Nevanlinna and algebraic hyperbolicity

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair [Formula: see text], where [Formula: see text] is a projective variety and [Formula: see text] is an effective Cartier divisor on [Formula: see text]. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by Păun and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596 ].

Successive minima and asymptotic slopes in Arakelov geometry

Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\bar {D}$ be an adelic ${\mathbb {R}}$ -Cartier divisor on $X$ . We prove a conjecture of Chen, showing that the essential minimum $\zeta _{\mathrm {ess}}(\bar {D})$ of $\bar {D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $\zeta _{\mathrm {ess}}(\bar {D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom–Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space $X = {\mathbb {P}}_K^{d}$ , our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and Rémond.

Complements on adic spaces

This chapter analyzes a collection of complements in the theory of adic spaces. These complements include adic morphisms, analytic adic spaces, and Cartier divisors. It turns out that there is a very general criterion for sheafyness. In general, uniformity does not guarantee sheafyness, but a strengthening of the uniformity condition does. Moreover, sheafyness, without any extra assumptions, implies other good properties. Ultimately, it is not immediately clear how to get a good theory of coherent sheaves on adic spaces. The chapter then considers Cartier divisors on adic spaces. The term closed Cartier divisor is meant to evoke a closed immersion of adic spaces.

Triviality Properties of Principal Bundles on Singular Curves. II

For $G$ a split semi-simple group scheme and $P$ a principal $G$-bundle on a relative curve $X\rightarrow S$, we study a natural obstruction for the triviality of $P$ on the complement of a relatively ample Cartier divisor $D\subset X$. We show, by constructing explicit examples, that the obstruction is nontrivial if $G$ is not simply connected, but it can be made to vanish by a faithfully flat base change, if $S$ is the spectrum of a dvr (and some other hypotheses). The vanishing of this obstruction is shown to be a sufficient condition for étale local triviality if $S$ is a smooth curve, and the singular locus of $X-D$ is finite over $S$.

The Noether–Lefschetz locus of surfaces in toric threefolds

The Noether–Lefschetz theorem asserts that any curve in a very general surface [Formula: see text] in [Formula: see text] of degree [Formula: see text] is a restriction of a surface in the ambient space, that is, the Picard number of [Formula: see text] is [Formula: see text]. We proved previously that under some conditions, which replace the condition [Formula: see text], a very general surface in a simplicial toric threefold [Formula: see text] (with orbifold singularities) has the same Picard number as [Formula: see text]. Here we define the Noether–Lefschetz loci of quasi-smooth surfaces in [Formula: see text] in a linear system of a Cartier ample divisor with respect to a [Formula: see text]-regular, respectively 0-regular, ample Cartier divisor, and give bounds on their codimensions. We also study the components of the Noether–Lefschetz loci which contain a line, defined as a rational curve which is minimal in a suitable sense.

RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus, those homotopy groups – called higher Chow groups with modulus – generalize additive higher Chow groups of Bloch–Esnault, Rülling, Park and Krishna–Levine, and that sheafified on $\overline{X}_{\text{Zar}}$ gives a candidate definition for a relative motivic complex of the pair, that we compute in weight $1$ . When $\overline{X}$ is smooth over $k$ and $D$ is such that $D_{\text{red}}$ is a normal crossing divisor, we construct a fundamental class in the cohomology of relative differentials for a cycle satisfying the modulus condition, refining El Zein’s explicit construction of the fundamental class of a cycle. This is used to define a natural regulator map from the relative motivic complex of $(\overline{X},D)$ to the relative de Rham complex. When $\overline{X}$ is defined over $\mathbb{C}$ , the same method leads to the construction of a regulator map to a relative version of Deligne cohomology, generalizing Bloch’s regulator from higher Chow groups. Finally, when $\overline{X}$ is moreover connected and proper over $\mathbb{C}$ , we use relative Deligne cohomology to define relative intermediate Jacobians with modulus $J_{\overline{X}|D}^{r}$ of the pair $(\overline{X},D)$ . For $r=\dim \overline{X}$ , we show that $J_{\overline{X}|D}^{r}$ is the universal regular quotient of the Chow group of $0$ -cycles with modulus.

Zero cycles with modulus and zero cycles on singular varieties

Given a smooth variety$X$and an effective Cartier divisor$D\subset X$, we show that the cohomological Chow group of 0-cycles on the double of$X$along$D$has a canonical decomposition in terms of the Chow group of 0-cycles$\text{CH}_{0}(X)$and the Chow group of 0-cycles with modulus$\text{CH}_{0}(X|D)$on$X$. When$X$is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of$\text{CH}_{0}(X|D)$. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that$\text{CH}_{0}(X|D)$is torsion-free and there is an injective cycle class map$\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$if$X$is affine. For a smooth affine surface$X$, this is strengthened to show that$K_{0}(X,D)$is an extension of$\text{CH}_{1}(X|D)$by$\text{CH}_{0}(X|D)$.

Log-canonical pairs and Gorenstein stable surfaces with

We classify log-canonical pairs $(X,{\rm\Delta})$ of dimension two such that $K_{X}+{\rm\Delta}$ is an ample Cartier divisor with $(K_{X}+{\rm\Delta})^{2}=1$, giving some applications to stable surfaces with $K^{2}=1$. A rough classification is also given in the case where ${\rm\Delta}=0$.

An approach of the minimal model program for horospherical varieties via moment polytopes

We describe the minimal model program in the family of ℚ-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP for toric varieties due to M. Reid, and we complete the results on MMP for spherical varieties due to M. Brion in the case of horospherical varieties.

Cartier and Weil divisors on varieties with quotient singularities

It is well-known that the notions of Weil and Cartier Q-divisors coincide for V-manifolds. The main goal of this paper is to give a direct constructive proof of this result providing a procedure to express explicitly a Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.