Di Proietto, Valentina; Shiho, Atsushi

On the Homotopy Exact Sequence for Log Algebraic Fundamental Groups

Doc. Math. 23, 543-597 (2018)
DOI: 10.25537/dm.2018v23.543-597


We construct a log algebraic version of the homotopy sequence for a normal crossing log variety over a log point of characteristic zero and prove some exactness properties of it. Our proofs are purely algebraic.

Mathematics Subject Classification

14F35, 14F40


log scheme, fundamental group, homotopy exact sequence, module with integrable connection


  • 1. F. Andreatta, A. Iovita, and M. Kim, A $p$-adic non-abelian criterion for good reduction of curves, Duke Math. J. 164 (2015), no. 13, 2597--2642. DOI 10.1215/00127094-3146817; zbl 1347.11051; MR3405595.
  • 2. R. Crew, Finiteness theorems for the cohomology of an overconvergent isocrystal on a curve, Ann. Sci. Ecole Norm. Sup. (4) 31 (1998), no. 6, 717--763. DOI 10.1016/S0012-9593(99)80001-9; zbl 0943.14008; MR1664230.
  • 3. P. Deligne, Le groupe fondamental de la droite projective moins trois points, in Galois groups over ${\mathbb{Q}}$, Math. Sci. Res. Inst. Publ. 16, Springer, New York (1989), pp. 79--297. MR1012168.
  • 4. P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA (1990), pp. 111--195. zbl 0727.14010; MR1106898.
  • 5. P. Deligne and J. S. Milne, Tannakian categories, Hodge cycles, motives, and Shimura varieties (Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, eds.), Lecture Notes in Mathematics, vol. 900, Springer-Verlag (1982). zbl 0465.00010; MR0654325.
  • 6. J. P. Dos Santos, The homotopy exact sequence for the fundamental group scheme and infinitesimal equivalence relations, Algebraic Geometry 2 (2015), no. 5, 535--590. DOI 10.14231/AG-2015-024; zbl 1336.14017; MR3421782.
  • 7. H. Esnault and P. H. Hai, The Gauss-Manin connection and Tannaka duality, Int. Math. Res. Not. (2006), 35 pp. MR2211153 arxiv 0509111.
  • 8. H. Esnault, P. H. Hai, and X. Sun, On Nori's fundamental group scheme, Geometry and dynamics of groups and spaces, Progr. Math., vol. 265, Birkhäuser (2008), pp. 377--398. zbl 1137.14035; MR2402410; arxiv math/0605645.
  • 9. A. Grothendieck, Revêtements étales et groupe fondamental, Séminaire de Géométrie Algébrique, vol. 1960/61, Institut des Hautes Études Scientifiques, Paris (1963). MR0217088.
  • 10. P. H. Hai, Gauss-Manin stratification and stratified fundamental group schemes, Ann. Inst. Fourier (Grenoble) 63 (2013), no. 6, 2267--2285. DOI 10.5802/aif.2829; zbl 1298.14022; MR3237447.
  • 11. J. C. Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA (1987). zbl 0654.20039; MR0899071.
  • 12. N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Etudes Sci. Publ. Math. 39 (1970), 175--232. DOI 10.1007/BF02684688; zbl 0221.14007; MR0291177.
  • 13. K. Kato, Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD (1989), pp. 191--224. zbl 0776.14004; MR1463703.
  • 14. F. Kato, Log smooth deformation theory, Tôhoku Math. J. (2) 48 (1996), no. 3, 317--354. DOI 10.2748/tmj/1178225336; zbl 0876.14007; MR1404507; arxiv alg-geom/9406004.
  • 15. K. S. Kedlaya, Semistable reduction for overconvergent ${F}$-isocrystals. I. Unipotence and logarithmic extensions, Compositio Math. 143 (2007), no. 5, 1164--1212. DOI 10.1112/S0010437X07002886; zbl 1144.14012; MR2360314; arxiv math/0405069.
  • 16. M. Kim and R. M. Hain, A de Rham-Witt approach to crystalline rational homotopy theory, Compos. Math. 140 (2004), no. 5, 1245--1276. DOI 10.1112/S0010437X04000442; zbl 1086.14021; MR2081156; math/0105008.
  • 17. C. Lazda, Relative fundamental groups and rational points, Rend. Sem. Mat. Univ. Padova 134 (2015), 1--45. DOI 10.4171/RSMUP/134-1; zbl 1335.11050; MR3428414; arxiv 1303.6484.
  • 18. C.  Lazda and A.  Pál, A homotopy exact sequence for overconvergent isocrystals, arXiv:1704.07574.
  • 19. J. S. Milne, Basic theory of affine group scheme, Ver. 1.00, avalaible at http://www.jmilne.org/math/CourseNotes/AGS.pdf.
  • 20. J. S. Milne, Algebraic Groups, An introduction to the theory of algebraic group schemes over fields, Ver. 2.00, available at http://www.jmilne.org/math/CourseNotes/iAG.pdf.
  • 21. A. Ogus, $F$-crystals, Griffiths transversality, and the Hodge decomposition, Astérisque (1994), no. 221. zbl 0801.14004; MR1280543.
  • 22. A. Ogus, On the logarithmic Riemann-Hilbert correspondence, Doc. Math. (2003), no. Extra Vol., 655--724 (electronic), Kazuya Kato's fiftieth birthday. https://www.elibm.org/article/10011541 zbl 1100.14507; https://www.elibm.org/article/10011541 ---newline--- zbl 1100.14507 ---newline---.
  • 23. A. Shiho, Crystalline fundamental groups. I. Isocrystals on log crystalline site and log convergent site, J. Math. Sci. Univ. Tokyo 7 (2000), no. 4, 509--656. zbl 0984.14009; MR1800845.
  • 24. W. C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics, vol. 66, Springer-Verlag, New York-Berlin (1979). zbl 0442.14017; MR0547117.
  • 25. L. Zhang, The homotopy exact sequence of the algebraic fundamental group, Int. Math. Res. Not. (2014), no. 22, 6155--6174. DOI 10.1093/imrn/rnt163; zbl 1329.14044; MR3283001; arxiv 1608.00384.


Di Proietto, Valentina
College of Engineering, Mathematics and Physical Sciences, University of Exeter, EX4 4RN, Exeter, United Kingdom
Shiho, Atsushi
Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan