# Motivic Cell Structure of Toric Surfaces

Wednesday, April 17th, 2013 | Author:

In this post I'll do a few very explicit computations for motivic cell structures of smooth projective toric varieties coming from the Białynicki-Birula decomposition, namely $\mathbb{P}^1, \mathbb{P}^1 \times \mathbb{P}^1, \mathbb{P}^2$ and Hirzebruch surfaces. It is a bit lengthy but maybe helpful to anyone who wants to do some explicit calculations with BB-decompositions. I hope you're accustomed to toric varieties, but I won't do anything fancy. You can safely skip the motivic part of this post.

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# Białynicki-Birula and Motivic Decompositions

Thursday, April 04th, 2013 | Author:

This is about Białynicki-Birula's paper from '72 on actions of reductive linear algebraic groups on non-singular varieties, in particular Gm-operations on smooth projective varieties. I give a proof sketch of Theorem 4.1 therein and explain a little bit how Brosnan applied these results in 2005 to get decompositions of the Chow motive of smooth projective varieties with Gm-operation. Wendt used these methods in 2010 to lift such a decomposition on the homotopy-level, to prove that smooth projective spherical varieties admit stable motivic cell decompositions. Most of this blogpost consists of an outline of the B-B paper.

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# Invariants of projective space III: Motives

Monday, December 10th, 2012 | Author:

I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. The discussion started with cohomology (part 1), continued with bundles and cycles (part 2) and in this part 3, we discuss motivic stuff.

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# Invariants of projective space II: Cycles and Bundles

Thursday, December 06th, 2012 | Author:

I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. The discussion started with cohomology (part 1) and in this part 2, we discuss intersection-theoretic and bundle-theoretic invariants. In part 3 we will see the motivic stuff.

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# Invariants of projective space I: Cohomology

Tuesday, December 04th, 2012 | Author:

I want to explain a particularly easy example of a motivic cellular decomposition: That of $n$-dimensional projective space. We will have a look at the cohomology, the Chow groups and the algebraic K-theory of projective space -- a discussion probably interesting to non-motivic people as well. After these invariants, I will look at the motive and the A¹-homotopy type. Then I want to describe the decomposition of the motive (and the homotopy type) homotopy-theoretically, by means of cofiber sequences. (We will see that projective space is not isomorphic to a coproduct of motivic spheres with the same motive). Of course, nothing is new, I'm just working out exercises here.

In this part 1, I discuss only the cohomology of $\mathbb{P}^n$. Part 2 contains a discussion of the intersection theory and bundles and part 3 contains the motivic stuff. I intentionally left out usage of projective bundle formulas, as I will discuss them separately.

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# Cellular objects in the motivic model category

Friday, November 16th, 2012 | Author:

This article explains the notion of cellular objects in the motivic model category, which are like CW complexes for the algebraic/motivic world.

(last edit on 2014-04-08, added a remark on realizations and a Thom construction)