a real mixed Hodge structure.

We associate to a test configuration for a polarized variety a filtration of the section ring of the line bundle. Using the recent work of Boucksom and Chen we get a concave function on the Okounkov body whose law with respect to Lebesgue measure determines the asymptotic distribution of the weights of the test configuration. We show that this is a generalization of a well-known result in toric geometry. As an application, we prove that the pushforward of the Lebesgue measure on the Okounkov body is equal to a Duistermaat–Heckman measure of a certain deformation of the manifold. Via the Duisteraat–Heckman formula, we get as a corollary that in the special case of an effective ℂ×-action on the manifold lifting to the line bundle, the pushforward of the Lebesgue measure on the Okounkov body is piecewise polynomial.

We consider the Prym map from the space of double coverings of a curve of genus g with r branch points to the moduli space of abelian varieties. We prove that 𝒫:ℛg,r→𝒜δg−1+r/2 is generically injective if We also show that a very general Prym variety of dimension at least 4 is not isogenous to a Jacobian.

Let S be a Riemann surface of type (p,n) with 3p+n>4 and n≥1. We investigate products of some pseudo-Anosov maps θ and Dehn twists tα on S, and prove that under certain conditions the products tkα∘θ are pseudo-Anosov for all integers k. We also give examples that show that tkα∘θ are not pseudo-Anosov for some integers k.

We prove that the infinitesimal variations of Hodge structure arising in a number of geometric situations are non-generic. In particular, we consider the case of generic hypersurfaces in complete smooth projective toric varieties, generic hypersurfaces in weighted projective spaces and generic complete intersections in projective space and show that, for sufficiently high degrees, the corresponding infinitesimal variations are non-generic.

Let S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.