Volume 2019, Issue 748

Hodge correlators are complex numbers given by certain integrals assigned to a smooth complex curve. We show that they are correlators of a Feynman integral, and describe the real mixed Hodge structure on the pronilpotent completion of the fundamental group of the curve. We introduce motivic correlators, which are elements of the motivic Lie algebra and whose periods are the Hodge correlators. They describe the motivic fundamental group of the curve. We describe variations of real mixed Hodge structures on a variety by certain connections on the product of the variety by twistor plane. We call them twistor connections. In particular, we define the canonical period map on variations of real mixed Hodge structures. We show that the obtained period functions satisfy a simple Maurer–Cartan type non-linear differential equation. Generalizing this, we suggest a DG-enhancement of the subcategory of Saito’s Hodge complexes with smooth cohomology. We show that when the curve varies, the Hodge correlators are the coefficients of the twistor connection describing the corresponding variation of real MHS. Examples of the Hodge correlators include classical and elliptic polylogarithms, and their generalizations. The simplest Hodge correlators on the modular curves are the Rankin–Selberg integrals. Examples of the motivic correlators include Beilinson’s elements in the motivic cohomology, e.g. the ones delivering the Beilinson–Kato Euler system on modular curves.

We show how to deduce the standard sign conjecture (a weakening of the Künneth standard conjecture) for Shimura varieties from some statements about discrete automorphic representations (Arthur’s conjectures plus a bit more). We also indicate what is known (to us) about these statements.

We show that if ϕ is a homeomorphism of a closed, orientable surface of genus g , and ϕ has large translation distance in the curve complex, then the fundamental group of the mapping torus Mϕ{M_{\phi}} has rank 2⁢g+1{2g+1}.

In this article we initiate research on locally compact C*\mathrm{C}^{*}-simple groups. We first show that every C*\mathrm{C}^{*}-simple group must be totally disconnected. Then we study C*\mathrm{C}^{*}-algebras and von Neumann algebras associated with certain groups acting on trees. After formulating a locally compact analogue of Powers’ property, we prove that the reduced group C*\mathrm{C}^{*}-algebra of such groups is simple. This is the first simplicity result for C*\mathrm{C}^{*}-algebras of non-discrete groups and answers a question of de la Harpe. We also consider group von Neumann algebras of certain non-discrete groups acting on trees. We prove factoriality, determine their type and show non-amenability. We end the article by giving natural examples of groups satisfying the hypotheses of our work.

We investigate the spaces of rational curves on a general hypersurface. In particular, we show that for a general degree d hypersurface in ℙn{\mathbb{P}^{n}} with n≥d+2{n\geq d+2}, the space ℳ¯0,0⁢(X,e){\overline{\mathcal{M}}_{0,0}(X,e)} of degree e Kontsevich stable maps from a rational curve to X is an irreducible local complete intersection stack of dimension e⁢(n-d+1)+n-4{e(n-d+1)+n-4}. This resolves all but one case of a conjecture of Coskun, Harris and Starr, and also proves that the Gromov–Witten invariants of these hypersurfaces are enumerative.

Associated to a Mukai flop X⇢X′{X\dashrightarrow X^{\prime}} is on the one hand a sequence of equivalences Db⁢(X)→Db⁢(X′){D^{b}(X)\to D^{b}(X^{\prime})}, due to Kawamata and Namikawa, and on the other hand a sequence of autoequivalences of Db⁢(X){D^{b}(X)}, due to Huybrechts and Thomas. We work out a complete picture of the relationship between the two. We do the same for standard flops, relating Bondal and Orlov’s derived equivalences to spherical twists, extending a well-known story for the Atiyah flop to higher dimensions.

We construct a new 20-dimensional family of projective six-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension-three subvarieties of the Grassmannian G⁢(3,6)G(3,6). These codimension-three subvarieties are defined as Lagrangian degeneracy loci and their construction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds of K⁢3K3-type, Beauville–Bogomolov degree 4 and divisibility 2 is unirational.

In this paper we prove some general results for constant mean curvature lamination limits of certain sequences of compact surfaces MnM_{n} embedded in ℝ3\mathbb{R}^{3} with constant mean curvature HnH_{n} and fixed finite genus, when the boundaries of these surfaces tend to infinity. Two of these theorems generalize to the non-zero constant mean curvature case, similar structure theorems by Colding and Minicozzi in [6, 8] for limits of sequences of minimal surfaces of fixed finite genus.

Let 𝐆\mathbf{G} be a tamely ramified connected reductive group defined over a non-archimedean local field k . We show that the Bernstein center of a tame supercuspidal block of 𝐆⁢(k)\mathbf{G}(k) is isomorphic to the Bernstein center of a depth-zero supercuspidal block of 𝐆0⁢(k)\mathbf{G}^{0}(k) for some twisted Levi subgroup of 𝐆0\mathbf{G}^{0} of 𝐆\mathbf{G}.