Timeline for Motivic DT-Invariants for the Algebro-Geophobic
Current License: CC BY-SA 3.0
8 events
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| Jun 27, 2012 at 17:31 | comment | added | Jim Bryan | There are a bunch of papers out there taking this approach and I'm not sure where exactly I recommend you start, but you could try this one of Mozgovoy: arxiv.org/pdf/1103.2902v1.pdf. The references in that paper may also lead you to some stuff that will help you (for example Reineke's papers). All the ideas go back to Kontsevich-Soibelmann, but there is a small cottage industry of people (including myself) who have narrowed the level of generality of KS theory in order to make everything more concrete and explicit. | |
| Jun 26, 2012 at 14:36 | comment | added | Steve | @Jim: this sounds like what I was looking for. Do you know of any references for these aspects of the theory? | |
| Jun 25, 2012 at 23:57 | comment | added | Jim Bryan | There is one in-road to this subject which is more algebraic than geometric and does not require so much machinery. I'm talking about categories of representations of a quiver with super-potential. These are CY3 categories and admit motivic DT invariants, wall-crossing, etcetera. But everything is fairly concrete, stability comes from ordinary GIT stability, and since there is a global super-potential, you don't need all the $A_{\infty}$ business to generate local super-potentials. The definition of the motivic invariants is fairly easy to understand. | |
| Jun 25, 2012 at 11:42 | comment | added | Arend Bayer | I am afraid there is no easy path to the definition of motivic DT-invariants. You need moduli stacks of stable objects, the Grothendieck group of (classes of) motives, the construction of the motivic Milnor fibre. Having said that, I think there are two silver linings: 1. None of these concepts are quite as scary as they are sometimes made out to be. (For example, the Grothendieck group of classes of motives is a simpler concept than any category of motives.) 2. The wall-crossing behaviour of DT-invariants is the interesting and purely algebraic part of the story. See e.g. Keller's notes. | |
| Jun 25, 2012 at 8:12 | comment | added | Chris Brav | Bernhard Keller has notes from an algebraic perspective here: arXiv:1102.4148. | |
| Jun 25, 2012 at 3:02 | comment | added | Yosemite Sam | I wish such material existed! (although I am quite algebrophobic to be honest, so it's the gentleness I'd be interested in) For non-refined DT invariants I find Bridgeland's Introduction to Motivic Hall Algebras great. In the end etale just means local diffeomorphism (or relative Kahler differentials zero if that's clearer to you o_O) and stacks are just a fact of life and one ends up thinking about them just as any other space (I guess). At least the idea of a motivic Hall algebra is quite simple. | |
| Jun 25, 2012 at 1:59 | comment | added | David Corwin | Props for inventing a new term "Algebro-Geophobic". | |
| Jun 25, 2012 at 0:54 | history | asked | Steve | CC BY-SA 3.0 |