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67

Classically, Grothendieck's motives are only the pure motives, meaning abelian-ish things which capture the (Weil-cohomology-style) $H^i$ of smooth, projective varieties. To see the relationship with motivic cohomology, one should extend the notion of motive so that non-pure (i.e. "mixed") motives are allowed, these mixed motives being abelian-ish things ...


33

Torsten Ekedahl proposed a definition of higher Grothendieck groups of varieties. Unfortunately it seems that he never wrote anything down on this topic before passing away. Torsten had quite a large number of unfinished mathematical manuscripts and projects. I don't know what happened to them, although surely someone at Stockholm University has taken care ...


25

Two clarifications: For anabelian geometry, you should ask how much information about a variety is contained in the Galois action on its etale fundamental group. While it's true that the motivic Galois group is a higher-dimensional analogue of the Galois group, it also should be true that motives are "just" a special kind of Galois representation, i.e. ...


24

Here are some comments about the use of topologies in motivic homotopy theory. This is based on the discussion in Morel-Voevodsky's "A^1-homotopy theory of schemes" p.94-95 (MV below), I only add some background and references. I also comment on the differences between the development of the unstable and stable theories. I am not an expert, so please comment/...


24

For future references. Feel free to edit to include new cases, or any improvements. As for 2015, the standard conjectures on algebraic cycles is unconditionally (at lest) known for $X$: Lefschetz standard conjecture (Grothendieck conjectures $A(X)$ and $B(X)$) a curve (trivial). a surface with $H^1(X)=2\cdot\mathrm{Pic}^0(X)$ (Grothendieck). an abelian ...


22

This circle of topics is certainly one of my favourite surprising connections in mathematics. I will try to outline what little I understand of the big picture. Apologies for the length. Hilbert's 3rd problem and Dehn complexes: As is well-known, Hilbert's 3rd problem asked for examples of tetrahedra of equal volume which are not scissors congruent, and ...


22

Here is a guess about the remark of Orlov. Suppose that one wants to define a good notion of noncommutative scheme, given that an affine noncommutative scheme is an associative algebra. Trying to define the spectrum of an associative algebra leads to various problems (c.f. this answer), so a different approach is needed. On the other hand there are ...


21

I wanted to contribute something because nobody's really explained why a t-structure on a stable $\infty$-category is the same thing as a t-structure on its homotopy category. It's not just because someone defined it as such in Definition 1.2.1.4 of Higher Algebra; it's because the most natural generalization is no generalization at all. Roughly speaking, a ...


21

As for the first two questions (papers, results, and applications): For motivation, I'd recommend understanding the content of Batyrev's paper "Birational Calabi-Yau n-folds have equal Betti numbers" which proves the claim in its title. Using motivic integration techniques analogous to the $p$-adic techniques in Batyrev's paper, Kontsevich proved that $K$-...


21

[I've incorporated or addressed comments of Dan Petersen and Daniel Litt into this. My thanks to them.] One sometimes says that $X$ admits a cellular decomposition if it admits a stratification by affine spaces. The isomorphism of the type you mention was known before Totaro, cf Fulton's Intersection Theory 19.1.11 (in my edition). This applies to flag ...


21

There's a very deep connection between motives and Grothendieck-Teichmüller theory but it isn't well-understood yet. I can't even frame it precisely in higher genus, but at least I can frame a precise conjecture in genus zero. It has to do with motives that are connected to periods of moduli spaces on the one hand, and Grothendieck-Teichmüller being ...


20

This phenomenon is specific to the fields $\mathbb F_q$ (in this case you need the Artin motives as well) and $\bar{\mathbb F}_q$. Roughly, the outline is as follows. If the Tate conjecture is true and the Frobenius action is semisimple, then the homological standard conjecture is true. Then Jannsen's Motives, numerical equivalence, and semi-simplicity ...


19

The question seems fine to me. Off the top of my head: 1) The Jacobian is a group, and in fact an abelian variety, whereas the curve usually isn't. This gives you a lot of structure to play with that you didn't have initially. For example, to show that a general curve doesn't map onto a curve of smaller positive genus, you can use the fact that the ...


18

I think the presence of Voevodsky's category of (mixed) motives is a red herring here. Let me briefly explain why I think that, and then say why any "real" notion of motives (say, pure Chow motives, as in Vivek Shende's comment) or, I guess, Voevodsky's motives, are much more serious than the so-called naive motives. The (partially conjectural) motivic ...


15

The first question (applied to $\mathrm{GL}(2)$-abelian varieties over $\mathbf{Q}$) seems to include the following problem: what totally real fields $F$ occur as the field of coefficients of a classical weight $2$ modular form? This seems a totally impossible question to answer. For example, it includes the question of which Hilbert modular surfaces $X_F$ ...


15

1) Counterexamples were found in the paper Brown, Francis; Schnetz, Oliver: "A $K3$ in $\phi^4$". Duke Math. J. 161 (2012), no. 10, 1817–1862. It is now the general feeling that most $\phi^4$-Feynman integrals are not mixed Tate. 2) In 2008, Schnetz has compiled a list of Feynman integrals in Quantum periods: A census of $\phi^4$-transcendentals. These were ...


15

I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint. Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with integral coefficients. Furthermore, motives with transfers(!; see below) with rational coefficients "do not depend on the choice of the topology" (whether one ...


15

I hope someone can provide a better answer with more of an eye towards the motivic world, but for now let me outline that the exact same phenomenon exists in classical stable homotopy theory. Here the analogue of $SH$ is the category $Sp$ of spectra. There are plenty of ways to define it, so let me assume it exists for now. One way to think about them is as ...


14

Isn't this very easy? If the varieties are $\mathbb{A}^1$-homotopy equivalent, then their Voevodsky motives are isomorphic also (since there is a connecting functor making the obvious diagram commutative). So it remains to note that all the ingredients of your question are "motivic". For this purpose one may recall that Chow motives embed into Voevodsky one ...


14

While waiting for someone more competent than me to answer, let me turn the question right back to you. Why should motivic cohomology be finitely generated? The answer is, of course, that there's no reason for it. And it is not. Let us take a look at a special example The Picard group Let us fix the ground field to be $\mathbb{C}$. The Picard group of a ...


13

(1) is true if $char(k)=0$. This follows from a combination of results. First of all, it is true over any field that the spectrum $MGL$ is connective, which means that $$MGL^{p,q}(X)=0$$ if $p>q+dim(X)$, $X\in Sm/k$ [1, Cor. 2.9]. (Slightly more is true: for any $p\geq q+dim(X)$, the orientation map $MGL\to H\mathbb{Z}$ induces an isomorphism $MGL^{p,q}(...


13

The survey of Nekovar tells you what was known about the Beilinson conjectures in the early 90s. Other surveys/introductions from that time include Scholl-Deninger, Soulé, Ramakrishnan (in Contemporary mathematics 83), and the volume edited by Rapoport, Schappacher and Schneider (introduction here, all articles here). Since then, not a lot has happened I ...


13

In the spirit of You Could Have Invented Spectral Sequences by T.Chow, I claim you could have invented $\operatorname{Ext}^{1}(\mathbb Q(0),M)$ as group of "rational points" of a motive. Here is how. The motivation comes from conjectures on special values of $L$-function. Before the general conjectures were formulated by S.Bloch and K.Kato, there were two ...


13

The most general functor of this form was constructed by Ayoub in La réalisation étale et les opérations de Grothendieck. Ayoub considers the ∞-category $DA^{et}(S,\Lambda)$ which is defined exactly as $SH(S)$ except that (1) spectra are replaced by chain complexes of $\Lambda$-modules and (2) the Nisnevich topology is replaced by the étale topology. So ...


13

Not sure about later developments, but the idea is mentioned in a famous passage of Grothendieck's Récoltes et Semailles. I quote from Roy Lisker's translation: Thus, the motive presents itself as the deepest "form invariant" which one has been able to associate up to the present moment with an algebraic variety, setting aside its "motivic fundamental ...


13

I'm very interested myself on a better answer to this question, but let me point out the obvious: the main problem is that there is no Hodge theory on positive characteristic. The proof in characteristic zero simply says that it is enough to consider $\mathbb{C}$ (via Lefschetz principle), and that there you can use Hodge theory, the Hodge index theorem in ...


13

Write $X=\mathrm{Spec}\, k$, which is a $0$-dimensional variety. Motives of $0$-dimensional varieties are called Artin motives, and they are pure. The Betti realization is the Betti cohomology of $X$, which is $$ H^0(X(\mathbb{C}),\mathbb{Q})=\mathbb{Q}^{X(\mathbb{C})}=\mathbb{Q}^{\mathrm{Hom}(k,\mathbb{C})}. $$ There is an involution induced by complex ...


12

Hi Mikhail, I honestly don't have a good answer, but I'll share my thoughts on this and related things, since this is potentially quite interesting. I don't see a problem in formally defining the Chow group of a compact Kähler manifold as the group of cycles modulo rational equivalence. However, if you want to compose correspondences, then you would need a ...


12

Partially inspired by an unpublished work of Wojtkoviak for $\zeta (3)$, all of this follows from this result by Deligne: Theorem. $\pi_1(\mathbb{P}^1-\{0,1,\infty\})$ is a smooth mixed Tate motive over $\mathrm{Spec}\mathbb{Z}$. Now, over $\mathbb{P}^1_\mathbb{Z}-\{0,1,\infty\})$ he can define motivic (polylogarithm) local system $P$, each fiber of which ...


12

Alain Connes: "a noncommutative algebra creates its own intrinsic time". First of all, as Yemon Choi commented, this quote of Alain Connes is a slogan, not a theorem. "Most of the NC algebras create their own intrinsic time" would be a bit more correct, more precisely: Theorem: a von Neumann algebra of type $\rm III$ creates its own intrinsic time up to ...


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