Inventiones mathematicae最新文献

Let (X) be a smooth, complex Fano 4-fold, and (rho _{X}) its Picard number. We show that if (rho _{X}>12), then (X) is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions (fcolon Xto Y) such that (dim f(operatorname{Exc}(f))=2), together with the author’s previous work on Fano 4-folds. In particular, given (fcolon Xto Y) as above, under suitable assumptions we show that (S:=f(operatorname{Exc}(f))) is a smooth del Pezzo surface with (-K_{S}=(-K_{Y})_{|S}).

Abstract

A (C^{infty }) smooth surface diffeomorphism admits an SRB measure if and only if the set ({ x, limsup _{n}frac{1}{n}log |d_{x}f^{n}|>0}) has positive Lebesgue measure. Moreover the basins of the ergodic SRB measures are covering this set Lebesgue almost everywhere. We also obtain similar results for (C^{r}) surface diffeomorphisms with (+infty >r>1) .

We construct a non-full exceptional collection of maximal length consisting of line bundles on the blow-up of the projective plane in 10 general points. As a consequence, the orthogonal complement of this collection is a universal phantom category. This provides a counterexample to a conjecture of Kuznetsov and to a conjecture of Orlov.

Let (k) be a number field and (G) be a finite group. Let (mathfrak{F}_{k}^{G}(Q)) be the family of number fields (K) with absolute discriminant (D_{K}) at most (Q) such that (K/k) is normal with Galois group isomorphic to (G). If (G) is the symmetric group (S_{n}) or any transitive group of prime degree, then we unconditionally prove that for all (Kin mathfrak{F}_{k}^{G}(Q)) with at most (O_{varepsilon }(Q^{varepsilon })) exceptions, the (L)-functions associated to the faithful Artin representations of (mathrm{Gal}(K/k)) have a region of holomorphy and non-vanishing commensurate with predictions by the Artin conjecture and the generalized Riemann hypothesis. This result is a special case of a more general theorem. As applications, we prove that:

1. (1)

   there exist infinitely many degree (n) (S_{n})-fields over ℚ whose class group is as large as the Artin conjecture and GRH imply, settling a question of Duke;

2. (2)

   for a prime (p), the periodic torus orbits attached to the ideal classes of almost all totally real degree (p) fields (F) over ℚ equidistribute on (mathrm{PGL}_{p}(mathbb{Z})backslash mathrm{PGL}_{p}(mathbb{R})) with respect to Haar measure;

3. (3)

   for each (ell geq 2), the (ell )-torsion subgroups of the ideal class groups of almost all degree (p) fields over (k) (resp. almost all degree (n) (S_{n})-fields over (k)) are as small as GRH implies; and

4. (4)

   an effective variant of the Chebotarev density theorem holds for almost all fields in such families.

This belongs to a series of papers motivated by Ballmann’s Higher Rank Rigidity Conjecture. We prove the following. Let (X) be a CAT(0) space with a geometric group action (Gamma curvearrowright X). Suppose that every geodesic in (X) lies in an (n)-flat, (ngeq 2). If (X) contains a periodic (n)-flat which does not bound a flat ((n+1))-half-space, then (X) is a Riemannian symmetric space, a Euclidean building or non-trivially splits as a metric product. This generalizes the Higher Rank Rigidity Theorem for Hadamard manifolds with geometric group actions.

Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective asymptotic equidistribution result for one-parameter unipotent orbits in ({mathrm{SL}}(3, {mathbb{R}})/{mathrm{SL}}(3,{mathbb{Z}})). We also provide a complementary convergence statement, by developing the structural theory of dual Bohr sets: at the cost of a slightly stronger Diophantine assumption, this sharpens a result of Kleinbock’s from 2003. Finally, we refine the theory of logarithm laws in homogeneous spaces.

It has long been conjectured that for nonlinear wave equations that satisfy a nonlinear form of the null condition, the low regularity well-posedness theory can be significantly improved compared to the sharp results of Smith-Tataru for the generic case. The aim of this article is to prove the first result in this direction, namely for the time-like minimal surface equation in the Minkowski space-time. Further, our improvement is substantial, namely by (3/8) derivatives in two space dimensions and by (1/4) derivatives in higher dimensions.

We prove a scalar curvature rigidity theorem for convex polytopes. The proof uses the Fredholm theory for Dirac operators on manifolds with boundary. A variant of a theorem of Fefferman and Phong plays a central role in our analysis.

This work justifies the linear response formula for the Hall conductance of a two-dimensional disordered system. The proof rests on controlling the dynamics associated with a random time-dependent Hamiltonian.

The principal challenge is related to the fact that spectral and dynamical localization are intrinsically unstable under perturbation, and the exact spectral flow - the tool used previously to control the dynamics in this context - does not exist. We resolve this problem by proving a local adiabatic theorem: With high probability, the physical evolution of a localized eigenstate (psi ) associated with a random system remains close to the spectral flow for a restriction of the instantaneous Hamiltonian to a region (R) where the bulk of (psi ) is supported. Allowing (R) to grow at most logarithmically in time ensures that the deviation of the physical evolution from this spectral flow is small.

To substantiate our claim on the failure of the global spectral flow in disordered systems, we prove eigenvector hybridization in a one-dimensional Anderson model at all scales.

We construct special cycles on the moduli stack of hermitian shtukas. We prove an identity between (1) the (r^{mathrm{th}}) central derivative of non-singular Fourier coefficients of a normalized Siegel–Eisenstein series, and (2) the degree of special cycles of “virtual dimension 0” on the moduli stack of hermitian shtukas with (r) legs. This may be viewed as a function-field analogue of the Kudla-Rapoport Conjecture, that has the additional feature of encompassing all higher derivatives of the Eisenstein series.