In this paper, we study the theory of complements, introduced by Shokurov, for Calabi–Yau type varieties with the coefficient set [0, 1]. We show that there exists a finite set of positive integers $\mathcal {N}$, such that if a threefold pair $(X/Z\ni z,B)$ has an $\mathbb {R}$-complement which is klt over a neighborhood of z, then it has an n-complement for some $n\in \mathcal {N}$. We also show the boundedness of complements for $\mathbb {R}$-complementary surface pairs.
We study the three-dimensional many-particle quantum dynamics in mean-field setting. We forge together the hierarchy method and the modulated energy method. We prove rigorously that the compressible Euler equation is the limit as the particle number tends to infinity and the Planck’s constant tends to zero. We improve the previous sufficient small time hierarchy argument to any finite time via a new iteration scheme and Strichartz bounds first raised by Klainerman and Machedon in this context. We establish strong and quantitative microscopic to macroscopic convergence of mass and momentum densities up to the 1st blow up time of the limiting Euler equation. We justify that the macroscopic pressure emerges from the space-time averages of microscopic interactions via the Strichartz-type bounds. We have hence found a physical meaning for Strichartz-type bounds.
We prove $L^p$ bounds for the Fourier extension operators associated to smooth surfaces in ${\mathbb {R}}^3$ with negative Gaussian curvatures for every $p>3.25$.
We formulate a local analogue of the ghost conjecture of Bergdall and Pollack, which essentially relies purely on the representation theory of ${{\,\textrm{GL}\,}}_2({\mathbb {Q}}_p)$. We further study the combinatorial properties of the ghost series as well as its Newton polygon, in particular, giving a characterization of the vertices of the Newton polygon and proving an integrality result of the slopes. In a forthcoming sequel, we will prove this local ghost conjecture under some mild hypothesis and give arithmetic applications.
We introduce an algebraicity criterion. It has the following form: Consider an analytic subvariety of some algebraic variety X over a global field K. Under certain conditions, if X contains many K-points, then X is algebraic over K. This gives a way to show the transcendence of points via the transcendence of analytic subvarieties. Such a situation often appears when we have a dynamical system, because we can often produce infinitely many points from one point via iterates. Combining this criterion and the study of invariant subvarieties, we get some results on the transcendence in arithmetic dynamics. We get a characterization for products of Böttcher coordinates or products of multiplicative canonical heights for polynomial dynamical pairs to be algebraic. For this, we study the invariant subvarieties for products of endomorphisms. In particular, we partially generalize Medvedev–Scanlon’s classification of invariant subvarieties of split polynomial maps to separable endomorphisms on $({\mathbb P}^1)^N$ in any characteristic. We also get some high dimensional partial generalization via introducing a notion of independence. We then study dominant endomorphisms f on ${\mathbb A}^N$ over a number field of algebraic degree $d\ge 2$. We show that in most cases (e.g. when such an endomorphism extends to an endomorphism on ${\mathbb P}^N$), there are many analytic curves centered at infinity which are periodic. We show that for most of them, it is algebraic if and only if it contains at least one algebraic point. We also study the periodic curves. We show that for most f, all periodic curves have degree at most 2. When $N=2$, we get a more precise classification result. We show that under a condition which is satisfied for a general f, if f has infinitely many periodic curves, then f is homogenous up to change of origin.