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On holomorphic distributions on Fano threefolds. (English) Zbl 1439.57047
Summary: This paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds $$X$$ which is threedimensional and with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on $$X$$. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.
MSC:
 57R30 Foliations in differential topology; geometric theory 14J45 Fano varieties 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 32S65 Singularities of holomorphic vector fields and foliations
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References:
 [1] Araujo, C.; Corrêa, M., On degeneracy scheme of maps of vector bundles and application to holomorphic foliations, Math. Z., 276, 505-515 (2013) [2] Araujo, C.; Corrêa, M.; Massarenti, A., Codimension one Fano distributions on Fano manifolds, Commun. Contemp. Math., 20 (2018) [3] Arrondo, E.; Costa, L., Vector bundles on Fano 3-folds without intermediate cohomology, Commun. Algebra, 28, 3899-3911 (2000) [4] Ballico, E., On Buchsbaum bundles on quadric hypersurfaces, Cent. Eur. J. Math., 10, 1361-1379 (2012) [5] Calvo-Andrade, O.; Corrêa, M.; Jardim, M., Codimension one holomorphic distributions on the projective three-space, Int. Math. Res. Not., 251 (2018) [6] Casanellas, M.; Hartshorne, R., Gorenstein biliason and ACM sheaves, J. Algebra, 278, 314-341 (2004) [7] Cerveau, D.; Lins Neto, A., Irreducible components of the space of holomorphic foliations of degree two in $$\mathbb{C} \mathbb{P}(n)$$, Ann. Math., 143, 577-612 (1996) [8] Corrêa, M.; Jardim, M.; Vidal Martins, R., On the singular scheme of split foliations, Indiana Univ. Math. J., 64, 1359-1381 (2015) [9] Corrêa, M.; Maza, L. G.; Soares, M. G., Hypersurfaces invariant by Pfaff equations, Commun. Contemp. Math., 17 (2015) [10] Corrêa, M.; Maza, L. G.; Soares, M. G., Algebraic integrability of polynomial differential r-forms, J. Pure Appl. Algebra, 215, 9, 2290-2294 (2011) [11] Corti, A.; Reid, M., Explicit Birational Geometry of 3-Folds, London Mathematical Society Lecture Note Series (2000), Cambridge University Press [12] Dolgachev, I., Weighted Projective Varieties, Lecture Notes in Mathematics, vol. 956 (1982), Springer Verlag [13] Esteves, E.; Kleiman, S. L., Bounding solutions of Pfaff equations, Commun. Algebra, 31 (2003) [14] Flenner, H., Divisorenklassengruppen quasihomogener Singularitäten, J. Reine Angew. Math., 328, 128-160 (1981) [15] Fulton, W., Intersection Theory, Ergeb. Math. Grenz. (1983), Springer-Verlag [16] Giraldo, L.; Pan-Collantes, A. J., On the singular scheme of codimension one holomorphic foliations in $$\mathbb{P}^3$$, Int. J. Math., 7, 843-858 (2010) [17] Griffiths, P., Hermitian Differential Geometry, Chern Classes, and Positive Vector Bundles. Global Analysis (Papers in Honor of K. Kodaira), 185-251 (1969), Univ. Tokyo Press: Univ. Tokyo Press Tokyo [18] Hartshorne, R., Stable reflexive sheaves, Math. Ann., 254, 121-176 (1980) [19] Iskovskikh, V. A., Fano threefolds I, Math. USSR, Izv., 11, 485-527 (1977) [20] Iskovskikh, V. A., Fano threefolds II, Math. USSR, Izv., 12, 469-506 (1978) [21] Kobayashi, S.; Ochiai, T., Characterizations of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ., 13, 31-47 (1973) [22] Loray, F.; Pereira, J. V.; Touzet, F., Foliations with trivial canonical bundle on Fano 3-folds, Math. Nachr., 286, 921-940 (2013) [23] Madonna, C., ACM vector bundles on prime Fano threefolds and complete intersection Calabi-Yau threefolds, Rev. Roum. Math. Pures Appl., 47, 211-222 (2002) [24] Mukai, S., Fano 3-folds, Lond. Math. Soc. Lect. Notes Ser., 179, 255-263 (1992) [25] Ottaviani, G., Spinor bundles on quadrics, Trans. Am. Math. Soc., 307, 301-316 (1988) [26] Ottaviani, G., Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl., 317-341 (1989) [27] Shepherd-Barron, N. I., Fano threefolds in positive characteristic, Compos. Math., 105, 237-265 (1997) [28] Snow, D. M., Cohomology of twisted holomorphic forms on Grassmann manifolds and quadric hypersurfaces, Math. Ann., 276, 159-176 (1986) [29] Stückrad, J.; Vogel, W., Buchsbaum Rings and Applications (1986), Springer: Springer New York
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