scholarly journals Moduli of Bridgeland semistable objects on 3-folds and Donaldson–Thomas invariants

2019 ◽  
Vol 2019 (747) ◽  
pp. 175-219 ◽  
Author(s):  
Dulip Piyaratne ◽  
Yukinobu Toda
Keyword(s):  
Finite Type ◽  
Equivalent Form ◽  
Stability Conditions ◽  
Weak Stability ◽  
Algebraic Stacks ◽  
Moduli Stacks ◽  
Étale Quotients

Abstract In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.

scholarly journals Uniformizing the Moduli Stacks of Global G-Shtukas

2019 ◽  
Author(s):  
E Arasteh Rad ◽  
Urs Hartl
Keyword(s):  
Finite Type ◽  
Moduli Spaces ◽  
Function Field ◽  
Group Scheme ◽  
Base Change ◽  
Generic Fiber ◽  
Smooth Projective Curve ◽  
Langlands Program ◽  
Moduli Stacks ◽  
Divisible Groups

Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

scholarly journals A fiber dimension theorem for essential and canonical dimension

2012 ◽  
Vol 149 (1) ◽  
pp. 148-174 ◽  
Author(s):  
Roland Lötscher
Keyword(s):  
Algebraic Geometry ◽  
Lower Bound ◽  
Finite Type ◽  
Algebraic Groups ◽  
General Version ◽  
Essential Dimension ◽  
Algebraic Stacks ◽  
Algebraic Tori ◽  
Canonical Dimension

AbstractThe well-known fiber dimension theorem in algebraic geometry says that for every morphism f:X→Y of integral schemes of finite type the dimension of every fiber of f is at least dim X−dim Y. This has recently been generalized by Brosnan, Reichstein and Vistoli to certain morphisms of algebraic stacks f:𝒳→𝒴, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover, we will prove a variant of this theorem, where essential dimension and canonical dimension are linked. These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by MacDonald, Meyer, Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

RECURSIVE COMPUTATION OF THE INVARIANT DISTRIBUTION OF A DIFFUSION: THE CASE OF A WEAKLY MEAN REVERTING DRIFT

2003 ◽  
Vol 03 (04) ◽  
pp. 435-451 ◽  
Author(s):  
DAMIEN LAMBERTON ◽  
GILLES PAGÈS
Keyword(s):  
Diffusion Process ◽  
Invariant Distribution ◽  
Brownian Diffusion ◽  
Stability Conditions ◽  
Euler Scheme ◽  
Weak Stability ◽  
Recursive Procedure ◽  
Recursive Computation

We study a recursive procedure, based on the Euler scheme with decreasing step, for the computation of the invariant distribution of a Brownian diffusion process satisfying weak stability conditions. We are able to extend some of the results of [5] to diffusions for which the drift size at infinity is very small.

scholarly journals Stability conditions on product varieties

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yucheng Liu
Keyword(s):  
Stability Condition ◽  
Projective Variety ◽  
Smooth Projective Variety ◽  
Stability Conditions ◽  
Projective Curve ◽  
Weak Stability ◽  
Smooth Projective Curve

AbstractGiven a stability condition on a smooth projective variety X, we construct a family of stability conditions on {X\times C}, where C is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macrì and weak stability conditions on the Abramovich-Polishchuk heart of a bounded t-structure in {D(X\times C)}.

scholarly journals Modular compactifications of the space of pointed elliptic curves I

2010 ◽  
Vol 147 (3) ◽  
pp. 877-913 ◽  
Author(s):  
David Ishii Smyth
Keyword(s):  
Stability Conditions ◽  
Model Program ◽  
Arithmetic Genus ◽  
Minimal Model Program ◽  
Stable Curves ◽  
Moduli Stacks ◽  
Log Minimal Model Program ◽  
Curve Singularities ◽  
Definition Of ◽  
Genus One

AbstractWe introduce a sequence of isolated curve singularities, the elliptic m-fold points, and an associated sequence of stability conditions, generalizing the usual definition of Deligne–Mumford stability. For every pair of integers 1≤m<n, we prove that the moduli problem of n-pointed m-stable curves of arithmetic genus one is representable by a proper irreducible Deligne–Mumford stack $\overline {\mathcal {M}}_{1,n}(m)$. We also consider weighted variants of these stability conditions, and construct the corresponding moduli stacks $\overline {\mathcal {M}}_{1,\mathcal {A}}(m)$. In forthcoming work, we will prove that these stacks have projective coarse moduli and use the resulting spaces to give a complete description of the log minimal model program for $\overline {M}_{1,n}$.

Stability Conditions Related to a Previously Unrecognized Form of Thrombin

1972 ◽  
Vol 27 (02) ◽  
pp. 361-362 ◽  
Author(s):  
Walter H. Seegers ◽  
Lowell E. McCoy
Keyword(s):  
Stability Conditions
Sign in / Sign up

Export Citation Format

Share Document