Showing papers in "Crelle's Journal in 2021"

Rotational symmetry of Weingarten spheres in homogeneous three-manifolds

Abstract: Let $M$ be a simply connected homogeneous three-manifold with isometry group of dimension $4$, and let $\Sigma$ be any compact surface of genus zero immersed in $M$ whose mean, extrinsic and Gauss curvatures satisfy a smooth elliptic relation $\Phi(H,K_e,K)=0$. In this paper we prove that $\Sigma$ is a sphere of revolution, provided that the unique inextendible rotational surface $S$ in $M$ that satisfies this equation and touches its rotation axis orthogonally has bounded second fundamental form. In particular, we prove that: (i) any elliptic Weingarten sphere immersed in $\mathbb{H}^2\times \mathbb{R}$ is a rotational sphere. (ii) Any sphere of constant positive extrinsic curvature immersed in $M$ is a rotational sphere, and (iii) Any immersed sphere in $M$ that satisfies an elliptic Weingarten equation $H=\phi(H^2-K_e)\geq a>0$ with $\phi$ bounded, is a rotational sphere. As a very particular case of this last result, we recover the Abresch-Rosenberg classification of constant mean curvature spheres in $M$.

Conformality for a robust class of non-conformal attractors

Abstract: In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that the Hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent. In the appendix, in collaboration with M. Bridgeman, we extend a classical result on the Hessian of the Hausdorff dimension on purely imaginary directions.

Eigenfunction concentration via geodesic beams

Abstract: In this article we develop new techniques for studying concentration of Laplace eigenfunctions $\phi_\lambda$ as their frequency, $\lambda$, grows. The method consists of controlling $\phi_\lambda(x)$ by decomposing $\phi_\lambda$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $\lambda^{-\frac{1}{2}}$. We control $\phi_\lambda(x)$ by the $L^2$-mass of $\phi_\lambda$ on each geodesic tube and derive a purely dynamical statement through which $\phi_\lambda(x)$ can be studied. In particular, we obtain estimates on $\phi_\lambda(x)$ by decomposing the set of geodesic tubes into those that are non self-looping for time $T$ and those that are. This approach allows for quantitative improvements, in terms of $T$, on the available bounds for $L^\infty$ norms, $L^p$ norms, pointwise Weyl laws, and averages over submanifolds.

On the rotational symmetry of 3-dimensional $κ$-solutions

Abstract: Author(s): Bamler, Richard H; Kleiner, Bruce | Abstract: In a recent paper, Brendle showed the uniqueness of the Bryant soliton among 3-dimensional $\kappa$-solutions. In this paper, we present an alternative proof for this fact and show that compact $\kappa$-solutions are rotational symmetric. Our proof arose from independent work relating to our Strong Stability Theorem for singular Ricci flows.

Geometric and spectral estimates based on spectral Ricci curvature assumptions

Abstract: We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold (Mn,g) for which the lowest eigenvalue of the Ricci tensor ρ is such that the Schr\"odinger operator (n−2)Δ+ρ is positive has finite fundamental group. As a continuation of our earlier results, we obtain isoperimetric inequalities from a Kato condition on the Ricci curvature. Furthermore, we obtain the Kato condition for the Ricci curvature under purely geometric assumptions

Brauer’s Height Zero Conjecture for principal blocks

Abstract: We prove \emph{the other half} of Brauer's Height Zero Conjecture in the case of principal blocks.

On the Frobenius functor for symmetric tensor categories in positive characteristic

Abstract: We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor $F: \mathcal{C}\to \mathcal{C}\boxtimes {\rm Ver}_p$, where ${\rm Ver}_p$ is the Verlinde category (the semisimplification of ${\rm Rep}_{\bf k}(\mathbb{Z}/p)$). This generalizes the usual Frobenius twist functor in modular representation theory and also one defined in arXiv:1503.01492, where it is used to show that if $\mathcal{C}$ is finite and semisimple then it admits a fiber functor to ${\rm Ver}_p$. The main new feature is that when $\mathcal{C}$ is not semisimple, $F$ need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor $\mathcal{C}\to {\rm Ver}_p$. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of $F$, and use it to show that for categories with finitely many simple objects $F$ does not increase the Frobenius-Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which $F$ is exact, and define the canonical maximal Frobenius exact subcategory $\mathcal{C}_{\rm ex}$ inside any STC $\mathcal{C}$ with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius-Perron dimension is preserved by $F$. We prove that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to ${\rm Ver}_p$. We also show that a sufficiently large power of $F$ lands in $\mathcal{C}_{\rm ex}$. Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category and show that a STC with Chevalley property is (almost) Frobenius exact.

Single-valued integration and double copy

Abstract: We study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulas for them. This implies an elementary 'double copy' formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Neron--Tate heights on curves, single-valued multiple zeta values and polylogarithms. In a sequel to this paper \cite{BD2} we apply this formalism to the moduli space of curves of genus zero with marked points, to deduce a recent conjecture due to Stieberger in string perturbation theory, which states that closed string amplitudes are the single-valued projections of open string amplitudes.

Stability conditions on product varieties

Abstract: Given a stability condition on a smooth projective variety $X$, we construct a family of stability conditions on $X\times C$, 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 Macri and weak stability conditions on the Abramovich-Polishchuk heart of a bounded t-structure in $D(X\times C)$.

The Hodge-FVH correspondence

Abstract: The Hodge-FVH correspondence establishes a relationship between the special cubic Hodge integrals and an integrable hierarchy, which is called the fractional Volterra hierarchy. In this paper we prove this correspondence. As an application of this result, we prove a gap condition for certain special cubic Hodge integrals and give an algorithm for computing the coefficients that appear in the gap condition.

Motivic decompositions for the Hilbert scheme of points of a K3 surface

Abstract: We construct an explicit, multiplicative Chow-K\"unneth decomposition for the Hilbert scheme of points of a K3 surface. We further refine this decomposition with respect to the action of the Looijenga-Lunts-Verbitsky Lie algebra.

Bergman–Einstein metrics, a generalization of Kerner’s theorem and Stein spaces with spherical boundaries

Abstract: Abstract 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.

Area minimizing surfaces of bounded genus in metric spaces

Abstract: 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 Holder 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.

Fano manifolds and stability of tangent bundles

Abstract: We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds.

On the regular-convexity of Ricci shrinker limit spaces

Abstract: In this paper, we study the structure of the pointed-Gromov-Hausdorff limits of sequences of Ricci shrinkers. We define a regular-singular decomposition following the work of Cheeger-Colding for manifolds with a uniform Ricci curvature lower bound, and prove that the regular part of any Ricci shrinker limit space is convex, inspired by Colding-Naber's original idea of parabolic smoothing of the distance functions.

Local existence and uniqueness of skew mean curvature flow

Abstract: The Skew Mean Curvature Flow(SMCF) is a Schrodinger-type geometric flow canonically defined on a co-dimension two submanifold, which generalizes the famous vortex filament equation in fluid dynamics. In this paper, we prove the local existence and uniqueness of general dimensional SMCF in Euclidean spaces.

On birational boundedness of foliated surfaces

Abstract: 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: \mathbb Z_{\geq 0}\to \mathbb Z $, then there exists an integer $N_1>0$ such that if $(X,\mathcal F)$ is a canonical or nef model of a foliation of general type with Hilbert polynomial $\chi (X, mK_{\mathcal F})=P(m)$ for all $m\in \mathbb Z_{\geq 0}$, then $|mK_{\mathcal F}|$ defines a birational map for all $m\geq N_1$. We also prove a Grauert-Riemannschneider type vanishing theorem for foliated surfaces with canonical singularities.

Singular tuples of matrices is not a null cone (and, the symmetries of algebraic varieties)

Abstract: The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: SINGn,m{{\rm SING}_{n,m}}, consisting of all m-tuples of n×n{n\times n} complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in SINGn,m{{\rm SING}_{n,m}} will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear group actions. Can this be used for the problem above? Our main result is negative: SINGn,m{{\rm SING}_{n,m}} is not the null cone of any (reductive) group action! This stands in stark contrast to a non-commutative analog of this variety, and points to an inherent structural difficulty of SINGn,m{{\rm SING}_{n,m}}. To prove this result, we identify precisely the group of symmetries of SINGn,m{{\rm SING}_{n,m}}. We find this characterization, and the tools we introduce to prove it, of independent interest. Our work significantly generalizes a result of Frobenius for the special case m=1{m=1}, and suggests a general method for determining the symmetries of algebraic varieties.

On derived equivalences and homological dimensions

Abstract: Abstract 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.

Special functions and Gauss–Thakur sums in higher rank and dimension

Abstract: Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 80s where they were at the heart of comparison isomorphisms, further important applications e.g. to transcendence theory have only been discovered recently. The Anderson-Thakur special function interpolates L-values via Pellarin-type identities, and its values at algebraic elements recover Gauss-Thakur sums, as shown by Angles and Pellarin. For Drinfeld-Hayes modules, generalizations of Anderson generating functions have been introduced by Green-Papanikolas and -- under the name of `special functions' -- by Angles-Ngo Dac-Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A-module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A-motive. We also undertake the study of Gauss-Thakur sums for Anderson A-modules, and show that the result of Angles-Pellarin relating values of the special functions to Gauss-Thakur sums holds in this generality.

Higher-page Bott–Chern and Aeppli cohomologies and applications

Abstract: For every positive integer $r$, we introduce two new cohomologies, that we call $E_r$-Bott-Chern and $E_r$-Aeppli, on compact complex manifolds. When $r=1$, they coincide with the usual Bott-Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when $r\geq 2$. They provide analogues in the Bott-Chern-Aeppli context of the $E_r$-cohomologies featuring in the Frolicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page-$(r-1)$-$\partial\bar\partial$-manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott-Chern and Aeppli cohomologies and for the spaces featuring in the Frolicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

Smooth rational affine varieties with infinitely many real forms

Abstract: We construct smooth rational real algebraic varieties of every dimension ≥ 4 {\\geq 4} which admit infinitely many pairwise non-isomorphic real forms.

Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation

Abstract: An absolute parallelism for $2$-nondegenerate CR manifolds $M$ of hypersurface type was recently constructed independently by Isaev-Zaitsev, Medori-Spiro, and Pocchiola in the minimal possible dimension ($\dim M=5$), and for $\dim M=7$ in certain cases by the first author. We develop a bigraded analog of Tanaka's prolongation procedure to construct a canonical absolute parallelism for these CR structures in arbitrary (odd) dimension with Levi kernel of arbitrary admissible dimension. We introduce the notion of a bigraded Tanaka symbol. Under regularity assumption that the symbol is a Lie algebra, we define a bigraded analog of the Tanaka universal algebraic prolongation and prove that for any CR structure with a given regular symbol there exists a canonical absolute parallelism on a bundle whose dimension is that of this bigraded prolongation. We show that there is a unique (up to local equivalence) such CR structure whose algebra of infinitesimal symmetries has maximal possible dimension, and the latter algebra is isomorphic to the real part of the bigraded prolongation of the symbol. In the case of $1$-dimensional Levi kernel we classify all regular symbols and calculate their bigraded prolongations. In this case the regular symbols can be subdivided into nilpotent, strongly non-nilpotent and weakly non-nilpotent. The bigraded prolongation of strongly non-nilpotent symbols is isomorphic to $\mathfrak{so}\left(m,\mathbb C\right)$ where $m=\tfrac{1}{2}(\dim M+5)$. Any real form of this algebra, except $\mathfrak{so}\left(m\right)$ and $\mathfrak{so}\left(m-1,1\right)$, corresponds to the real part of the bigraded prolongation of exactly one strongly non-nilpotent symbol. However, for a fixed $\dim M\geq 7$ the dimension of the bigraded prolongations achieves its maximum on one of the nilpotent regular symbols, and this maximal dimension is equal to $\tfrac{1}{4}(\dim M-1)^2+7$.

The resolution property via Azumaya algebras

Abstract: Using formal-local methods, we prove that a separated and normal tame Artin surface has the resolution property. By proving that normal tame Artin stacks can be rigidified, we ultimately reduce our analysis to establishing the existence of Azumaya algebras. Our construction passes through the case of tame Artin gerbes, tame Artin curves, and algebraic space surfaces, each of which we establish independently.

Lagrangian cobordism and tropical curves

Abstract: We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.