Volume 2021, Issue 770

In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L -derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative “period polynomials” in the case of cusp forms. The unimodularity of the roots seems to be a very subtle property which is special to our “period polynomials”. This is suggested by numerical experiments on families of perturbed “period polynomials” (Section 5.3) suggested by Zagier. We prove a special case of our conjecture in the case of Eisenstein series. Although not much is currently known about derivatives higher than first order ones for general modular forms, celebrated recent work of Yun and Zhang established the analogues of the Gross–Zagier formula for higher L -derivatives in the function field case. A critical role in their work was played by a notion of “super-positivity”, which, as recently shown by Goldfeld and Huang, holds in infinitely many cases for classical modular forms. As will be discussed, this is similar to properties which were required by Jin, Ma, Ono, and Soundararajan in their proof of the Riemann Hypothesis for Period Polynomials, thus suggesting a connection between the analytic nature of our conjectures here and the framework of Yun and Zhang.

The Cremona group is the group of birational transformations of the complex projective plane. In this paper we classify its subgroups that consist only of elliptic elements using elementary model theory. This yields in particular a description of the structure of torsion subgroups. As an application, we prove the Tits alternative for arbitrary subgroups of the Cremona group, generalizing a result of Cantat. We also describe solvable subgroups of the Cremona group and their derived length, refining results from Déserti.

Unlike Hochschild (co)homology and K -theory, global and dominant dimensions of algebras are far from being invariant under derived equivalences in general. We show that, however, global dimension and dominant dimension are derived invariant when restricting to a class of algebras with anti-automorphisms preserving simples. Such anti-automorphisms exist for all cellular algebras and in particular for many finite-dimensional algebras arising in algebraic Lie theory. Both dimensions then can be characterised intrinsically inside certain derived categories. On the way, a restriction theorem is proved, and used, which says that derived equivalences between algebras with positive ν-dominant dimension always restrict to derived equivalences between their associated self-injective algebras, which under this assumption do exist.

The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in ℝ n {\mathbb{R}^{n}} , whose level set { u = 0 } {\{u=0\}} is contained in a half-space. If n ≤ 3 {n\leq 3} , we prove that the solution must be one-dimensional. In dimension n ≥ 4 {n\geq 4} , we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.

Given a stability condition on a smooth projective variety X , we construct a family of stability conditions on X × C {X\times C} , where C is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macrì and weak stability conditions on the Abramovich-Polishchuk heart of a bounded t-structure in D ⁢ ( X × C ) {D(X\times C)} .

In this article we show that for every finite area hyperbolic surface X of type ( g , n ) {(g,n)} and any harmonic Beltrami differential μ on X , then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to - g {-g} as the genus g goes to infinity.

We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in ℂ n , n ≥ 2 {\mathbb{C}^{n},n\geq 2} , is Kähler–Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic metric over a Stein space with a spherical boundary.

In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function P : ℤ ≥ 0 → ℤ {P:\mathbb{Z}_{\geq 0}\to\mathbb{Z}} , then there exists an integer N > 0 {N>0} such that if ( X , ℱ ) {(X,{\mathcal{F}})} is a canonical or nef model of a foliation of general type with Hilbert polynomial χ ⁢ ( X , 𝒪 X ⁢ ( m ⁢ K ℱ ) ) = P ⁢ ( m ) {\chi(X,{\mathcal{O}}_{X}(mK_{\mathcal{F}}))=P(m)} for all m ∈ ℤ ≥ 0 {m\in\mathbb{Z}_{\geq 0}} , then | m ⁢ K ℱ | {|mK_{\mathcal{F}}|} defines a birational map for all m ≥ N {m\geq N} . On the way, we also prove a Grauert–Riemenschneider-type vanishing theorem for foliated surfaces with canonical singularities.