/pdf/introduction-to-the-minimal-model-problem-2rzka1dyt7.pdf

Introduction to the Minimal Model Problem

Higher direct images of dualizing sheaves. II

Dans son travail sur la log-concavite des multiplicites, Okounkov montre au passage que l'on peut associer un corps convexe a un systeme lineaire sur une variete projective, puis utiliser la geometrie convexe pour etudier ces systemes lineaires. Bien qu'Okounkov travaille essentiellement dans le cadre classique des fibres en droites amples, il se trouve que sa construction s'etend au cas d'un grand diviseur arbitraire. De plus, ce point de vue permet de rendre transparentes de nombreuses proprietes de base des invariants asymptotiques des systemes lineaires, et ouvre la porte a de nombreuses extensions. Le but de cet article est d'initier un developpement systematique de la theorie et de donner quelques applications et exemples.

/pdf/convex-bodies-associated-to-linear-series-2k3c3rzc0k.pdf

Convex bodies associated to linear series

We prove Maruyama’s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring. Our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field. This result implies that in mixed characteristic the moduli spaces of Gieseker semistable sheaves are projective schemes of finite type. The proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant. This inequality also leads to effective restriction theorems in all characteristics, improving earlier results in characteristic zero.

http://annals.math.princeton.edu/wp-content/uploads/annals-v159-n1-p04.pdf

Semistable sheaves in positive characteristic

In his work on log-concavity of multiplicities, Okounkov showed in passing that one could associate a convex body to a linear series on a projective variety, and then use convex geometry to study such linear systems. Although Okounkov was essentially working in the classical setting of ample line bundles, it turns out that the construction goes through for an arbitrary big divisor. Moreover, this viewpoint renders transparent many basic facts about asymptotic invariants of linear series, and opens the door to a number of extensions. The purpose of this paper is to initiate a systematic development of the theory, and to give a number of applications and examples.

Convex Bodies Associated to Linear Series

In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture).

Arithmetic height functions over finitely generated fields

An interesting point of the above theorem is that even if the weak positivity of disx/y (E) at y is a global property on Y, it can be derived from the local assumption "the goodness of Xg and the semistability of Eg". This gives a great advantage to our applications. In order to understand the intuition underlying the theorem, let us consider a simulated case. Namely, we suppose that f: X -* Y is a smooth surface fibred over a curve and the fiber is general. Bogomolov's instability theorem [1] says that if E is semistable, then the codimension two cycle 2rc2 (E) (r 1)cl (E)2 has nonnegative degree. So if we push it down to a codimension one cycle on Y, then one can rephrase Bogomolov's theorem as saying that the semistability of Ey implies the non-negativity of disx/y(E).

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Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves

We introduce the volume function for C∞-hermitian invertible sheaves on an arithmetic variety as an analogue of the geometric volume function. The main result of this paper is the continuity of the arithmetic volume function. As a consequence, we have the arithmetic Hilbert-Samuel formula for a nef C∞-hermitian invertible sheaf. We also give another applications, for example, a generalized Hodge index theorem, an arithmetic Bogomolov-Gieseker’s inequality, etc. INTRODUCTION Let X be a d-dimensional projective arithmetic variety and Pic(X) the group of isomorphism classes of C∞-hermitian invertible sheaves onX . For L ∈ Pic(X), the volume vol(L) of L is defined by vol(L) = lim sup m→∞ log#{s ∈ H(X,mL) | ∥s∥sup ≤ 1} md/d! . For example, if L is ample, then vol(L) = deg(ĉ(L)·d) (cf. Lemma 3.1). This is an arithmetic analogue of the volume function for invertible sheaves on a projective variety over a field. The geometric volume function plays a crucial role for the birational geometry via big invertible sheaves. In this sense, to introduce the arithmetic analogue of it is very significant. The first important property of the volume function is the characterization of a big C∞hermitian invertible sheaf by the positivity of its volume (cf. Theorem 4.5). The second one is the homogeneity of the volume function, namely, vol(nL) = nvol(L) for all nonnegative integers n (cf. Proposition 4.7). By this property, it can be extended to Pic(X)⊗ Q. From viewpoint of arithmetic analogue, the most important and fundamental question is the continuity of vol : Pic(X)⊗Q → R, that is, the validity of the formula: lim e1,...,en∈Q e1→0,...,en→0 vol(L+ e1A1 + · · ·+ enAn) = vol(L) for any L,A1, . . . , An ∈ Pic(X) ⊗ Q. The main purpose of this paper is to give an affirmative answer for the above question (cf. Theorem 5.4). As a consequence, we have the following arithmetic Hilbert-Samuel formula for a nef C∞-hermitian invertible sheaf: Date: 5/January/2007, 17:30(JP), (Version 2.0). 1991Mathematics Subject Classification. 14G40, 11G50. 1

/pdf/continuity-of-volumes-on-arithmetic-varieties-4bv6auyv6n.pdf

Continuity of volumes on arithmetic varieties

In this paper, we propose a new height function for a variety defined over a finitely generated field over Q. For this height function, we will prove Northcott's theorem and Bogomolov's conjecture, so that we can recover the original Raynaud's theorem (Manin-Mumford's conjecture).

Let f : X --> Y be a projective morphism of smooth algebraic varieties over an algebraically closed field of characteristic zero with dim f = 1. Let E be a vector bundle of rank r on X. In this paper, we would like to show that if X_y is smooth and E_y is semistable for some point y of Y, then f_* (2r c_2(E) - (r-1) c_1(E)^2) is weakly positive at y. We apply this result to obtain the following description of the cone of weakly positive $\QQ$-Cartier divisors on the moduli space of stable curves. Let M_g (resp. M_g^0) be the moduli space of stable (resp. smooth) curves of genus g >= 2. Let h be the Hodge class and d_i's (i = 0,...,[g/2]) the boundary classes. A Q-Cartier divisor x h + y_0 d_0 + ... + y_[g/2] d_[g/2] is weakly positive over M_g^0 if and only if x >= 0, g x + (8g + 4) y_0>= 0, and i(g-i) x + (2g+1) y_i>= 0 for all 1 <= i <= [g/2].

Atsushi Moriwaki

Papers

Arithmetic height functions over finitely generated fields

Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves

Continuity of volumes on arithmetic varieties

Arithmetic height functions over finitely generated fields

Relative Bogomolov's inequality and the cone of positive divisors on the moduli space of stable curves