Mathematische Annalen

http://link.springer.com/journal/208

List of Papers (Total 240)

On K3 double planes covering Enriques surfaces

The moduli space of nodal Enriques surfaces is irreducible of dimension 9. A nodal Enriques surface is shown to be the quotient of a double cover of the plane by a lift of the Cremona involution. We also show that this gives a straightforward proof of the known description of the automorphism group for the generic such surface.

Equidistribution in shrinking sets and \(L^4\) -norm bounds for automorphic forms

We study two closely related problems stemming from the random wave conjecture for Maaß forms. The first problem is bounding the \(L^4\)-norm of a Maaß form in the large eigenvalue limit; we complete the work of Spinu to show that the \(L^4\)-norm of an Eisenstein series \(E(z,1/2+it_g)\) restricted to compact sets is bounded by \(\sqrt{\log t_g}\). The second problem is quantum...

Hyperbolic systems with non-diagonalisable principal part and variable multiplicities, I: well-posedness

In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the...

Misiurewicz parameters and dynamical stability of polynomial-like maps of large topological degree

Given a family of polynomial-like maps of large topological degree, we relate the presence of Misiurewicz parameters to a growth condition for the volume of the iterates of the critical set. This generalizes to higher dimensions the well-known equivalence between stability and normality of the critical orbits in dimension one. We also introduce a notion of holomorphic motion of...

Dynamics of transcendental Hénon maps

The dynamics of transcendental functions in the complex plane has received a significant amount of attention. In particular much is known about the description of Fatou components. Besides the types of periodic Fatou components that can occur for polynomials, there also exist so-called Baker domains, periodic components where all orbits converge to infinity, as well as wandering...

SU(2) \(^2\) -invariant \(G_2\) -instantons

We initiate the systematic study of \(G_2\)-instantons with SU(2)\(^2\)-symmetry. As well as developing foundational theory, we give existence, non-existence and classification results for these instantons. We particularly focus on \(\mathbb {R}^4\times S^3\) with its two explicitly known distinct holonomy \(G_2\) metrics, which have different volume growths at infinity...

Moduli of non-commutative polarized schemes

We construct, using geometric invariant theory, a quasi-projective Deligne–Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded \(A_\infty \)-algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a...

Univalent completion

We show by elementary means that every Kan fibration in simplicial sets can be embedded in a univalent Kan fibration.

Piecewise-regular maps

Let V, W be real algebraic varieties (that is, up to isomorphism, real algebraic sets), and \(X \subseteq V\) some subset. A map from X into W is said to be regular if it can be extended to a regular map defined on some Zariski locally closed subvariety of V that contains X. Furthermore, a continuous map \(f :X \rightarrow W\) is said to be piecewise-regular if there exists a...

Spectral shift functions and Dirichlet-to-Neumann maps

The spectral shift function of a pair of self-adjoint operators is expressed via an abstract operator-valued Titchmarsh–Weyl m-function. This general result is applied to different self-adjoint realizations of second-order elliptic partial differential operators on smooth domains with compact boundaries and Schrödinger operators with compactly supported potentials. In these...

Rationality results for the exterior and the symmetric square L-function (with an appendix by Nadir Matringe)

Let \(G=\mathrm{GL}_{2n}\) over a totally real number field F and \(n\ge 2\). Let \(\Pi \) be a cuspidal automorphic representation of \(G(\mathbb {A})\), which is cohomological and a functorial lift from SO\((2n+1)\). The latter condition can be equivalently reformulated that the exterior square L-function of \(\Pi \) has a pole at \(s=1\). In this paper, we prove a rationality...

Relative K-stability for Kähler manifolds

We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general \(L^p\) lower bound on the Calabi functional involving test configurations and their associated numerical...

Hyperbolic geometry and moduli of real curves of genus three

The moduli space of smooth real plane quartic curves consists of six connected components. We prove that each of these components admits a real hyperbolic structure. These connected components correspond to the six real forms of a certain hyperbolic lattice over the Gaussian integers. We will study this Gaussian lattice in detail. For the connected component that corresponds to...

Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings

We construct and study the moduli of continuous representations of a profinite group with integral p-adic coefficients. We present this moduli space over the moduli space of continuous pseudorepresentations and show that this morphism is algebraizable. When this profinite group is the absolute Galois group of a p-adic local field, we show that these moduli spaces admit Zariski...

Rational points of varieties with ample cotangent bundle over function fields

Let K be the function field of a smooth curve over an algebraically closed field k. Let X be a scheme, which is smooth and projective over K. Suppose that the cotangent bundle \(\Omega _{X/K}\) is ample. Let \(R:=\mathrm{Zar}(X(K)\cap X)\) be the Zariski closure of the set of all K-rational points of X, endowed with its reduced induced structure. We prove that for each...

Non-properly embedded H-planes in \({\mathbb H}^2\times {\mathbb R}\)

For any \(H \in (0,\frac{1}{2})\), we construct complete, non-proper, stable, simply-connected surfaces embedded in \({\mathbb H}^2\times {\mathbb R}\) with constant mean curvature H.

The gradient flow of O’Hara’s knot energies

Jun O’Hara invented a family of knot energies \(E^{j,p}\), \(j,p \in (0, \infty )\), O’Hara in Topology Hawaii (Honolulu, HI, 1990). World Science Publication, River Edge 1992. We study the negative gradient flow of the sum of one of the energies \(E^\alpha = E^{\alpha ,1}\), \(\alpha \in (2,3)\), and a positive multiple of the length. Showing that the gradients of these knot...

Covariants of binary sextics and vector-valued Siegel modular forms of genus two

We extend Igusa’s description of the relation between invariants of binary sextics and Siegel modular forms of degree 2 to a relation between covariants and vector-valued Siegel modular forms of degree 2. We show how this relation can be used to effectively calculate the Fourier expansions of Siegel modular forms of degree 2.

An upper bound for the pseudoisotopy stable range

We prove that the pseudoisotopy stable range for manifolds of dimension 2n can be no better than \((2n-2)\). In order to do so, we define new characteristic classes for block bundles, extending our earlier work with Ebert, and prove their non-triviality. We also explain how similar methods show that \(\mathrm {Top}(2n)/\mathrm {O}(2n)\) is rationally \((4n-5)\)-connected.

A necessary condition of potential blowup for the Navier–Stokes system in half-space

Assuming that T is a potential blow up time for the Navier–Stokes system in \(\mathbb {R}^{3}_{+}\), we show that the norm of the velocity field in the Lorenz space \(L^{3,q}\) with \(q<\infty \) goes to \(\infty \) as time t approaches T.

Strongly nonnegative curvature

We prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition: their curvature operator can be modified with a 4-form to become positive-semidefinite.

Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence

We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was also...

Well-posedness of hyperbolic systems with multiplicities and smooth coefficients

We study hyperbolic systems with multiplicities and smooth coefficients. In the case of non-analytic, smooth coefficients, we prove well-posedness in any Gevrey class and when the coefficients are analytic, we prove \(C^\infty \) well-posedness. The proof is based on a transformation to block Sylvester form introduced by D’Ancona and Spagnolo (Boll UMI 8(1B):169–185, 1998) which...