Volume 2021, Issue 775

We prove a priori bounds for all derivatives of non-relativistic Coulombic eigenfunctions ψ, involving negative powers of the distance to the singularities of the many-body potential. We use these to derive bounds for all derivatives of the corresponding one-electron densities ρ, involving negative powers of the distance from the nuclei. The results are both natural and optimal, as seen from the ground state of Hydrogen.

Let A be a connection of a principal bundle P over a Riemannian manifold M , such that its curvature FA∈Lloc2⁢(M){F_{A}\in L_{\mathrm{loc}}^{2}(M)} satisfies the stationarity equation. It is a consequence of the stationarity that θA⁢(x,r)=ec⁢r2⁢r4-n⁢∫Br⁢(x)|FA|2⁢𝑑Vg{\theta_{A}(x,r)=e^{cr^{2}}r^{4-n}\int_{B_{r}(x)}|F_{A}|^{2}\,dV_{g}} is monotonically increasing in r , for some c depending only on the local geometry of M . We are interested in the singular set defined by S⁢(A)={x:limr→0⁡θA⁢(x,r)≠0}{S(A)=\{x:\lim_{r\to 0}\theta_{A}(x,r)\neq 0\}}, and its stratification Sk⁢(A)={x:no tangent measure of A at x is (k+1)-symmetric}{S^{k}(A)=\{x:\text{no tangent measure of $A$ at $x$ is $(k+1)$-symmetric}\}}. We then introduce the quantitative stratification Sϵk⁢(A){S^{k}_{\epsilon}(A)}; roughly speaking Sϵk⁢(A){S^{k}_{\epsilon}(A)} is the set of points at which no ball Br⁢(x){B_{r}(x)} is ϵ-close to being (k+1){(k+1)}-symmetric. In the main theorems, we show that Sϵk{S^{k}_{\epsilon}} is k -rectifiable and satisfies the Minkowski volume estimate Vol⁡(Br⁢(Sϵk)∩B1)≤C⁢rn-k{\operatorname{Vol}(B_{r}(S^{k}_{\epsilon})\cap B_{1})\leq Cr^{n-k}}. Lastly, we apply the main theorems to the stationary Yang–Mills connections to obtain a rectifiability theorem that extends some previously known results in [G. Tian, Gauge theory and calibrated geometry. I, Ann. of Math. (2) 151 2000, 1, 193–268].

For a connected reductive group G defined over a non-archimedean local field F , we consider the Bernstein blocks in the category of smooth representations of G⁢(F){G(F)}. Bernstein blocks whose cuspidal support involves a regular supercuspidal representation are called regular Bernstein blocks. Most Bernstein blocks are regular when the residual characteristic of F is not too small. Under mild hypotheses on the residual characteristic, we show that the Bernstein center of a regular Bernstein block of G⁢(F){G(F)} is isomorphic to the Bernstein center of a regular depth-zero Bernstein block of G0⁢(F){G^{0}(F)}, where G0{G^{0}} is a certain twisted Levi subgroup of G . In some cases, we show that the blocks themselves are equivalent, and as a consequence we prove the ABPS Conjecture in some new cases.

We classify all minimal models X of dimension n , Kodaira dimension n-1{n-1} and with vanishing Chern number c1n-2⁢c2⁢(X)=0{c_{1}^{n-2}c_{2}(X)=0}. This solves a problem of Kollár. Completing previous work of Kollár and Grassi, we also show that there is a universal constant ϵ>0{\epsilon>0} such that any minimal threefold satisfies either c1⁢c2=0{c_{1}c_{2}=0} or -c1⁢c2>ϵ{-c_{1}c_{2}>\epsilon}. This settles completely a conjecture of Kollár.

In the present paper, we generalize the celebrated classical lemma of Birch and Heegner on quadratic twists of elliptic curves over ℚ{{\mathbb{Q}}}. We prove the existence of explicit infinite families of quadratic twists with analytic ranks 0 and 1 for a large class of elliptic curves, and use Heegner points to explicitly construct rational points of infinite order on the twists of rank 1. In addition, we show that these families of quadratic twists satisfy the 2-part of the Birch and Swinnerton-Dyer conjecture when the original curve does. We also prove a new result in the direction of the Goldfeld conjecture.

In this paper, 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 formulae 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 Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves ℳ0,n{\mathcal{M}_{0,n}} of genus zero with n marked points. We also study a morphism between certain rings of “motivic” periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest.

We develop new techniques for studying concentration of Laplace eigenfunctions ϕλ{\phi_{\lambda}} as their frequency, λ, grows. The method consists of controlling ϕλ⁢(x){\phi_{\lambda}(x)} by decomposing ϕλ{\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 λ-12{\lambda^{-\frac{1}{2}}}. We control ϕλ⁢(x){\phi_{\lambda}(x)} by the L2{L^{2}}-mass of ϕλ{\phi_{\lambda}} on each geodesic tube and derive a purely dynamical statement through which ϕλ⁢(x){\phi_{\lambda}(x)} can be studied. In particular, we obtain estimates on ϕλ⁢(x){\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∞{L^{\infty}}-norms, Lp{L^{p}}-norms, pointwise Weyl laws, and averages over submanifolds.

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.