We prove an upper bound for the $L^4$-norm and for the $L^2$-norm restricted to the vertical geodesic of a holomorphic Hecke cusp form $f$ of large weight $k$. The method is based on Watson’s formula and estimating a mean value of certain $L$-functions of degree 6. Further applications to restriction problems of Siegel modular forms and subconvexity bounds of degree 8 $L$-functions are given.

We study the curvature of metric spaces and branched covers of Riemannian manifolds, with applications in topology and algebraic geometry. Here curvature bounds are expressed in terms of the $CAT\left(\chi \right)$ inequality. We prove a general $CAT\left(\chi \right)$ extension theorem, giving sufficient conditions on and near the boundary of a locally $CAT\left(\chi \right)$ metric space for the completion to be $CAT\left(\chi \right)$. We use this to prove that a branched cover of a complete Riemannian manifold is locally $CAT\left(\chi \right)$ if and only if all tangent spaces are $CAT\left(0\right)$ and the base has sectional curvature bounded above by $\chi$. We also show that the branched cover is a geodesic space. Using our curvature bound and a local asphericity assumption we give a sufficient condition for the branched cover to be globally $CAT\left(\chi \right)$ and the complement of the branch locus to be contractible.

We conjecture that the universal branched cover of ${\mathbb{C}}^n$ over the mirrors of a finite Coxeter group is $CAT\left(0\right)$. This is closely related to a conjecture of Charney and Davis, and we combine their work with our machinery to show that our conjecture implies the Arnol$'$d–Pham–Thom conjecture on $K(\pi ,1)$ spaces for Artin groups. Also conditionally on our conjecture, we prove the asphericity of moduli spaces of amply lattice-polarized K3 surfaces and of the discriminant complements of all the unimodal hypersurface singularities in Arnol$'$d’s hierarchy.

We show that every smooth closed curve $\Gamma$ immersed in Euclidean space ${\mathbf{R}}^3$ satisfies the sharp inequality $2(\mathcal{P}+\mathcal{I})+\mathcal{V}\ge 6$ which relates the numbers $\mathcal{P}$ of pairs of parallel tangent lines, $\mathcal{I}$ of inflections (or points of vanishing curvature), and $\mathcal{V}$ of vertices (or points of vanishing torsion) of $\Gamma$. We also show that $2({\mathcal{P}}^{+}+\mathcal{I})+\mathcal{V}\ge 4$, where ${\mathcal{P}}^{+}$ is the number of pairs of concordant parallel tangent lines. The proofs, which employ curve-shortening flow with surgery, are based on corresponding inequalities for the numbers of double points, singularities, and inflections of closed curves in the real projective plane ${\mathbf{RP}}^2$ and the sphere ${\mathbf{S}}^2$ which intersect every closed geodesic. These findings extend some classical results in curve theory from works of Möbius, Fenchel, and Segre, including Arnold’s “tennis ball theorem.”

Building on our previous joint work with A. Schmitt, we explain a recursive algorithm to determine the cohomology of moduli spaces of Higgs bundles on any given curve (in the coprime situation). As an application of the method, we compute the $y$-genus of the space of ${PGL}_n$-Higgs bundles for any rank $n$, confirming a conjecture of T. Hausel.

We describe Somekawa’s $K$-group associated to a finite collection of semiabelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky’s category of motives. While Somekawa’s definition is based on Weil reciprocity, Voevodsky’s category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.

We show that any smooth complex projective variety whose fundamental group has a complex representation with infinite image must have a nonzero symmetric differential (a section of a symmetric power of the cotangent bundle). Along the way, we produce many symmetric differentials on the base of a variation of Hodge structures.