Definable categories and 𝕋-motives (jointly with M. Prest) to appear on Rendiconti del Seminario Matematico della Università di Padova
Ogus realization of 1-motives (jointly with F. Andreatta & A. Bertapelle) Journal of Algebra Vol. 487 (2017) 294-316
𝕋-motives Journal of Pure and Applied Algebra Vol. 221, Issue 7 (2017) 1495-1898 [0]
On the derived category of 1-motives (jointly with B. Kahn) Astérisque Vol. 381 (2016), xi + 254 pages [1]
On the Deligne-Beilinson cohomology sheaves Annals of K-Theory Vol. 1, Issue 1 (2016) 3-17. [2]
Nori 1-motives (jointly with J. Ayoub) Math. Ann. Vol. 361, Issue 1-2 (2015) 367-402
Sharp de Rham realization (jointly with A. Bertapelle) Advances in Math. Vol. 222, Issue 4 (2009) 1308-1338
1-motivic sheaves and the Albanese functor (jointly with J. Ayoub) Journal of Pure and Applied Algebra Vol. 213, No. 5 (2009) 809-839
The Néron-Severi group of a proper seminormal complex variety (jointly with A. Rosenschon & V. Srinivas) Math. Zeitschrift, Vol. 261, N. 2 (2009) 261-276.
A note on relative duality for Voevodsky motives (jointly with B. Kahn) Tohoku Math. Journal Vol. 60, No. 3 (2008) 349-356
Formal Hodge Theory Math. Research Letters Vol. 14 Issue 3 (2007) 385-394.
On the theory of 1-motives in Algebraic Cycles and Motives London Mathematical Society Lecture Note Series, Vol. 343, Cambridge University Press, London, 2007. [3]
A pamphlet on motivic cohomology Milan Journal of Math. Vol. 73 (2005) 53-73
Crystalline realizations of 1-motives (jointly with F. Andreatta) Math. Ann. Vol. 331, N. 1, (2005) 111-172
Deligne's conjecture on 1-motives (jointly with A. Rosenschon & M. Saito) Annals of Math. Vol. 158, N. 2, (2003) 593-633
On algebraic 1-motives related to Hodge cycles in Algebraic Geometry Walter de Gruyter, Berlin/New York, 2002, 25-60. [4]
Albanese and Picard 1-motives (jointly with V. Srinivas) Mémoires Société Mathématique de France Vol. 87 (2001) vi + 104 pages
On the motivic topos & Nori motives.[5] A letter & some remarks on the motivic topos (pdf) is a "road map" on the motivic topos, 𝕋-motives & Nori motives. The video of my talk 𝕋-motives (video) for the conference Topos à l'IHES is posted on IHES Youtube channel. I also gave a talk titled Nori motives and the motivic topos (beamer pdf) on August 21, 2015 at the Conference K-theory, Cyclic Homology and Motives - a conference in celebration of C. A. Weibel's 65th year. The first talk I gave on the motivic topos was on March 13, 2014: it was titled "Cohomology theories in algebraic geometry and the motivic topos" and it was given in the framework of the seminar The problem of classifying cohomology theories : a topos-theoretic approach organized in Milano, jointly with O. Caramello & L. Lafforgue. I gave a similar talk titled "Motivic Topos" at the Conference Around forms, cycles and motives on the occasion of Albrecht Pfister's 80th birthday (Mainz, September 8-12, 2014).
On sharp cohomologies & formal Hodge structures.[6] Slides for the lectures Sharp cohomology (beamer pdf) at the Brixen Summer School on Cohomology theories (September 12-17, 2011). Notes of the talks Sharp cohomology (pdf) at Paris 7 (January 10, 2007) & Paris 13 (May 18, 2007) on "sharp" theories along with a list of references. The first talk I gave on the subject of "sharp" singular cohomology was on June 16, 2005 at the meeting Current Geometry, the international conference on problems and trends of contemporary geometry (Palazzo Serra di Cassano, Naples). However, this project was presented very roughly in an email letter to M. Saito on July 1, 2001. I drafted a general theory in a covering letter to A. Beilinson (pdf) on November 28, 2005, when submitting the paper Formal Hodge Theory.
[0] Special issue dedicated to Chuck Weibel on the occasion of his 65th birthday.
[1] The published version contains relevant changes & the old ArXiv versions are obsolete. However, the last revised version dated 13 Sep 2016 (v2 pdf file) arXiv:1009.1900v2 [math.AG] is updated. This project started (jointly with B. Kahn) on February 2002. We planned a paper to be split (at least) in two Parts titled "On the derived category of 1-motives: I, II" A preliminary version of Part I was published on line on the K-Theory archive as a preprint N. 800 (October 5, 2006) & updated to N. 851 (June 11, 2007). In Part I we constructed the functor LAlb and we computed it completely for smooth and to a large extent for general schemes (in characteristic 0). One application is a new proof of Roitman's torsion theorem and its generalisations: we provided a new motivic version which extends this theorem to any algebraic scheme (in characteristic 0). The proof of Deligne's conjectures via realisations applied to LAlb was an achievement of Part II of our project. Actually, starting from the 2010 arXiv version we titled "On the derived category of 1-motives" Part I substantially expanded to contain Part II. Part I is also available in the arXiv & as an IHÉS prepublication (including an index of notations already) Prépublication Mathématique de l'IHÉS (M/07/22), June 2007, 144 pages.
[2] This paper for the new journal Annals of K-Theory is an updated version of arXiv:alg-geom/9412006 which is now obsolete.
[3] Proceedings of the workshop on the occasion of the 75-th birthday of J.P. Murre (Lorentz Center, Leiden, August 30-September 3, 2004) The workshop was organized by S.J. Edixhoven, J. Nagel and C. Peters.
[4] A Volume in Memory of P. Francia.
[5] I can tell that on 11 April 2013 exactly I had the idea of a "motivic topos". I remember it because I was traveling on a train from Brussels to Ghent after an early morning flight from Milano going to the EUA General Assembly 2013 hosted by Ghent University, Belgium, on 11 April 2013 as an official representative of the Rector of Milan University. I was a Rector's Delegate and I was also very busy with administrative duties those days. However, the picture of a "motivic topos" was so clear for me that I drafted in an email to L. Lafforgue the day after I got it.
[6] The concept of formal Hodge structure is pointing to that of "sharp" singular cohomology of a complex algebraic variety (also "sharp" de Rham over a field of zero characteristic and "sharp" crystalline in positive characteristics). Sharp singular cohomology should be a formal Hodge structure containing, in the underlying algebraic structure, a formal group which is an extension of ordinary singular cohomology mixed Hodge structure. Following Grothendieck strategy to construct a cohomology one should take care of the 1-motivic cohomology via Laumon 1-motives and "sharp" realizations. Also "sharp" mixed motives are expected.