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The report of Mikhail Bondarko for 2007 1 Scientic activity; publications
The main scientic goal of the pro ject is the study of categories of motives. Recall that (Voevodsky's) motives correspond to a certain "universal cohomology theory for smooth algebraic varieties". Note that in contrast with "classical"categories of motives (i.e. Chow, homological and numerical ones) Voef evodsky's DMgmf DMgm are endowed with natural (and highly non-trivial) structures of triangulated categories. One of my main results is the study of the relation of DMgm with C how; this sheds some light on the celebrated standard motivic conjectures. In 2007 I nished my preprint Dierential graded motives: weight

was accepted for publication in the Journal of the Institute of Mathematics of ef Jussieu. It contains a full description of Voevodsky's DMgmf in terms of 'twisted' Suslin cubical complexes (in the sense of Kapranov and Bondal). In particular, for any motivic complex M (for instance, the Suslin complex of an arbitrary variety) there exists a quasi-isomorphic complex M 'constructed from' the Suslin complexes of smooth pro jective varieties; M is unique up to a homotopy. I also proved the following results.

complex, weight ltrations and spectral sequences for realizations; Voevodsky vs. Hanamura (see http://arxiv.org/abs/math.AG/0601713); it

Theorem 1.1. I There exist a conservative exact weight complex functor t : ef DMgmf DMgm K b (C howef f ) K b (C how). ef II t induces isomorphisms K0 (DMgmf ) K0 (C how) and K0 (DMgm ) K0 (C how); they are isomorphisms of rings. III For any cohomological realization H : DMgm DB (A) (here A is an ef abelian category and X Obj DMgmf ) there exists a natural weight spectral i i+j sequence S : H (P-j ) H (X ) where (Pi ) is a representative of t(X ). S is canonical and motivical ly functorial starting from E1 . It yields the usual weight spectral sequences and weight ltrations for mixed Hodge and etale cohomology of varieties. IV Voeovodsky's DMgm Q is antiequivalent to the Hanamura's motivic category. V A motif (an object of Voevodsky's DMgm ) is a mixed Tate one whenever its weight complex is.
A new method of attaching weights to cohomology functors was developed. In particular, a certain weight ltration for motivic cohomology was dened; note that this ltration is non-trivial, new, and universal for the important class of Bloch-Ogus cohomology theories. I also wrote a preprint Weight structures, weight ltrations, weight

spectral sequences, and weight complexes for triangulated categories (including motives and spectra) (electronic, http://arxiv.org/abs/0704.4003).
I showed that parts I-III of Theorem 1.1 follow from a very general relevant formalism for triangulated categories; this situation was not described in literature. One considers a set of axioms that are (in a certain sense) "dual" to the axioms of t-structures; I call this a weight structure. Several properties of weight structures are similar to those of t-structures; yet other ones are quite distinct. 1


Each triangulated category C with a weight structure has an additive heart with the property that there are no morphisms of positive degrees between ob jects of the heart in C . Any weight structure denes a conservative 'weight complex' functor to a certain "weak" homotopy category of complexes over the heart. Moreover, the weight structure gives a Postnikov tower of any ob ject which is canonical and functorial up to homotopy. In particular, for any (co)homological functor one obtains a 'weight spectral sequence' whose terms are (co)homology of the corresponding ob jects of the heart; this spectral sequence is canonical and functorial starting from E2 . Next, one can often obtain t-structures and weight structures from each other by passing to (left and right) 'adjacent subcategories'. The hearts of 'adjacent' structures are closely connected with each other. The most important examples of this formalism are Voevodsky's DMgm ef DMgmf and the stable homotopy category S H (of spectra). One of the main consequences of the weight structure formalism for motives is that canonical weight ltrations and spectral sequences exist for arbitrary realizations of motives (not necessarily having a dierential graded enhancement). In particular, weights should exist for the (conjectural!) "mixed motivic"cohomology (of varieties and motives). ef The adjacent structure formalism yields that Voevodsky's DM- f has a tstructure whose heart is the category of additive functors C howef f Ab. I also proved that (a certain version of ) the weight complex functor can be ef ef dened on DMgmf DM- f without using the resolution of singularities (so one can dene it for motives over any perfect eld). For spectra the weight spectral sequence specializes to the Atiyah-Hirzebruch sequence. The formalism allows to calculate K0 (S Hf in ) and certain K0 (End S Hf in ) (and K0 (Endn S Hf in ) for n N); here S Hf in is the category of nite spectra. The results on adjacent structure establishes (and allows to study) a connection of the coniveau ltration on cohomology of motives with the (Voevodsky's) ef homotopy t-structure on DM- f (this extends the seminal result of Bloch and Ogus). In particular, torsion motivic cohomology of motives can be expressed in terms certain etale cohomology (here the recently proved Beilinson-Lichtenbaum conjecture is used). In section 2.2 of the new preprint Artin's Vanishing for a formula for the "dierence"of the motivic cohomology with the etale one) is proved. In the latter preprint several interesting motivic problems are studied. Unfortunately, Theorem 2.1.1 (of version 2 of the preprint) is wrong in the form it is stated. This makes (most of ) the results of the preprint conditional modulo Theorem 2.3.2. Yet I hope to correct the proof of Theorem 2.3.2. Note also that the rational version of it follows from certain "standard"motivic conjectures; hence the preprint (at least) reveals certain new connections between motivic conjectures. This preprint is also related with a short preprint Explicit generators for ef (conjectural) mixed motives (in Voevodsky's DMgmf ). The Kunneth decomposition of pure (numerical) motives, http://arxiv.org/abs/math/0703499. In the latter preprint I showed (briey) that if certain "standard"conjectures are fullled then the Kunneth decomposition of the diagonal and a certain gen-

torsion motivic homology; numerical motives form a tannakian category (see http://arxiv.org/abs/0711.3918) a nice formula of this sort (and also

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erating set of mixed motives could be described quite explicitly. In 2007 I made two talks at international mathematical conferences on the topics described. 1. Arithmetic Geometry, June 1319, 2007, Saint-Petersburg, Russia: "The universal Euler characteristic for motives". 2. International Algebraic Conference dedicated to the 100th anniversary of D. K. Faddeev, September 2429, 2007, Saint-Petersburg, Russia: "Weights for cohomology: weight structures, ltrations, spectral sequences, and weight complexes (for motives and spectra)".

2 Pedagogical activity
In 2007 I led student's practice in higher algebra and number theory and read lectures on this sub ject (in St. Petersburg State University). Besides I actively participated in the composition of a book of problems in Number theory; it will be published and used for teaching students.

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