We study horizontal subvarieties $Z$ of a Griffiths period domain $\mathbf{D}$. If $Z$ is defined by algebraic equations, and if $Z$ is also invariant under a large discrete subgroup in an appropriate sense, we prove that $Z$ is a Hermitian symmetric domain $\mathcal{D}$, embedded via a totally geodesic embedding in $\mathbf{D}$. Next we discuss the case when $Z$ is in addition of Calabi–Yau type. We classify the possible variations of Hodge structure (VHS) of Calabi–Yau type parameterized by Hermitian symmetric domains $\mathcal{D}$ and show that they are essentially those found by Gross and Sheng and Zuo, up to taking factors of symmetric powers and certain shift operations. In the weight $3$ case, we explicitly describe the embedding $Z\hookrightarrow \mathbf{D}$ from the perspective of Griffiths transversality and relate this description to the Harish-Chandra realization of $\mathcal{D}$ and to the Korányi–Wolf tube domain description. There are further connections to homogeneous Legendrian varieties and the four Severi varieties of Zak.

The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras, and integrable systems.

Integrability of the pentagram map was conjectured by Schwartz and proved by the present authors for a larger space of twisted polygons. In this article, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasiperiodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.

We generalize the familiar notions of overtwistedness and Giroux torsion in $3$-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k\ge 0$ can be interpreted as measuring a gradation in “degrees of tightness” of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact-type embeddings into any closed symplectic $4$-manifold, and has vanishing contact invariant in embedded contact homology, and we give examples of contact manifolds that have planar $k$-torsion for any $k\ge 2$ but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness, and compactness theorems for certain classes of $J$-holomorphic curves in blown-up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains.

We construct a continuous image of a Radon–Nikodým compact space which is not Radon–Nikodým compact, solving the problem posed in the 1980s by Isaac Namioka.

Let $k$ be a field finitely generated over $\mathbb{Q}$, and let $X$ be a curve over $k$. Fix a prime $\ell$. A representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$ is said to be geometrically Lie perfect if any open subgroup of $\rho \left({\pi }_1\right(X_k\left)\right)$ has finite abelianization. Let $G$ denote the image of $\rho$. Any closed point $x$ on $X$ induces a splitting $x:{\Gamma }_{\kappa \left(x\right)}:={\pi }_1(Spec(\kappa \left(x\right)\left)\right)\to {\pi }_1\left(X_{\kappa \left(x\right)}\right)$ of the restriction epimorphism ${\pi }_1\left(X_{\kappa \left(x\right)}\right)\to {\Gamma }_{\kappa \left(x\right)}$ (here, $\kappa \left(x\right)$ denotes the residue field of $X$ at $x$) so one can define the closed subgroup $G_x:=\rho \circ x\left({\Gamma }_{\kappa \left(x\right)}\right)\subset G$. The main result of this paper is the following uniform open image theorem. Under the above assumptions, for any geometrically Lie perfect representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$ and any integer $d\ge 1$, the set $X_{\rho ,d}$ of all closed points $x\in X$ such that $G_x$ is not open in $G$ and $\left[\kappa \right(x):k]\le d$ is finite and there exists an integer $B_{\rho ,d}\ge 1$ such that $[G:G_x]\le B_{\rho ,d}$ for any closed point $x\in X\setminus X_{\rho ,d}$ with $\left[\kappa \right(x):k]\le d$.

A key ingredient of our proof is that, for any integer $\gamma \ge 1$, there exists an integer $\nu =\nu \left(\gamma \right)\ge 1$ such that, given any projective system $\cdots \to Y_{n+1}\to Y_n\to \cdots \to Y_0$ of curves (over an algebraically closed field of characteristic $0$) with the same gonality $\gamma$ and with $Y_{n+1}\to Y_n$ a Galois cover of degree greater than $1$, one can construct a projective system of genus $0$ curves $\cdots \to B_{n+1}\to B_n\to \cdots \to B_{\nu }$ and degree $\gamma$ morphisms $f_n:Y_n\to B_n$, $n\ge \nu$, such that $Y_{n+1}$ is birational to $B_{n+1}{\times }_{B_n,f_n}{Y}_n$, $n\ge \nu$. This, together with the case for $d=1$ (which is the main result of part I of this paper), gives the proof for general $d$.

Our method also yields the following unconditional variant of our main result. With the above assumptions on $k$ and $X$, for any $\ell$-adic representation $\rho :{\pi }_1\left(X\right)\to {GL}_m\left({\mathbb{Z}}_{\ell }\right)$ and integer $d\ge 1$, the set of all closed points $x\in X$ such that $G_x$ is of codimension at least $3$ in $G$ and $\left[\kappa \right(x):k]\le d$ is finite.