Related papers: Canonical key formula for projective abelian schem…

Let $\pi:\CA\ra S$ be an abelian scheme over a scheme $S$ which is quasi-projective over an affine noetherian scheme and let $\CL$ be a symmetric, rigidified, relatively ample line bundle on $\CA$. We show that there is an isomorphism…

A good canonical projection of a surface $S$ of general type is a morphism to the 3-dimensional projective space P^3 given by 4 sections of the canonical line bundle. To such a projection one associates the direct image sheaf F of the…

In this paper, we explore when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. Given an infinite field $k$ and a sequence of relatively prime integers $a_0 = 0 < a_1 <…

We investigate sheaves supported on the zero section of the total space of a locally-free sheaf E on a smooth, projective variety X when the top exterior power of E is isomorphic to the canonical bundle of X. We rephrase this construction…

Let G=SL(E) be the special linear algebraic group on E where E is a finite dimensional vector space over a field K of characteristic zero. In this paper we study the canonical filtration of the dual G-module of global sections of a…

Let $k$ be a field, let ${\sf C}$ be a $k$-linear abelian category, let $\underline{\mathcal{L}}:=\{\mathcal{L}_{i}\}_{i \in \mathbb{Z}}$ be a sequence of objects in ${\sf C}$, and let $B_{\underline{\mathcal{L}}}$ be the associated orbit…

In this paper we show that the global (log) canonical threshold of $d$-sheeted covers of the $M$-dimensional projective space of index 1, where $d\geqslant 4$, is equal to one for almost all families (except for a finite set). The varieties…

Canonical heights and Arakelov geometry on semi-abelian varieties. In this paper, we propose a construction of the canonical heights on an extension of an abelian variety by the multiplicative group, in the framework of Arakelov geometry.…

An outstanding folklore conjecture asserts that, for any prime $p$, up to isomorphism the projective plane $PG(2,\mathbb{F}_p)$ over the field $\mathbb{F}_p := \mathbb{Z}/p\mathbb{Z}$ is the unique projective plane of order $p$. Let $\pi$…

We prove that a 1-dimnl family of abelian varieties with an ample sheaf defining principal polarization can be canonically compactified (after a finite base change) to a projective family with an ample sheaf. We show that the central fiber…

Let Y be a non-singular projective manifold with an ample canonical sheaf, and let V be a rational variation of Hodge structures of weight one on Y with Higgs bundle E(1,0) + E(0,1), coming from a family of Abelian varieties. If Y is a…

Given an essentially finite type morphism of schemes f: X --> Y and a positive integer d, let f^{d}: X^{d} --> Y denote the natural map from the d-fold fiber product, X^{d}, of X over Y and \pi_i: X^{d} --> X the i'th canonical projection.…

This paper investigates the transfer of classical geometric structures from a smooth manifold $M$ to its Weil bundle $(M^\mathbf A, \tilde\pi_M, M)$ associated with a Weil algebra $\mathbf A$. We show that various structures including…

Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing…

We prove that the canonical bundle of any holomorphic family of compact complex algebraic manifolds carries a singular Hermitian metric having non-negative curvature current and such that every holomorphic section of the canonical bundle of…

We define and construct integral canonical models for automorphic vector bundles over Shimura varieties of abelian type. More precisely, we first build on Kisin's work to construct integral canonical models over rings of integers of number…

Let $\pi\cln X\to \Delta^m$ be a proper smooth K\"ahler morphism from a complex manifold $X$ to the unit polydisc $\Delta^m$. Suppose the fibers over the complement of a proper analytic subset are biholomorphic to a fixed projective…

We propose a sheaf-theoretic approach to the theory of differential calculi on quantum principal bundles over non-affine bases. After recalling the affine case we define differential calculi on sheaves of comodule algebras as sheaves of…

This paper introduces an abelian category of logarithmic coherent sheaves that arranges coherent sheaves across all expansions and root stacks of a simple normal crossing degeneration. Formally, logarithmic coherent sheaves are coherent…

We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that…