Big line bundles over arithmetic varieties

In this paper, we consider two questions about topological entropy of dynamical systems. We propose to resolve these questions by the same approach of using étale analogs of topological and algebraic dynamical systems. The first question is to define topological entropy for a topological dynamical system ( f , X , Ω ) (f,X,\Omega ) . The main idea is to make use of, in addition to invariant compact subspaces of ( X , Ω ) (X,\Omega ) , compactifications of étale covers π : ( f ′ , X ′ , Ω ′ ) → ( f , X , Ω ) \pi :(f’,X’,\Omega ’)\rightarrow (f,X,\Omega ) ; that is, π ∘ f ′ = f ∘ π \pi \circ f’=f\circ \pi and the fibers of π \pi are all finite. We prove some basic results and propose a conjecture, whose validity allows us to prove further results. The second question is to define topological entropy for algebraic dynamical systems, with the requirement that it should be as close to the pullback on cohomology groups as possible. To this end, we develop an étale analog of algebraic dynamical systems.

The boundary quotient for algebraic dynamical systems

The paper introduces a new method for finding optimal control of algebraic dynamic systems. The structure of algebraic dynamical systems is nonlinear with quadratic and bilinear terms. A new hybrid extended Fourier series is introduced, and state and control variables of the system are expanded by this series. Moreover, properties of new series are presented, and integration and product operational matrices are obtained. Using operational matrices, optimal control of the systems is converted to a set of simultaneous nonlinear algebraic relations. An illustrative example is included to compare our results with those in the literature.

A Fej\'{e}r-type theorem is proved within the framework of $C^*$-algebras associated with certain irreversible algebraic dynamical systems. This makes it possible to strengthen a result on the structure of the relative commutant of a family of generating isometries in a boundary quotient.

The equidistribution conjecture is proved for general semiabelian varieties over number fields. Previously, this conjecture was only known in the special case of almost split semiabelian varieties through work of Chambert-Loir. The general case has remained intractable so far because the height of a semiabelian variety is negative unless it is almost split. In fact, this places the conjecture outside the scope of Yuan’s equidistribution theorem on algebraic dynamical systems. To overcome this, an asymptotic adaption of the equidistribution technique invented by Szpiro, Ullmo, and Zhang is used here. It also allows a new proof of the Bogomolov conjecture and hence a self-contained proof of the strong equidistribution conjecture in the same general setting.

Given a nef and big line bundle L L on a projective variety X X of dimension d ≥ 2 d \geq 2 , we prove that the Seshadri constant of L L at a very general point is larger than ( d + 1 ) 1 d − 1 (d+1)^{\frac {1}{d}-1} . This slightly improves the lower bound 1 / d 1/d established by Ein, Küchle and Lazarsfeld [J. Differential Geom. 42 (1995), pp. 193–219]. The proof relies on the concept of successive minima for line bundles recently introduced by Ambro and Ito [Adv. Math. 365 (2020), 38 pp.].

Let $X$ denote a smooth projective variety of dimension $n$ defined over the field of complex numbers such that the anti-canonical line bundle $-K_X$ of $X$ is nef and big with $h^{0}(-K_{X})>0$, and let $L$ be a nef and big line bundle on $X$. In this paper, we consider the dimension of the global sections of $K_{X}+mL$ with $m\geq n-1$ for this case. In particular, under the assumption that $K_{X}+(n-1)L$ is nef, we prove that $h^{0}(K_{X}+(n-1)L)>0$ if $6\leq n\leq 9$ and $L$ is ample.

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

In this chapter we start some basic facts on analytic and complex geometry (divisors, blowing-up, big line bundles), we prove the theorem of Siegel-Remmert-Thimm, that the field of meromorphic functions on a connected compact complex manifold is an algebraic field of transcendence degree less than the dimension of the manifold. Then we study in more detail Moishezon manifolds and their relation to projective manifolds. In particular we prove that a Moishezon manifold is projective if and only if it carries a Kähler metric. We end the section 2.2 by giving the solution of the Grauert-Riemenschneider conjecture as application of the holomorphic Morse inequalities from Theorem 1.7.1.KeywordsLine BundleComplex ManifoldAhler ManifoldHolomorphic Line BundleCompact Complex ManifoldThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We establish an equidistribution theorem for the common zeros of random sections of high powers of several singular Hermitian big line bundles associated to moderate measures.

Let L be a big line bundle on a compact complex manifold X. Given a non-pluripolar compact subset K of X and a continuous Hermitian metric e −φ on L, we define the energy at equilibrium of (K,φ) as the Monge-Ampère energy of the extremal psh weight associated to (K,φ). We prove the differentiability of the energy at equilibrium with respect to φ, and we show that this energy describes the asymptotic behaviour as k→∞ of the volume of the sup-norm unit ball induced by (K,k φ) on the space of global holomorphic sections H 0(X,kL). As a consequence of these results, we recover and extend Rumely’s Robin-type formula for the transfinite diameter. We also obtain an asymptotic description of the analytic torsion, and extend Yuan’s equidistribution theorem for algebraic points of small height to the case of a big line bundle.

Traces of algebraic integers and dynamical systems

Given a matroid M on the ground set E, the Bergman fan e B(M), or space of M- ultrametrics, is a polyhedral complex in R E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situ- ation, we study the following problem: Given a point ω in R E , we wish to nd an M-ultrametric which is closest to it in the '¥-metric. The solution to this problem follows easily from the existence of the subdominant M-ultramet- ric: a componentwise maximum M-ultrametric which is componentwise smaller than ω. A pro- cedure for computing it is given, which brings together the points of view of matroid theory and tropical geometry. When the matroid in question is the graphical matroid of the complete graph Kn, the Bergman fan e B(Kn) parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet's method for computing the tree that most closely matches measured data.

This is the second of a series of papers dealing with an analog in Arakelov geometry of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula "`a la Bott" for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; recent results of Bismut-Goette on the equivariant (Ray-Singer) analytic torsion play a key role in the proof.

C*-algebraic partial dynamical systems