We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ('number of BPS states with given charge' in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit 'as the motive of affine line approaches to 1' we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

What do you do to start reading dimension theory? Searching the book that you love to read first or find an interesting book that will make you want to read? Everybody has difference with their reason of reading a book. Actuary, reading habit must be from earlier. Many people may be love to read, but not a book. It's not fault. Someone will be bored to open the thick book with small words to read. In more, this is the real condition. So do happen probably with this dimension theory.

Aspects of topology: On dimension theory

We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula for birational morphisms and we prove a geometric analogue of a result of Oesterle.

/pdf/germs-of-arcs-on-singular-algebraic-varieties-and-motivic-3eaatwy3o8.pdf

Germs of arcs on singular algebraic varieties and motivic integration

Basic Hodge Theory.- Compact Kahler Manifolds.- Pure Hodge Structures.- Abstract Aspects of Mixed Hodge Structures.- Mixed Hodge Structures on Cohomology Groups.- Smooth Varieties.- Singular Varieties.- Singular Varieties: Complementary Results.- Applications to Algebraic Cycles and to Singularities.- Mixed Hodge Structures on Homotopy Groups.- Hodge Theory and Iterated Integrals.- Hodge Theory and Minimal Models.- Hodge Structures and Local Systems.- Variations of Hodge Structure.- Degenerations of Hodge Structures.- Applications of Asymptotic Hodge Theory.- Perverse Sheaves and D-Modules.- Mixed Hodge Modules.

http://barbra-coco.dyndns.org/student/Peters_Steenbrinck.pdf

Mixed Hodge Structures

We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2011/0005/0002/CNTP-2011-0005-0002-a001.pdf

Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

We define motivic analogues of Igusa's local zeta functions. These functions take their values in a Grothendieck group of Chow motives. They specialize to p-adic Igusa local zeta functions and to the topological zeta functions we introduced several years ago. We study their basic properties, such as functional equations, and their relation with motivic nearby cycles. In particular the Hodge spectrum of a singular point of a function may be recovered from the Hodge realization of these zeta functions.

Motivic Igusa zeta functions

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical invariants.

Geometry on Arc Spaces of Algebraic Varieties

1.1. In this paper, intended to be the first in a series, we lay new general foundations for motivic integration and give answers to some important issues in the subject. Since its creation by Maxim Kontsevich [23], motivic integration developed quickly and has spread out in many directions. In a nutshell, in motivic integration, numbers are replaced by geometric objects, like virtual varieties, or motives. But, classicaly, not only numbers are defined using integrals, but also interesting classes of functions. The previous constructions of motivic integration were all quite geometric, and it was quite unclear how they could be generalized to handle integrals depending on parameters. The new approach we present here, based on cell decomposition, allows us to develop a very general theory of motivic integration taking parameters into account. More precisely, we define a natural class of functions – constructible motivic functions – which is stable under integration. The basic idea underlying our approach is to construct more generally push-forward morphisms f! which are functorial – they satisfy ( f ◦ g)! = f! ◦ g! – so that performing motivic integration corresponds to taking the push-forward to the point. This strategy has many technical advantages. In essence, it allows to reduce the construction of f! to the case of closed immersions and projections, and in the latter case we can perform induction on the relative dimension, the basic case being that of relative dimension 1,

/pdf/constructible-motivic-functions-and-motivic-integration-3qn81fdpl5.pdf

Constructible motivic functions and motivic integration

(1.1) Throughout this paper k always denotes a finite field Fq with q elements, and E a prime number not dividing q. The algebraic closure of a field K is denoted by / ( . Let ~b: k--+ C • be a nontrivial additive character, and ~ , the Qt-sheaf on A~ associated to ~ and the Artin-Schreier covering t q t = x. For a morphism f : X --+ A~, with X a scheme of finite type over k, one considers the exponential sum S(f ) = ~Xtk)~b(f(x)). By Grothendieck's trace formula we have

François Loeser

Papers

Germs of arcs on singular algebraic varieties and motivic integration

Motivic Igusa zeta functions

Geometry on Arc Spaces of Algebraic Varieties

Constructible motivic functions and motivic integration

Weights of exponential sums, intersection cohomology, and Newton polyhedra