Volume 2016, Issue 721

We give a precise formulation of the modularity principle for the log canonical models M¯g⁢(α):=Proj⊕d≥0H0⁢(ℳ¯g,⌊d⁢(Kℳ¯g+α⁢δ)⌋)${\overline{\text{M}}_{g}(\alpha):=\text{Proj}\oplus_{d\geq 0}\text{H}^{0}(% \overline{\mathcal{M}}_{g},\lfloor d(K_{\overline{\mathcal{M}}_{g}}+\alpha% \delta)\rfloor)}$ of the moduli space of stable curves. We define a new invariant of Gorenstein curve singularities with 𝔾m${\mathbb{G}_{m}}$-action which can be used to predict the critical α-value at which a singularity first arises in the modular interpretation of M¯g⁢(α)${\overline{\text{M}}_{g}(\alpha)}$. We compute these critical α-values for large classes of singularities with 𝔾m${\mathbb{G}_{m}}$-action, including all ADE, toric planar, and unibranch Gorenstein singularities, and use these results to give a conjectural outline of the log MMP for ℳ¯g${\overline{\mathcal{M}}_{g}}$.

Using Gauss–Manin derivatives of generalized normal functions, we arrive at results on the non-triviality of the transcendental regulator for Km$K_{m}$ of a very general projective algebraic manifold. Our strongest results are for the transcendental regulator for K1$K_{1}$ of a very general K⁢3$K3$ surface and its self-product. We also construct an explicit family of K1$K_{1}$ cycles on H⊕E8⊕E8$H\oplus E_{8}\oplus E_{8}$-polarized K⁢3$K3$ surfaces, and show they are indecomposable by a direct evaluation of the real regulator. Critical use is made of natural elliptic fibrations, hypersurface normal forms, and an explicit parametrization by modular functions.

In this paper we give a new proof of the André–Oort conjecture under the generalised Riemann hypothesis. In fact, we generalise the strategy pioneered by Edixhoven, and implemented by Klingler and Yafaev, to all special subvarieties. Thus, we remove ergodic theory from the proof of Klingler, Ullmo and Yafaev and replace it with tools from algebraic geometry. Our key ingredient is a lower bound for the degrees of strongly special subvarieties coming from Prasad’s volume formula for S-arithmetic quotients of semisimple groups.

We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kähler metrics which are toric with respect to a common 2-torus action. In the generic case, these “ambitoric” structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable. We use this description to classify 4-dimensional Einstein metrics which are hermitian with respect to both orientations, as well as a class of solutions to the Einstein–Maxwell equations including riemannian analogues of the Plebański–Demiański metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kähler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kähler metrics. For both the Einstein–Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kähler metrics on toric 4-orbifolds with second Betti number b2=2${b_{2}=2}$.

The spectral curve correspondence for finite-type solutions of the sinh- Gordon equation describes how they arise from and give rise to hyperelliptic curves with a real structure. Constant mean curvature (CMC) 2-tori in 𝕊3${\mathbb{S}^{3}}$ result when these spectral curves satisfy periodicity conditions. We prove that the spectral curves of CMC tori are dense in the space of smooth spectral curves of finite-type solutions of the sinh-Gordon equation. One consequence of this is the existence of countably many real n -dimensional families of CMC tori in 𝕊3${\mathbb{S}^{3}}$ for each positive integer n .

Let f∈Sk0+2⁢(Γ0⁢(N⁢p))${f\in S_{k_{0}+2}(\Gamma_{0}(Np))}$ be a normalized N -new eigenform with p∤N${p\,{\nmid}\,N}$ and such that ap2≠pk0+1${a_{p}^{2}\neq p^{k_{0}+1}}$ and ordp⁡(ap)<k0+1${\operatorname{ord}_{p}(a_{p})<k_{0}+1}$. By Coleman’s theory, there is a p -adic family 𝐅${\mathbf{F}}$ of eigenforms whose weight k0+2${k_{0}+2}$ specialization is f . Let K be a real quadratic field and let ψ be an unramified character of Gal⁡(K¯/K)${\operatorname{Gal}(\overline{K}/K)}$. Under mild hypotheses on the discriminant of K and the factorization of N , we construct a p -adic L -function ℒ𝐅/K,ψ${\mathcal{L}_{\mathbf{F}/K,\psi}}$ interpolating the central critical values of the Rankin L -functions associated to the base change to K of the specializations of 𝐅${\mathbf{F}}$ in classical weight, twisted by ψ. When the character ψ is quadratic, ℒ𝐅/K,ψ${\mathcal{L}_{\mathbf{F}/K,\psi}}$ factors into a product of two Mazur-Kitagawa p -adic L -functions. If, in addition, 𝐅${\mathbf{F}}$ has p -new specialization in weight k0+2${k_{0}+2}$, then under natural parity hypotheses we may relate derivatives of each of the Mazur-Kitagawa factors of ℒ𝐅/K,ψ${\mathcal{L}_{\mathbf{F}/K,\psi}}$ at k0${k_{0}}$ to Bloch–Kato logarithms of Heegner cycles. On the other hand the derivatives of our p -adic L -functions encodes the position of the so called Darmon cycles. As an application we prove rationality results about them, generalizing theorems of Bertolini–Darmon, Seveso, and Shahabi.

Let G be a group that admits a cocompact classifying space for proper actions X . We derive a formula for the Bredon cohomological dimension for proper actions of G in terms of the relative cohomology with compact support of certain pairs of subcomplexes of X . We use this formula to compute the Bredon cohomological dimension for proper actions of fundamental groups of non-positively curved simple complexes of finite groups. As an application we show that if a virtually torsion-free group acts properly and chamber transitively on a building, its virtual cohomological dimension coincides with its Bredon cohomological dimension. This covers the case of Coxeter groups and graph products of finite groups.