Iitaka dimension for cycles

Lehmann, Brian

doi:10.1090/tran/7491

Iitaka dimension for cycles
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Trans. Amer. Math. Soc. 371 (2019), 4815-4835

Abstract:

We define the Iitaka dimension of a numerical cycle class and develop its theory. We conjecture that the Iitaka dimension is integer valued and give some evidence in this direction. We focus on two cases of geometric interest: Schubert cycles on Grassmannians and cycles contracted by morphisms.

References

Thomas Bauer, Frédéric Campana, Thomas Eckl, Stefan Kebekus, Thomas Peternell, Sławomir Rams, Tomasz Szemberg, and Lorenz Wotzlaw, A reduction map for nef line bundles, Complex geometry (Göttingen, 2000) Springer, Berlin, 2002, pp. 27–36. MR 1922095
Robert Bryant, Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces, arXiv:math/0006186 [math.DG], 2005.
F. Campana, Coréduction algébrique d’un espace analytique faiblement kählérien compact, Invent. Math. 63 (1981), no. 2, 187–223 (French). MR 610537, DOI 10.1007/BF01393876
Dawei Chen and Izzet Coskun, Extremal higher codimension cycles on moduli spaces of curves, Proc. Lond. Math. Soc. (3) 111 (2015), no. 1, 181–204. MR 3404780, DOI 10.1112/plms/pdv029
Izzet Coskun, Rigid and non-smoothable Schubert classes, J. Differential Geom. 87 (2011), no. 3, 493–514. MR 2819546
Izzet Coskun, Rigidity of Schubert classes in orthogonal Grassmannians, Israel J. Math. 200 (2014), no. 1, 85–126. MR 3219572, DOI 10.1007/s11856-014-0009-3
Izzet Coskun and Colleen Robles, Flexibility of Schubert classes, Differential Geom. Appl. 31 (2013), no. 6, 759–774. MR 3130568, DOI 10.1016/j.difgeo.2013.09.003
Mihai Fulger, János Kollár, and Brian Lehmann, Volume and Hilbert function of $\Bbb R$-divisors, Michigan Math. J. 65 (2016), no. 2, 371–387. MR 3510912, DOI 10.1307/mmj/1465329018
Mihai Fulger and Brian Lehmann, Morphisms and faces of pseudo-effective cones, Proc. Lond. Math. Soc. (3) 112 (2016), no. 4, 651–676. MR 3483128, DOI 10.1112/plms/pdw008
Mihai Fulger and Brian Lehmann, Positive cones of dual cycle classes, Algebr. Geom. 4 (2017), no. 1, 1–28. MR 3592463, DOI 10.14231/AG-2017-001
Mihai Fulger and Brian Lehmann, Zariski decompositions of numerical cycle classes, J. Algebraic Geom. 26 (2017), no. 1, 43–106. MR 3570583, DOI 10.1090/jag/677
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Jaehyun Hong, Rigidity of singular Schubert varieties in $\textrm {Gr}(m,n)$, 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 10.1090/S0002-9947-06-04041-4
Shigeru Iitaka, On $D$-dimensions of algebraic varieties, Proc. Japan Acad. 46 (1970), 487–489. MR 285532
Shigeru Iitaka, On $D$-dimensions of algebraic varieties, J. Math. Soc. Japan 23 (1971), 356–373. MR 285531, DOI 10.2969/jmsj/02320356
Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
János Kollár, Yoichi Miyaoka, and Shigefumi Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992), no. 3, 429–448. MR 1158625
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, DOI 10.1007/978-3-642-18808-4
Brian Lehmann, Volume-type functions for numerical cycle classes, Duke Math. J. 165 (2016), no. 16, 3147–3187. MR 3566200, DOI 10.1215/00127094-3673772
C. Robles, Schur flexibility of cominuscule Schubert varieties, Comm. Anal. Geom. 21 (2013), no. 5, 979–1013. MR 3152970, DOI 10.4310/CAG.2013.v21.n5.a5
C. Robles and D. The, Rigid Schubert varieties in compact Hermitian symmetric spaces, Selecta Math. (N.S.) 18 (2012), no. 3, 717–777. MR 2960030, DOI 10.1007/s00029-011-0082-y
Beniamino Segre, Sulle $V_n$ contenenti più di $\infty ^{n-k}S_k$. I, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 5 (1948), 193–197 (Italian). MR 36041
Beniamino Segre, Sulle $V_n$ contenenti più di $\infty ^{n-k}S_k$. II, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 5 (1948), 275–280 (Italian). MR 36042
Claire Voisin, Coniveau 2 complete intersections and effective cones, Geom. Funct. Anal. 19 (2010), no. 5, 1494–1513. MR 2585582, DOI 10.1007/s00039-009-0036-0
Maria F. Walters, Geometry and uniqueness of some extreme subvarieties in complex Grassmannians, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–University of Michigan. MR 2695855