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Betti numbers and pseudoeffective cones in 2-Fano varieties

  • Giosuè Emanuele Muratore EMAIL logo
Published/Copyright: June 17, 2021
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Advances in Geometry
From the journal Advances in Geometry Volume 21 Issue 4

Abstract

The 2-Fano varieties, defined by De Jong and Starr, satisfy some higher-dimensional analogous properties of Fano varieties. We consider (weak) k-Fano varieties and conjecture the polyhedrality of the cone of pseudoeffective k-cycles for those varieties, in analogy with the case k = 1. Then we calculate some Betti numbers of a large class of k-Fano varieties to prove some special case of the conjecture. In particular, the conjecture is true for all 2-Fano varieties of index at least n − 2, and we complete the classification of weak 2-Fano varieties answering Questions 39 and 41 in [2].

Keywords: Fano variety; Betti number; cycle
MSC 2010: Primary 14J45; Secondary 14M15

Acknowledgements

I thank Angelo Lopez for all the support he has shown me since the beginning of this work, and Gianluca Pacienza for his help. I also thank Carolina Araujo, Izzet Coskun and Enrico Fatighenti for answering many of my questions.

  1. Communicated by: I. Coskun

References

[1] C. Araujo, A.-M. Castravet, Polarized minimal families of rational curves and higher Fano manifolds. Amer. J. Math. 134 (2012), 87–107. MR2876140 Zbl 1248.1404510.1353/ajm.2012.0008Search in Google Scholar

[2] C. Araujo, A.-M. Castravet, Classification of 2-Fano manifolds with high index. In: A celebration of algebraic geometry, volume 18 of Clay Math. Proc., 1–36, Amer. Math. Soc. 2013. MR3114934 Zbl 1317.14092Search in Google Scholar

[3] A. Borel, F. Hirzebruch, Characteristic classes and homogeneous spaces. I. Amer. J. Math. 80 (1958), 458–538. MR102800 Zbl 0097.3640110.2307/2372795Search in Google Scholar

[4] N. Bourbaki, Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines. Hermann, Paris 1968. MR0240238 Zbl 0186.33001Search in Google Scholar

[5] I. Coskun, Restriction varieties and geometric branching rules. Adv. Math. 228 (2011), 2441–2502. MR2836127 Zbl 1262.1405910.1016/j.aim.2011.07.010Search in Google Scholar

[6] I. Coskun, Symplectic restriction varieties and geometric branching rules. In: A celebration of algebraic geometry, volume 18 of Clay Math. Proc., 205–239, Amer. Math. Soc. 2013. MR3114942 Zbl 1317.14107Search in Google Scholar

[7] R. L. de Arruda, On Varieties of Lines on Linear Sections of Grassmannians. Preprint 2015, arXiv:1505.06488 [math.AG].Search in Google Scholar

[8] A. J. de Jong, J. Starr, Higher Fano manifolds and rational surfaces. Duke Math. J. 139 (2007), 173–183. MR2322679 Zbl 1124.1403910.1215/S0012-7094-07-13914-0Search in Google Scholar

[9] A. J. de Jong, J. Starr, A note on Fano manifolds whose second Chern character is positive. Preprint 2006, arXiv:math/0602644 [math.AG].Search in Google Scholar

[10] O. Debarre, L. Ein, R. Lazarsfeld, C. Voisin, Pseudoeffective and nef classes on abelian varieties. Compos. Math. 147 (2011), 1793–1818. MR2862063 Zbl 1234.1400810.1112/S0010437X11005227Search in Google Scholar

[11] A. Dimca, Singularities and topology of hypersurfaces. Springer 1992. MR1194180 Zbl 0753.5700110.1007/978-1-4612-4404-2Search in Google Scholar

[12] C. Ehresmann, Sur la topologie de certains espaces homogènes. Ann. of Math. (2) 35 (1934), 396–443. MR1503170 Zbl 0009.3290310.2307/1968440Search in Google Scholar

[13] T. Fujita, Classification of projective varieties of Δ-genus one. Proc. Japan Acad. Ser. A Math. Sci. 58 (1982), 113–116. MR664549 Zbl 0568.1401810.3792/pjaa.58.113Search in Google Scholar

[14] T. Fujita, On polarized varieties of small Δ-genera. Tohoku Math. J. (2) 34 (1982), 319–341. MR676113 Zbl 0489.1400210.2748/tmj/1178229197Search in Google Scholar

[15] W. Fulton, Intersection theory. Springer 1998. MR1644323 Zbl 0885.1400210.1007/978-1-4612-1700-8Search in Google Scholar

[16] W. Fulton, R. MacPherson, F. Sottile, B. Sturmfels, Intersection theory on spherical varieties. J. Algebraic Geom. 4 (1995), 181–193. MR1299008 Zbl 0819.14019Search in Google Scholar

[17] R. Hartshorne, Algebraic geometry. Springer 1977. MR0463157 Zbl 0367.1400110.1007/978-1-4757-3849-0Search in Google Scholar

[18] A. Hatcher, Algebraic topology. Cambridge Univ. Press 2002. MR1867354 Zbl 1044.55001Search in Google Scholar

[19] V. A. Iskovskikh, Y. G. Prokhorov, Fano varieties. In: Algebraic geometry, V, volume 47 of Encyclopaedia Math. Sci., 1–247, Springer 1999. MR1668579 Zbl 0912.14013Search in Google Scholar

[20] S. L. Kleiman, The transversality of a general translate. Compositio Math. 28 (1974), 287–297. MR360616 Zbl 0288.14014Search in Google Scholar

[21] S. Kobayashi, T. Ochiai, Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13 (1973), 31–47. MR316745 Zbl 0261.3201310.1215/kjm/1250523432Search in Google Scholar

[22] B. Köck, Chow motif and higher Chow theory of G/PManuscripta Math. 70 (1991), 363–372. MR1092142 Zbl 0735.1400110.1007/BF02568384Search in Google Scholar

[23] O. Küchle, On Fano 4-folds of index 1 and homogeneous vector bundles over Grassmannians. Math. Z. 218 (1995), 563–575. MR1326986 Zbl 0826.1402410.1007/BF02571923Search in Google Scholar

[24] A. G. Kuznetsov, On Küchle varieties with Picard number greater than 1. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 79 (2015), 57–70. English translation: Izv. Math. 79 (2015), 698–709. MR3397419 Zbl 1342.1408710.1070/IM2015v079n04ABEH002758Search in Google Scholar

[25] R. Lazarsfeld, Positivity in algebraic geometry. I. Springer 2004. MR2095471 Zbl 1093.1450110.1007/978-3-642-18808-4Search in Google Scholar

[26] R. Lazarsfeld, Positivity in algebraic geometry. II. Springer 2004. MR2095472 Zbl 1093.1450010.1007/978-3-642-18810-7Search in Google Scholar

[27] S. Mukai, Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 3000–3002. MR995400 Zbl 0679.1402010.1073/pnas.86.9.3000Search in Google Scholar PubMed PubMed Central

[28] D. Mumford, An algebraic surface with K ample, (K2) = 9, pg = q = 0. Amer. J. Math. 101 (1979), 233–244. MR527834 Zbl 0433.1402110.2307/2373947Search in Google Scholar

[29] J. C. Ottem, Nef cycles on some hyperkähler fourfolds. Preprint 2015, arXiv:1505.01477 [math.AG]Search in Google Scholar

[30] M. Reid, The complete intersection of two or more quadrics. Dissertation Trinity College, Cambridge 1972.Search in Google Scholar

[31] D. M. Snow, Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces. Math. Ann. 276 (1986), 159–176. MR863714 Zbl 0596.3201610.1007/BF01450932Search in Google Scholar

[32] J. G. Spandaw, A Noether-Lefschetz theorem for vector bundles. Manuscripta Math. 89 (1996), 319–323. MR1378596 Zbl 0862.1401010.1007/BF02567520Search in Google Scholar

Received: 2018-07-17
Revised: 2020-05-05
Published Online: 2021-06-17
Published in Print: 2021-10-26

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. An 𝔽p2-maximal Wiman sextic and its automorphisms
  3. Pseudo-algebraic Ricci solitons on Einstein nilradicals
  4. The wobbly divisors of the moduli space of rank-2 vector bundles
  5. Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
  6. Betti numbers and pseudoeffective cones in 2-Fano varieties
  7. The generating rank of a polar Grassmannian
  8. The Beckman–Quarles theorem via the triangle inequality
  9. On Huisman’s conjectures about unramified real curves
  10. Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
  11. On the Segre invariant for rank two vector bundles on ℙ2
  12. Lifting coarse homotopies
  13. How to construct all metric f-K-contact manifolds
  14. An extremum problem for the power moment of a convex polygon contained in a disc
  15. Sharply transitive sets in PGL2(K)
https://doi.org/10.1515/advgeom-2021-0004
Keywords for this article
Fano variety; Betti number; cycle

Articles in the same Issue

  1. Frontmatter
  2. An 𝔽p2-maximal Wiman sextic and its automorphisms
  3. Pseudo-algebraic Ricci solitons on Einstein nilradicals
  4. The wobbly divisors of the moduli space of rank-2 vector bundles
  5. Symmetries of complex analytic vector fields with an essential singularity on the Riemann sphere
  6. Betti numbers and pseudoeffective cones in 2-Fano varieties
  7. The generating rank of a polar Grassmannian
  8. The Beckman–Quarles theorem via the triangle inequality
  9. On Huisman’s conjectures about unramified real curves
  10. Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition
  11. On the Segre invariant for rank two vector bundles on ℙ2
  12. Lifting coarse homotopies
  13. How to construct all metric f-K-contact manifolds
  14. An extremum problem for the power moment of a convex polygon contained in a disc
  15. Sharply transitive sets in PGL2(K)
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