We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold X equipped with a linear system V ∗ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on X . Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When X is an algebraic manifold having a sufficiently large automorphism group G and V ∗ is a linear representation of G , we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed in the paper \cite{LSY1}, joint with R. Song. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety.

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A1 structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given embeddings with isomorphic normal bundles of a closed manifold B into manifolds Q1 and Q2, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A1 category obtained by considering exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.

The superradiant scattering of a scalar field with frequency and angular momentum (ω, m) by a near-extreme Kerr black hole with mass and spin (M, J) was derived in the seventies by Starobinsky, Churilov, Press and Teukolsky. In this paper we show that for frequencies scaled to the superradiant bound the full functional dependence on (ω, m, M, J) of the scattering amplitudes is precisely reproduced by a dual two-dimensional conformal field theory in which the black hole corresponds to a specific thermal state and the scalar field to a specific operator. This striking agreement corroborates a conjectured Kerr/CFT correspondence.

We establish the existence of infinitely many$polynomial$progressions in the primes; more precisely, given any integer-valued polynomials$P$_{1}, …,$P$_{$k$}∈$Z$[$m$] in one unknown$m$with$P$_{1}(0) = … =$P$_{$k$}(0) = 0, and given any$ε$> 0, we show that there are infinitely many integers$x$and$m$, with $1 \leqslant m \leqslant x^\varepsilon$ , such that$x$+$P$_{1}($m$), …,$x$+$P$_{$k$}($m$) are simultaneously prime. The arguments are based on those in [18], which treated the linear case$P$_{$j$}= ($j$− 1)$m$and$ε$= 1; the main new features are a localization of the shift parameters (and the attendant Gowers norm objects) to both coarse and fine scales, the use of PET induction to linearize the polynomial averaging, and some elementary estimates for the number of points over finite fields in certain algebraic varieties.

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold$X$. Given a big (1, 1)-cohomology class$α$on$X$(i.e. a class that can be represented by a strictly positive current) and a positive measure$μ$on$X$of total mass equal to the volume of$α$and putting no mass on pluripolar sets, we show that$μ$can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in$α$. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if$μ$has$L$^{1+$ε$}-density with respect to Lebesgue measure. If$μ$is smooth and positive everywhere, we prove that$T$is smooth on the ample locus of$α$provided$α$is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.