arXiv subjects

Sven Meinhardt

Publications and source records attributed to Sven Meinhardt.

10 recordsLinked to original sources

arXiv2020

Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras

This paper concerns the cohomological aspects of Donaldson-Thomas theory for Jacobi algebras and the associated cohomological Hall algebra, introduced by Kontsevich and Soibelman. We prove the Hodge-theoretic categorification of the integrality conjecture and the wall crossing formula, and furthermore realise the isomorphism in both of these theorems as Poincaré-Birkhoff-Witt isomorphisms for the associated cohomological Hall algebra. We do this by defining a perverse filtration on the cohomological Hall algebra, a result of the "hidden properness" of the semisimplification map from the moduli stack of semistable representations of the Jacobi algebra to the coarse moduli space of polystable representations. This enables us to construct a degeneration of the cohomological Hall algebra, for generic stability condition and fixed slope, to a free supercommutative algebra generated by a mixed Hodge structure categorifying the BPS invariants. As a corollary of this construction we furthermore obtain a Lie algebra structure on this mixed Hodge structure - the Lie algebra of BPS invariants - for which the entire cohomological Hall algebra can be seen as the positive part of a Yangian-type quantum group.

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arXiv2018

On motivic vanishing cycles of critical loci

Let $U$ be a smooth scheme over an algebraically closed field $\mathbb K$ of characteristic zero and $f:U\to{\mathbb A}^1$ a regular function, and write $X=$Crit$(f)$, as a closed subscheme of $U$. The motivic vanishing cycle $MF_{U,f}^ϕ$ is an element of the $\hatμ$-equivariant motivic Grothendieck ring ${\mathcal M}^{\hatμ}_X$ defined by Denef and Loeser math.AG/0006050 and Looijenga math.AG/0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) $MF_{U,f}^ϕ$ depends only on the third-order thickenings $U^{(3)},f^{(3)}$ of $U,f$. (b) If $V$ is another smooth scheme, $g:V\to{\mathbb A}^1$ is regular, $Y=$Crit$(g)$, and $Φ:U\to V$ is an embedding with $f=g\circΦ$ and $Φ\vert_X:X\to Y$ an isomorphism, then $Φ\vert_X^*(MF_{V,g}^ϕ)$ equals $MF_{U,f}^ϕ$ "twisted" by a motive associated to a principal ${\mathbb Z}_2$-bundle defined using $Φ$, where now we work in a quotient ring $\bar{\mathcal M}^{\hatμ}_X$ of ${\mathcal M}^{\hatμ}_X$. (c) If $(X,s)$ is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive $MF_{X,s} \in\bar{\mathcal M}^{\hatμ}_X$, such that if $(X,s)$ is locally modelled on Crit$(f:U\to{\mathbb A}^1)$, then $MF_{X,s}$ is locally modelled on $MF_{U,f}^ϕ$. Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090.

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arXiv2016

The motivic Donaldson-Thomas invariants of (-2) curves

In this paper we calculate the motivic Donaldson-Thomas invariants for (-2)-curves arising from 3-fold flopping contractions in the minimal model programme. We translate this geometric situation into the machinery developed by Kontsevich and Soibelman, and using the results and framework developed previously by the authors we describe the monodromy on these invariants. In particular, in contrast to all existing known Donaldson-Thomas invariants for small resolutions of Gorenstein singularities these monodromy actions are nontrivial.

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arXiv2016

An introduction into (motivic) Donaldson-Thomas theory

The aim of the paper is to provide a rather gentle introduction into Donaldson-Thomas theory using quivers with potential. The reader should be familiar with some basic knowledge in algebraic or complex geometry. The text contains many examples and exercises to support the process of understanding the main concepts and ideas.

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arXiv2016

Donaldson-Thomas invariants versus intersection cohomology of quiver moduli

The main result of this paper is the statement that the Hodge theoretic Donaldson-Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson-Thomas "function" to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson-Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

math.AG↗

arXiv2015

Donaldson-Thomas theory for categories of homological dimension one with potential

The aim of the paper is twofold. Firstly, we give an axiomatic presentation of Donaldson-Thomas theory for categories of homological dimension at most one with potential. In particular, we provide rigorous proofs of all standard results concerning the integration map, wall-crossing, PT-DT correspondence, etc. following Kontsevich and Soibelman. We also show the equivalence of their approach and the one given by Joyce and Song. Secondly, we relate Donaldson-Thomas functions for such a category with arbitrary potential to those with zero potential under some mild conditions. As a result of this, we obtain a geometric interpretation of Donaldson-Thomas functions in all known realizations, i.e. mixed Hodge modules, perverse sheaves and constructible functions.

math.AG↗

arXiv2015

Donaldson-Thomas invariants vs. intersection cohomology for categories of homological dimension one

The present paper is an extension of a previous paper written in collaboration with Markus Reineke dealing with quiver representations. The aim of the paper is to generalize the theory and to provide a comprehensive theory of Donaldson-Thomas invariants for abelian categories of homological dimension one (without potential) satisfying some technical conditions. The theory will apply for instance to representations of quivers, coherent sheaves on smooth projective curves, and some coherent sheaves on smooth projective surfaces. We show that the (motivic) Donaldson-Thomas invariants satisfy the Integrality conjecture and identify the Hodge theoretic version with the (compactly supported) intersection cohomology of the corresponding moduli spaces of objects. In fact, we deal with a refined version of Donaldson-Thomas invariants which can be interpreted as classes in the Grothendieck group of some "sheaf" on the moduli space. In particular, we reproduce the intersection complex of moduli spaces using Donaldson-Thomas theory.

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arXiv2011

Motivic DT-invariants for the one loop quiver with potential

In this paper we compute the motivic Donaldson--Thomas invariants for the quiver with one loop and any potential. As the presence of arbitrary potentials requires the full machinery of \hat(μ)-equivariant motives, we give a detailed account of them. In particular, we will prove two results for the motivic vanishing cycle which might be of importance not only in Donaldson--Thomas theory.

math.AG↗

arXiv2008

Quotient categories, stability conditions, and birational geometry

This article deals with the quotient category of the category of coherent sheaves on an irreducible smooth projective variety by the full subcategory of sheaves supported in codimension greater than c. It turns out that this category has homological dimension c. As an application of this, we will describe the space of stability conditions on its derived category in the case c=1. Moreover, we describe all exact equivalences between these quotient categories in this particular case which is closely related to classification problems in birational geometry.

math.AG↗

arXiv2007

Stability conditions on generic complex tori

In this paper we describe a simply connected component of the complex manifold of stability conditions on the bounded derived category of a generic complex torus of any dimension. A torus is called generic if there are no nontrivial integral (p,p)-classes. We give an explicit description of all stability conditions in that connected component.

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