Valuative Criterion Research Articles

Effective Morphisms and Quotient Stacks

Abstract We give a valuative criterion for when a smooth algebraic stack with a separated good moduli space is the quotient of a separated Deligne–Mumford stack by a torus. For doing so, we introduce a new class of morphisms, the effective morphisms, which are a generalization of separated morphisms.

An arithmetic valuative criterion for proper maps of tame algebraic stacks

The valuative criterion for proper maps of schemes has many applications in arithmetic, e.g. specializing Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_{p}$$\\end{document}-points to Fp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{p}$$\\end{document}-points. For algebraic stacks, the usual valuative criterion for proper maps is ill-suited for these kind of arguments, since it only gives a specialization point defined over an extension of the residue field, e.g. a Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_{p}$$\\end{document}-point will specialize to an Fpn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{p^{n}}$$\\end{document}-point for some n. We give a new valuative criterion for proper maps of tame stacks which solves this problem and is well-suited for arithmetic applications. As a consequence, we prove that the Lang–Nishimura theorem holds for tame stacks.

The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties

We prove the following result: if a $$\,\,\,\,\,{\mathbb {Q}}\,\,\,\,\,$$ -Fano variety is uniformly K-stable, then it admits a Kähler–Einstein metric. This proves the uniform version of Yau–Tian–Donaldson conjecture for all (singular) Fano varieties with discrete automorphism groups. We achieve this by modifying Berman–Boucksom–Jonsson’s strategy in the smooth case with appropriate perturbative arguments. This perturbation approach depends on the valuative criterion and non-Archimedean estimates, and is motivated by our previous paper.

Properness criteria for families of coherent analytic spaces

We extend Langton's valuative criterion for families of coherent algebraic sheaves to a complex analytic set-up. As a consequence we derive a set of sufficient conditions for the compactness of a moduli space of semistable sheaves over a compact complex manifold. This applies also to some cases appearing in complex projective geometry not covered by previous results.

Completeness of Finite-Rank Differential Varieties

Differential algebraic geometry offers tantalizing similarities to the algebraic version as well as puzzling anomalies. This thesis builds on results of Kolchin, Blum, Morrison, van den Dries, and Pong to study the problem of completeness for projective differential varieties. The classical fundamental theorem of elimination theory asserts that if V is a projective algebraic variety defined over an algebraically closed field K and W is any algebraic variety defined over K, then the projection VxW -> W takes Zariski-closed sets to Zariski-closed sets. Differential varieties defined by differential polynomial equations over a differentially closed field are more complicated. We give the first example of an incomplete finite-rank differential variety, as well as new instances of complete differential varieties. We also explain how model theory yields multiple versions of Pong's valuative criterion for completeness and reduces the differential completeness problem to one involving algebraic varieties over the complex numbers.

Grothendieck Duality and $\mathbb Q$-Gorenstein Morphisms

The notions of a \mathbb Q -Gorenstein scheme and a \mathbb Q -Gorenstein morphism are introduced for locally Noetherian schemes by dualizing complexes and (relative) canonical sheaves. These cover all the previously known notions of a \mathbb Q -Gorenstein algebraic variety and a \mathbb Q -Gorenstein deformation satisfying the Kollar condition, over a field. By studying the relative S _2 -condition and base change properties, valuable results are proved for \mathbb Q -Gorenstein morphisms, which include the infinitesimal criteria, the valuative criterion, and \mathbb Q -Gorenstein refinements.

A valuative criterion for uniform K-stability of ℚ-Fano varieties

Abstract We give a simple necessary and sufficient condition for uniform K-stability of ℚ \mathbb{Q} -Fano varieties.

New examples (and counterexamples) of complete finite-rank differential varieties

ABSTRACTDifferential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their behavior can vary in interesting ways. Workers in both differential algebra and model theory have investigated the property of completeness of differential varieties. After reviewing their results, we extend that work by proving several versions of a “differential valuative criterion" and using them to give new examples of complete differential varieties. We conclude by analyzing the first examples of incomplete, finite-rank projective differential varieties, demonstrating a clear difference from projective algebraic varieties.

Completeness in partial differential algebraic geometry

This paper is part of the model theory of fields of characteristic 0, equipped with m commuting derivation operators (DCF0,m). It continues to partial differential fields work begun by Wai-Yan Pong, who treated the case m=1. We study the concept of completeness in differential algebraic geometry, applying methods of model theory and differential algebra. Our central tool in applying the valuative criterion developed in differential algebra by E.R. Kolchin, Peter Bloom, and Sally Morrison is a fundamental theorem in classical elimination theory due to the model theorist Lou van den Dries. We use this valuative criterion to give a new family of complete differential algebraic varieties. In addition to completeness, we prove some embedding theorems for differential algebraic varieties of arbitrary differential transcendence degree. As a special case, we show that every differential algebraic subvariety of the projective line which has Lascar rank less than ωm can be embedded in the affine line.

Existence of log canonical closures

Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U 0⊂U, such that (X,Δ)× U U 0 has a good minimal model over U 0 and the images of all the non-klt centers intersect U 0, then (X,Δ) has a good minimal model over U. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

Moduli of PT-semistable objects I

We show boundedness for PT-semistable objects of any Chern classes on a smooth projective three-fold X. Then we show that the stack of objects in the heart 〈 Coh ⩽ 1 ( X ) , Coh ⩾ 2 ( X ) [ 1 ] 〉 satisfies a version of the valuative criterion for completeness. In the remainder of the paper, we give a series of results on how to compute cohomology with respect to this heart.

Weak subintegral closure of ideals

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a new geometric interpretation of the Reid–Roberts–Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal I of a ring A an ideal I > , which consists of all elements of A such that v ( a ) > v ( I ) , for all Rees valuations v of I . The ideal I > plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition A f and is contained in every reduction of I . We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative weak closure. We illustrate the usefulness of our valuative criterion.

Sheaves of t-structures and valuative criteria for stable complexes

In this work more questions arise than answers given, for which of course we do not apologize. The core of this paper is concerned with the construction of a ``constant'' t-structure on the bounded derived category of coherent sheaves $D(X\times S)$, with $X$ and $S$ smooth varieties, given a nondegenerate t-structure on $D(X)$ with noetherian heart. While we believe this construction and the methods involved should be generally useful in studying derived categories, we present here but one application: we prove a valuative criterion for separation and properness for the collection P(1) of stable objects of phase 1 under a numerical, locally finite and noetherian Bridgeland-Douglas stability condition $(Z,\cP)$ on $D(X)$, where $X$ is a smooth projective variety. As an immediate result, the number of long-exact sequences in the first named author's output is no longer an embarrassment.

Discrete valuations centered on local domains

We study applications of discrete valuations to ideals in analytically irreducible domains, in particular, applications to zero divisors modulo powers of ideals. We prove a uniform version of Izumi's theorem and calculate several examples illustrating it, such as for rational singularities. The paper contains a new criterion of analytic irreducibility, a new criterion of one-fiberedness, and a valuative criterion for when the normal cone of an ideal in an integrally closed domain is reduced.

Completions of local morphisms and valuations

In this note we study injective local morphisms \(\varphi: (R,{\mathfrak m})\rightarrow (S,{\mathfrak n})\) of local excellent domains. In particular we are interested in the problem when the \({\mathfrak n}\)–adic topology onS restricts to a topology on R that is linearly equivalent to the \({\mathfrak m}\)–adic topology. Using a valuative criterion, we prove this in case R is analytically irreducible and \(S/R\) is essentially of finite type, and we recover and extend a weak version of Gabrielov's rank condition.