# Publications

### Lecture Notes and Slides

*Lectures on motivic cohomology.*

Conference on "Motives and Automorphic Forms" in Honour of Günter Harder's 85th Birthday, March 6 - 10, 2023.

*Three lectures on quadratic enumerative geometry.* "Moduli, Motives and Bundles – New Trends in Algebraic Geometry'' held at the BIRS Conference centre Oaxaca, Mexico, September 18-23, 2022.

*Motivic cohomology.* Lecture for the 2022 ICM. Proc. Int. Cong. Math. 2022, Vol. 3, pp. 2048–2106. EMS Press, Berlin, 2023.

*Welschinger invariants and quadratic degrees.* Slides for a talk at the conference "Periods and Motives", Berlin, July 10-12, 2019.

* Lectures on quadratic enumerative geometry.* Four lectures, given at the conference “Motivic homotopy theory and refined enumerative geometry”, May 14–18, 2018, at the Universität Duisburg-Essen.

*A calculus of characteristic classes in Witt cohomology.* Slides for a talk given at the Conference on Algebraic Geometry and Number Theory on the occasion of Jean-Louis Colliot-Thélène’s 70th birthday, December 4-6, 2017, Villa Finaly, Florence, Italy.

*The slice tower and the Adams-Novikov spectral sequence.* Slides for a talk, Sept. 2014.

Lausanne 2012 : Program for: The motivic fundamental group, October 1-5, 2012.

JAMI 2009: Tate motives and fundamental groups, March 25, 2009.

UCLA Colloquium: a survey of algebraic cobordism, Jan. 22, 2009.

BU Conference on motives QFT and pseudo-differential operators, June 2008

Lecture 1 Cycle complexes

Lecture 2 Motivic categories

Lecture 3 Tate motives.

ICTP school on algebraic K-theory and its applications, May 2007.

Six lectures on motives

French-Asian summer school in algebraic geometry and number theory, I.H.E.S. July 2006.

Six lectures on motives

Conference on Motives and Algebraic Cycles, in honor of Spencer Bloch,

Fields Institute-March 19-23, 2007

Motivic Postnikov towers

A lecture on Voevodsky's slice filtration, its relation to the homotopy coniveau tower,

with some applications to Severi-Brauer varieties and central simple algebras.

2nd German-Chinese conference on complex geometry, Shanghai-Sept. 2006

Algebraic cobordism

A lecture on algebraic cobordism, including a discussion of ``double-point" cobordism

and an application to Donaldson-Thomas invariants.

*Three lectures on algebraic cobordism*

Motives and Periods, June 5-12, 2006

UBC, Vancouver

Lecture 1, Lecture 2, Lecture 3

Introduction to algebraic cobordism

The first of 3 lectures on algebraic cobordism in the Distinguished Lecture Series

of the University of Western Ontario Mathematics Department,

Jan. 25-27, 2005.

A summer course in motivic homotopy theory

Algebraic geometry background, Nordfjordeid, 2003

A short course in K-theory

Morelia, 2002

### Preprints

(with H. Esnault) * Tate motives and the fundamental group*

Preprint (2007)

*K-theory and motivic cohomology of schemes, I *

Preprint (2004)

### Publications

* The intrinsic stable normal cone*

Algebr. Geom. 8 (2021), no. 5, 518–561.

(with Yaping Yang, Gufang Zhao) * Algebraic elliptic cohomology and flops II: SL-cobordism*

Adv. Math. 384 (2021), Paper No. 107726, 66 pp.

*Aspects of enumerative geometry with quadratic forms*

Doc. Math. 25 (2020), 2179–2239.

(with A. Raksit) * Motivic Gauss-Bonnet formulas*

Algebra Number Theory 14 (2020), no. 7, 1801–1851.

*Motivic Euler characteristics and Witt-valued characteristic classes*

Nagoya Math. J. 236 (2019), 251–310.

(with Yaping Yang, Gufang Zhao, Joël Riou) *Algebraic elliptic cohomology theory and flops I*

Math. Ann. 375 (2019), no. 3-4, 1823–1855.

(with A. Chatzistamatiou) * Torsion orders of complete intersections*

Algebra Number Theory 11 (2017), no. 8, 1779–1835.

(with A. Ananyevskiy and I. Panin) * Witt sheaves and the η -inverted sphere spectrum*

J. Topol. 10 (2017), no. 2, 370–385.

*Intersection theory in algebraic cobordism*

J. Pure Appl. Algebra 221 (2017), no. 7, 1645–1690.

*The Adams-Novikov spectral sequence and Voevodsky's slice tower*

Geom. Topol. 19 (2015), no. 5, 2691–2740.

(with G.S. Tripathi) *Quotients of MGL, their slices and their geometric parts*

Doc. Math.(2015-Extra Vol.-Alexander S. Merkurjev's Sixtieth Birthday), 407–442.

(with H. Esnault and O. Wittenberg) *Index of varieties over Henselian fields and Euler characteristic of coherent sheaves*

J. Algebraic Geom. 24 (2015), no. 4, 693–718.

*Motivic Landweber exact theories and their effective covers*

Homology Homotopy Appl. 17 (2015), no. 1, 377–400.

*A comparison of motivic and classical stable homotopy theories*

J. Topol. 7 (2014), no. 2, 327–362.

(with S. Dai) * Connective algebraic K-theory*

arXiv:1212.0228, J. K-Theory 13 (2014), no. 1, 9–56.

* Tate motives and the fundamental group*** Cycles, Motives and Shimura Varieties**, ed. V. Srinivas, Tata Institute of Fundamental Research, Mumbai, India, August 2010, 265--392.

*Slices and transfers*

Documenta Mathematica, Extra Volume: Andrei A. Suslin's Sixtieth Birthday (2010), 393--443.

(with J.-L. Colliot-Thélène) * Une version du théorème d'Amer et Brumer pour les zéro-cycles*

Quadratic forms, linear algebraic groups, and cohomology, 215–223, Dev. Math., **18**, Springer, New York, 2010.

(with Bruno Kahn) *Motives of Azumaya algebras*

Journal of the Inst. of Math. Jussieu **9**(3), 2010, 481–599.

*Smooth Motives *** Motives and algebraic cycles**, 175--231, Fields Inst. Commun.,**56**, Amer. Math. Soc., Providence, RI, 2009.

*Comparison of cobordism theories *

J. Algebra **322** (2009), no. 9, 3291--3317.

(with Rahul Pandharipande) *Algebraic Cobordism Revisited*

Invent. Math. **176** (2009), no. 1, 63--130.

* Six lectures on motives*

Some recent developments in algebraic *K*-theory, 131--227, ICTP Lect. Notes, **23**, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008.

*Motivic homotopy theory*

Milan J. Math. **76** (2008), 165--199.

*Oriented cohomology, Borel-Moore homology and algebraic cobordism *

Michigan Math. J., Special Issue in honor of Melvin Hochster, **57**, August 2008, 523-572

(with Amalendu Krishna) *Additive higher Chow groups of schemes*

J. Reine Angew. Math. **619** (2008), 75--140.

(with C. Serpé) * On a spectral sequence for equivariant K-theory**K*-Theory **38**(2008), no. 2, 177--222.

*The homotopy coniveau tower*

J Topology **1** (2008) 217--267.

* Steenrod operations, degree formulas and algebraic cobordism*

Pure and Applied Mathematics Quarterly Journal, volume in honor of the 60th birthday of Robert MacPherson **3**, No. 1 (2007) 283-306.

* Motivic Tubular Neighborhoods** *

Documenta Math. **12** (2007) 71--146.

(with Fabien Morel) **Algebraic Cobordism**

Monographs in Mathematics, 246 pp., Springer, Berlin 2007.

(with B. Dundas, P.A. Østvaer, O. Röndigs, V. Voevodsky) * Background from algebraic geometry* Link** Motivic homotopy theory** Lectures at a summer school in Nordfjordeid, Norway, August 2002. Universitext 220 pp. Springer, Berlin 2007, 70--145.

*Chow's moving lemma and the homotopy coniveau tower* *K*-theory **37** (1-2) 129-209 (2006).

*Mixed Motives*

An expository article in the ** Handbook of K-theory**, vol 1,

E.M. Friedlander, D.R. Grayson, eds.,

429-522. Springer Verlag, 2005.

(with S. Bloch and H. Esnault) * Divisibility of the eigenvalues of Frobenius acting on étale cohomology of singular projective varieties*

Amer. J. Math.

**127**(2005), no. 1, 193--207.

* A survey of algebraic cobordism*

Proceedings of the International Conference on Algebra.

Algebra Colloq. **11** (2004), no. 1, 79--90.

* Fundamental classes in algebraic cobordism*

K-Theory **30** (2) (2003), 129-135.

* Algebraic cobordism*

Proceedings of the 2002 ICM, August, 2002.

(with F. Morel) *Cobordisme algébrique, I*

C. R. Acad. Sci. Paris Sér. I Math. **332 **(2001), no. 8, 723--728.

(with F. Morel) *Cobordisme algébrique, II*

C. R. Acad. Sci. Paris Sér. I Math. **332** (2001), no. 9, 815--820.

*Techniques of localization in the theory of algebraic cycles*

J. Alg. Geom. **10** (2001) 299-363.

*Blowing up monomial ideals*

JPAA **160** (2001) 67-103.

(with T. Geisser) *The Bloch-Kato conjecture and a theorem of Suslin-Voevodsky*

Jour. f.d. Reine u. Ang. Math

**530**(2001), 55-103.

*Inverting the motivic Bott element*

K-Theory **19 **(2000), no. 1, 1--28.

(with H. Esnault) *The Steinberg curve*

Amer. J. Math. **122** (2000), no. 4, 783--804

(with T. Geisser) *The K-theory of fields in characteristic p*

Invent. Math. **139 **(2000), no. 3, 459--493.

** Mixed Motives** * On-line version*

Math. Surveys and Monographs **57**, x+515 pp., AMS, Prov. 1998.

(with H. Esnault, B. Kahn, E. Viehweg)

*The Arason invariant and mod 2 algebraic cycles*

J. of the AMS, **11 **no. 1, (1998), 73-118.

*Homology of algebraic varieties: an introduction to the works of Suslin and Voevodsky*

Bull. Amer. Math. Soc. (N.S.)

**34**(1997), no. 3, 293-312.

* Lambda operations, K-theory and motivic cohomology* ** Algebraic K-Theory,** (volume in memory of R. Thomason)

ed. V. Snaith, Fields Inst. Communications **16** (1997) 131-184.

(with H. Esnault, E. Viehweg)* Chow groups of projective varieties of very small degree*

Duke Math. J. **87** (1997), no. 1, 29-58.

*The weight two K-theory of fields*

K-theory **9** No. 5, (1995) 443-501.

*Bloch's higher Chow groups revisited***K-theory Strasbourg 1992** Astérisque vol. 226,

ed. C. Kassel, J.L. Loday, N. Schappacher,

Soc. Math. de France(1994) 235-320.

(with H. Esnault) *Surjectivity of cycle maps* **Journées de Geométrie Algébrique D'Orsay, Juillet 1992**,

Astérisque vol. 218(1993) 203-226.

*Tate motives and the vanishing conjectures for algebraic K-theory* **Algebraic K-Theory and Algebraic Topology**,

ed. P.G. Goerss and J.F. Jardine,

NATO ASI Series, Series C, Vol. **407 **(1993) 167-188.

* The algebraic K-theory of the classical groups and some twisted forms*

Duke Math. Journ.

**70**no. 2(1993) 405-443.

* Deligne-Beilinson cohomology for singular varieties*

Proceedings of the S.Margherita `K`-theory Conference, Cont. Math. **126** (1992) 113-146.

* Relative Milnor K-theory*

`K`-theory

**6**(1992) 113-175.

(with V. Srinivas and J. Weyman), `K`-theory of twisted Grassmannians`K`-theory **3** No. 2(1989) 99-122.

* The indecomposable K_{3} of fields*

Bulletin of the AMS, Nov. 1987.

* The indecomposable K_{3} of fields*

Ann. Sci. Ecole Norm. Sup.(4)

**22**No. 2(1989) 255-344.

* Localization on singular varieties*

Inv. Math. **91**(1988) 423-464.

* A remark on extremal Kähler metrics*

J. Diff. Geom. **21**(1985) 73-77.

* A K-theoretic approach to multiplicities*

Math. Ann.

**271**(1985) 451-458.

(with H. Gillet)* The relative form of Gersten's conjecture over a discrete valuation ring: the smooth case*

J. Pure and Appl. Alg. **46** (1987) 59-71.

* Pluri-canonical divisors on Käahler manifolds II*

Duke J. Math., **52** No. 1(1985) 61-65.

* K-groups of surface singularities and modules of finite length*

Compositio Math.

**59**No. 1(1986) 21-40.

* Bloch's formula for a singular surface*

Topology **124** No. 2(1985) 165-174.

*Zero-cycles and K-theory on singular varieties*

**Algebraic Geometry, Bowdoin 1985**S. Bloch et. al. ed. AMS Proc. Symp. Pure Math. Vol. 46 pt. 2(1987) 451-462.

(with C. Weibel)* Zero-cycles and complete intersections on affine surfaces*

J. Reine u. Ang. Math. **359**(1985) 106-120.

* Pluri-canonical divisors on Kähler manifolds*

Inventiones Math. **74**(1983) 293-303.

* Torsion zero-cycles on singular varieties*

Am. J. Math. **107**(1985) 737-757.

(with V. Srinivas) *Zero-cycles on certain singular elliptic surfaces*

Compositio Math. **52** fasc. 2(1984) 179-196.

* Deformations of irregular threefolds*** Algebraic Threefolds** Lecture Notes in Math. **947** 269-286, Springer 1982.

(with P. Blass) *Families of Zariski surfaces*

Duke J. Math. **48**(1981) 129-136.

* Deformations of uni-ruled varieties*

Duke J. Math. **48**(1981) 467-473.

* Some examples from the deformation theory of ruled varieties*

Am. J. Math. **103**(1981) 997-1020.

* Stability of irrational ruled threefolds*

Math. Ann. **262** (1983) 391-406.

* The stability of certain classes of uni-ruled varieties*(thesis)

Compositio Math. **52** fasc. 1(1984) 3-30.