Positivity of Chern classes of Schubert cells and varieties
Author:
June Huh
Journal:
J. Algebraic Geom. 25 (2016), 177-199
DOI:
https://doi.org/10.1090/jag/646
Published electronically:
August 18, 2015
MathSciNet review:
3419959
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We show that the Chern-Schwartz-MacPherson class of a Schubert cell in a Grassmannian is represented by a reduced and irreducible subvariety in each degree. This gives an affirmative answer to a positivity conjecture of Aluffi and Mihalcea.
References
- Paolo Aluffi, Differential forms with logarithmic poles and Chern-Schwartz-MacPherson classes of singular varieties, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 7, 619–624 (English, with English and French summaries). MR 1717120, DOI https://doi.org/10.1016/S0764-4442%2800%2980012-9
- Paolo Aluffi, Characteristic classes of singular varieties, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp. 1–32. MR 2143071, DOI https://doi.org/10.1007/3-7643-7342-3_1
- Paolo Aluffi, Classes de Chern des variétés singulières, revisitées, C. R. Math. Acad. Sci. Paris 342 (2006), no. 6, 405–410 (French, with English and French summaries). MR 2209219, DOI https://doi.org/10.1016/j.crma.2006.01.002
- Paolo Aluffi, Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes, Pure Appl. Math. Q. 2 (2006), no. 4, Special Issue: In honor of Robert D. MacPherson., 915–941. MR 2282409, DOI https://doi.org/10.4310/PAMQ.2006.v2.n4.a2
- Paolo Aluffi and Leonardo Constantin Mihalcea, Chern classes of Schubert cells and varieties, J. Algebraic Geom. 18 (2009), no. 1, 63–100. MR 2448279, DOI https://doi.org/10.1090/S1056-3911-08-00482-7
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012
- Robert Bryant, Rigidity and Quasi-Rigidity of Extremal Cycles in Hermitian Symmetric Spaces, Annals of Mathematics Studies 153, Princeton University Press, 2010.
- Jean-Paul Brasselet, José Seade, and Tatsuo Suwa, Vector fields on singular varieties, Lecture Notes in Mathematics, vol. 1987, Springer-Verlag, Berlin, 2009. MR 2574165
- Michel Brion, Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp. 33–85. MR 2143072, DOI https://doi.org/10.1007/3-7643-7342-3_2
- Michel Brion, Log homogeneous varieties, Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, pp. 1–39. MR 2500349
- Michel Brion, Vanishing theorems for Dolbeault cohomology of log homogeneous varieties, Tohoku Math. J. (2) 61 (2009), no. 3, 365–392. MR 2568260, DOI https://doi.org/10.2748/tmj/1255700200
- M. Brion and R. Joshua, Equivariant Chow ring and Chern classes of wonderful symmetric varieties of minimal rank, Transform. Groups 13 (2008), no. 3-4, 471–493. MR 2452601, DOI https://doi.org/10.1007/s00031-008-9020-2
- Michel Brion and Ivan Kausz, Vanishing of top equivariant Chern classes of regular embeddings, Asian J. Math. 9 (2005), no. 4, 489–496. MR 2216242, DOI https://doi.org/10.4310/AJM.2005.v9.n4.a3
- Izzet Coskun, Rigid and non-smoothable Schubert classes, J. Differential Geom. 87 (2011), no. 3, 493–514. MR 2819546
- Izzet Coskun and Colleen Robles, Flexibility of Schubert classes, Differential Geom. Appl. 31 (2013), no. 6, 759–774. MR 3130568, DOI https://doi.org/10.1016/j.difgeo.2013.09.003
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174
- Michel Demazure, Désingularisation des variétés de Schubert généralisées, Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 354697
- W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, Intersection theory on spherical varieties, J. Algebraic Geom. 4 (1995), no. 1, 181–193. MR 1299008
- William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. MR 1464693
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323
- Mark Goresky and William Pardon, Chern classes of automorphic vector bundles, Invent. Math. 147 (2002), no. 3, 561–612. MR 1893006, DOI https://doi.org/10.1007/s002220100184
- H. C. Hansen, On cycles in flag manifolds, Math. Scand. 33 (1973), 269–274 (1974). MR 376703, DOI https://doi.org/10.7146/math.scand.a-11489
- Jaehyun Hong, Rigidity of singular Schubert varieties in ${\rm Gr}(m,n)$, J. Differential Geom. 71 (2005), no. 1, 1–22. MR 2191767
- Jaehyun Hong, Rigidity of smooth Schubert varieties in Hermitian symmetric spaces, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2361–2381. MR 2276624, DOI https://doi.org/10.1090/S0002-9947-06-04041-4
- June Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc. 25 (2012), no. 3, 907–927. MR 2904577, DOI https://doi.org/10.1090/S0894-0347-2012-00731-0
- June Huh, $h$-vectors of matroids and logarithmic concavity, Adv. Math. 270 (2015), 49–59. MR 3286530, DOI https://doi.org/10.1016/j.aim.2014.11.002
- Benjamin F. Jones, On the singular Chern classes of Schubert varieties via small resolution, ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)–University of Notre Dame. MR 2736782
- Benjamin F. Jones, Singular Chern classes of Schubert varieties via small resolution, Int. Math. Res. Not. IMRN 8 (2010), 1371–1416. MR 2628830, DOI https://doi.org/10.1093/imrn/rnp174
- G. Kempf and D. Laksov, The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153–162. MR 338006, DOI https://doi.org/10.1007/BF02392111
- Gary Kennedy, MacPherson’s Chern classes of singular algebraic varieties, Comm. Algebra 18 (1990), no. 9, 2821–2839. MR 1063344, DOI https://doi.org/10.1080/00927879008824054
- Valentina Kiritchenko, Chern classes of reductive groups and an adjunction formula, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1225–1256 (English, with English and French summaries). MR 2266889
- Valentina Kiritchenko, On intersection indices of subvarieties in reductive groups, Mosc. Math. J. 7 (2007), no. 3, 489–505, 575 (English, with English and Russian summaries). MR 2343145, DOI https://doi.org/10.17323/1609-4514-2007-7-3-489-505
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. Classical setting: line bundles and linear series. MR 2095471
- R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432. MR 361141, DOI https://doi.org/10.2307/1971080
- Hideyuki Matsumura and Frans Oort, Representability of group functors, and automorphisms of algebraic schemes, Invent. Math. 4 (1967), 1–25. MR 217090, DOI https://doi.org/10.1007/BF01404578
- Leonardo Constantin Mihalcea, Sums of binomial determinants, non-intersecting lattice paths, and positivity of Chern-Schwartz-MacPherson classes, preprint, arXiv:0702566.
- Nicolas Perrin, Courbes rationnelles sur les variétés homogènes, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 1, 105–132 (French, with English and French summaries). MR 1881572
- C. P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Ann. 156 (1964), 25–33. MR 166198, DOI https://doi.org/10.1007/BF01359978
- Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291. MR 586450
- Jörg Schürmann, Lectures on characteristic classes of constructible functions, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp. 175–201. Notes by Piotr Pragacz and Andrzej Weber. MR 2143077, DOI https://doi.org/10.1007/3-7643-7342-3_7
- Marie-Hélène Schwartz, Classes caractéristiques définies par une stratification d’une variété analytique complexe. I, C. R. Acad. Sci. Paris 260 (1965), 3262–3264 (French). MR 212842
- Marie-Hélène Schwartz, Classes caractéristiques définies par une stratification d’une variété analytique complexe, C. R. Acad. Sci. Paris 260 (1965), 3535–3537 (French). MR 184254
- Judson P. Stryker III, Chern-Schwartz-MacPherson classes of graph hypersurfaces and Schubert varieties, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–The Florida State University. MR 2949825
- Patrice Tauvel and Rupert W. T. Yu, Lie algebras and algebraic groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2146652
- Andrzej Weber, Equivariant Chern classes and localization theorem, J. Singul. 5 (2012), 153–176. MR 2928940, DOI https://doi.org/10.5427/jsing.2012.5k
References
- Paolo Aluffi, Differential forms with logarithmic poles and Chern-Schwartz-MacPherson classes of singular varieties, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 7, 619–624 (English, with English and French summaries). MR 1717120 (2001d:14008), DOI https://doi.org/10.1016/S0764-4442%2800%2980012-9
- Paolo Aluffi, Characteristic classes of singular varieties, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp. 1–32. MR 2143071 (2006h:14010), DOI https://doi.org/10.1007/3-7643-7342-3_1
- Paolo Aluffi, Classes de Chern des variétés singulières, revisitées, C. R. Math. Acad. Sci. Paris 342 (2006), no. 6, 405–410 (French, with English and French summaries). MR 2209219 (2007g:14001), DOI https://doi.org/10.1016/j.crma.2006.01.002
- Paolo Aluffi, Limits of Chow groups, and a new construction of Chern-Schwartz-MacPherson classes, Pure Appl. Math. Q. 2 (2006), no. 4, 915–941. MR 2282409 (2007k:14007), DOI https://doi.org/10.4310/PAMQ.2006.v2.n4.a2
- Paolo Aluffi and Leonardo Constantin Mihalcea, Chern classes of Schubert cells and varieties, J. Algebraic Geom. 18 (2009), no. 1, 63–100. MR 2448279 (2009h:14086), DOI https://doi.org/10.1090/S1056-3911-08-00482-7
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012 (92d:20001)
- Robert Bryant, Rigidity and Quasi-Rigidity of Extremal Cycles in Hermitian Symmetric Spaces, Annals of Mathematics Studies 153, Princeton University Press, 2010.
- Jean-Paul Brasselet, José Seade, and Tatsuo Suwa, Vector fields on singular varieties, Lecture Notes in Mathematics, vol. 1987, Springer-Verlag, Berlin, 2009. MR 2574165 (2011d:32046)
- Michel Brion, Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp. 33–85. MR 2143072 (2006f:14058), DOI https://doi.org/10.1007/3-7643-7342-3_2
- Michel Brion, Log homogeneous varieties, Proceedings of the XVIth Latin American Algebra Colloquium (Spanish), Bibl. Rev. Mat. Iberoamericana, Rev. Mat. Iberoamericana, Madrid, 2007, pp. 1–39. MR 2500349 (2010m:14063)
- Michel Brion, Vanishing theorems for Dolbeault cohomology of log homogeneous varieties, Tohoku Math. J. (2) 61 (2009), no. 3, 365–392. MR 2568260 (2011c:14136), DOI https://doi.org/10.2748/tmj/1255700200
- M. Brion and R. Joshua, Equivariant Chow ring and Chern classes of wonderful symmetric varieties of minimal rank, Transform. Groups 13 (2008), no. 3-4, 471–493. MR 2452601 (2009h:14009), DOI https://doi.org/10.1007/s00031-008-9020-2
- Michel Brion and Ivan Kausz, Vanishing of top equivariant Chern classes of regular embeddings, Asian J. Math. 9 (2005), no. 4, 489–496. MR 2216242 (2007e:14006), DOI https://doi.org/10.4310/AJM.2005.v9.n4.a3
- Izzet Coskun, Rigid and non-smoothable Schubert classes, J. Differential Geom. 87 (2011), no. 3, 493–514. MR 2819546 (2012f:14094)
- Izzet Coskun and Colleen Robles, Flexibility of Schubert classes, Differential Geom. Appl. 31 (2013), no. 6, 759–774. MR 3130568, DOI https://doi.org/10.1016/j.difgeo.2013.09.003
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin, 1970 (French). MR 0417174 (54 \#5232)
- Michel Demazure, Désingularisation des variétés de Schubert généralisées, Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I. Ann. Sci. École Norm. Sup. (4) 7 (1974), 53–88 (French). MR 0354697 (50 \#7174)
- W. Fulton, R. MacPherson, F. Sottile, and B. Sturmfels, Intersection theory on spherical varieties, J. Algebraic Geom. 4 (1995), no. 1, 181–193. MR 1299008 (95j:14004)
- William Fulton, Young tableaux, with applications to representation theory and geometry, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. MR 1464693 (99f:05119)
- William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
- Mark Goresky and William Pardon, Chern classes of automorphic vector bundles, Invent. Math. 147 (2002), no. 3, 561–612. MR 1893006 (2003g:32047), DOI https://doi.org/10.1007/s002220100184
- H. C. Hansen, On cycles in flag manifolds, Math. Scand. 33 (1973), 269–274 (1974). MR 0376703 (51 \#12878)
- Jaehyun Hong, Rigidity of singular Schubert varieties in $\textrm {Gr}(m,n)$, J. Differential Geom. 71 (2005), no. 1, 1–22. MR 2191767 (2006i:14052)
- Jaehyun Hong, Rigidity of smooth Schubert varieties in Hermitian symmetric spaces, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2361–2381. MR 2276624 (2007m:32013), DOI https://doi.org/10.1090/S0002-9947-06-04041-4
- June Huh, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc. 25 (2012), no. 3, 907–927. MR 2904577, DOI https://doi.org/10.1090/S0894-0347-2012-00731-0
- June Huh, $h$-vectors of matroids and logarithmic concavity, Adv. Math. 270 (2015), 49–59. MR 3286530, DOI https://doi.org/10.1016/j.aim.2014.11.002
- Benjamin F. Jones, On the singular Chern classes of Schubert varieties via small resolution, Thesis (Ph.D.)–University of Notre Dame, 2007, ProQuest LLC, Ann Arbor, MI. MR 2736782
- Benjamin F. Jones, Singular Chern classes of Schubert varieties via small resolution, Int. Math. Res. Not. IMRN 8 (2010), 1371–1416. MR 2628830 (2011i:14079), DOI https://doi.org/10.1093/imrn/rnp174
- G. Kempf and D. Laksov, The determinantal formula of Schubert calculus, Acta Math. 132 (1974), 153–162. MR 0338006 (49 \#2773)
- Gary Kennedy, MacPherson’s Chern classes of singular algebraic varieties, Comm. Algebra 18 (1990), no. 9, 2821–2839. MR 1063344 (91h:14010), DOI https://doi.org/10.1080/00927879008824054
- Valentina Kiritchenko, Chern classes of reductive groups and an adjunction formula, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 4, 1225–1256 (English, with English and French summaries). MR 2266889 (2008a:14060)
- Valentina Kiritchenko, On intersection indices of subvarieties in reductive groups, Mosc. Math. J. 7 (2007), no. 3, 489–505, 575 (English, with English and Russian summaries). MR 2343145 (2008f:14011)
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 0360616 (50 \#13063)
- Robert Lazarsfeld, Positivity in algebraic geometry. I, Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. MR 2095471 (2005k:14001a)
- R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432. MR 0361141 (50 \#13587)
- Hideyuki Matsumura and Frans Oort, Representability of group functors, and automorphisms of algebraic schemes, Invent. Math. 4 (1967), 1–25. MR 0217090 (36 \#181)
- Leonardo Constantin Mihalcea, Sums of binomial determinants, non-intersecting lattice paths, and positivity of Chern-Schwartz-MacPherson classes, preprint, arXiv:0702566.
- Nicolas Perrin, Courbes rationnelles sur les variétés homogènes, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 1, 105–132 (French, with English and French summaries). MR 1881572 (2004c:14098)
- C. P. Ramanujam, A note on automorphism groups of algebraic varieties, Math. Ann. 156 (1964), 25–33. MR 0166198 (29 \#3475)
- Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291. MR 586450 (83h:32023)
- Jörg Schürmann, Lectures on characteristic classes of constructible functions, notes by Piotr Pragacz and Andrzej Weber, Topics in cohomological studies of algebraic varieties, Trends Math., Birkhäuser, Basel, 2005, pp. 175–201. MR 2143077 (2006g:32048), DOI https://doi.org/10.1007/3-7643-7342-3_7
- Marie-Hélène Schwartz, Classes caractéristiques définies par une stratification d’une variété analytique complexe. I, C. R. Acad. Sci. Paris 260 (1965), 3262–3264 (French). MR 0212842 (35 \#3707)
- Marie-Hélène Schwartz, Classes caractéristiques définies par une stratification d’une variété analytique complexe, C. R. Acad. Sci. Paris 260 (1965), 3535–3537 (French). MR 0184254 (32 \#1727)
- Judson P. Stryker III, Chern-Schwartz-MacPherson classes of graph hypersurfaces and Schubert varieties, Thesis (Ph.D.)–The Florida State University, 2011, ProQuest LLC, Ann Arbor, MI. MR 2949825
- Patrice Tauvel and Rupert W. T. Yu, Lie algebras and algebraic groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2146652 (2006c:17001)
- Andrzej Weber, Equivariant Chern classes and localization theorem, J. Singul. 5 (2012), 153–176. MR 2928940
Additional Information
June Huh
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Address at time of publication:
Institute for Advanced Study and Princeton University, Princeton, New Jersey 08540
MR Author ID:
974745
Email:
huh@princeton.edu
Received by editor(s):
April 4, 2013
Published electronically:
August 18, 2015
Article copyright:
© Copyright 2015
University Press, Inc.