Local and global applications of the minimal model program for co-rank 1 foliations on threefolds. (English) Zbl 1515.14025

Spicer, Calum; Svaldi, Roberto

J. Eur. Math. Soc. (JEMS) 24, No. 11, 3969-4025 (2022).

The study of minimal model programs (MMP) for foliations on threefolds has seen significant progress in recent works, notably in the series of papers [C. Spicer, Compos. Math. 156, No. 1, 1–38 (2020; Zbl 1428.14025); P. Cascini and C. Spicer, Invent. Math. 225, No. 2, 603–690 (2021; Zbl 1492.14025); P. Cascini and C. Spicer, “On the MMP for rank one foliations on threefolds”, Preprint, arXiv:2012.11433]. The present paper uses the techniques and ideas from the foliated MMP to make significant contributions on some local and global results for threefold foliations.
For the local results, the authors establish the integrability criterion under isolated canonical singularities, the existence of separatrix at log canonical germs, and foliated inversion of adjunction. To be precise, the following three theorems are proved:
First, the following result is a version of Malgrange’s integrability of singular foliations [B. Malgrange, Publ. Math., Inst. Hautes Étud. Sci. 46, 163–173 (1976; Zbl 0355.32013)].
{Theorem}. Let \(P \in X\) be a germ of an isolated (analytically) \(\mathbb Q\)-factorial threefold singularity with a co-rank 1 foliation \(\mathcal F\). Suppose that \(\mathcal F\) has an isolated canonical singularity at \(P\). Then \(\mathcal F\) admits a holomorphic first integral.
Second, recall that for a singular point of a foliation \(\mathcal F\), a separatrix is an \(\mathcal F\)-invariant hypersurface germ. It is a challenging problem to determine when a foliation singularity admits a separatrix. The authors show the existence of separatrix for the log canonical singularities.
{Theorem}. Let \(\mathcal F\) be a germ of a log canonical foliation singularity on \(0 \in \mathbb C^3\). Then \(\mathcal F\) admits a separatrix.
Third, the inversion of adjunction is a kind of result that describes the singularities of the ambient space through the singularities of restricted divisors.
{Theorem}. Let \(\mathcal F\) be a co-rank one foliation on a \(\mathbb Q\)-factorial threefold. For a prime divisor \(S\) and an effective \(\mathbb Q\)-divisor \(\Delta\) on \(X\) which does not contain \(S\) in its support. Let \(\nu : S^\nu \to S\) be the normalization and let \(\mathcal G\) be the restricted foliation to \(S^\nu\). Write \(\nu^*(K_{\mathcal F} +\Delta) =K_{\mathcal G}+\Theta\). Suppose that

1.

if \(S\) is transverse to \(\mathcal F\), then \((\mathcal G, \Theta)\) is log canonical;

2.

if \(S\) is \(\mathcal F\)-invariant, then \((S^\nu, \Theta)\) is log canonical.

Then \((\mathcal F, \epsilon(S)S + \Delta)\) is log canonical in a neighborhood of \(S\), where \(\epsilon(S)=0\) if \(S\) is not \(\mathcal F\)-invariant and \(\epsilon(S)=1\) otherwise. The proof of these results effectively utilizes F-dlt modifications, introduced in [P. Cascini and C. Spicer, Invent. Math. 225, No. 2, 603–690 (2021; Zbl 1492.14025)], as an analogous technique to the usual dlt modification in birational geometry.
For global results, the authors establish the terminations of flips, the non-vanishing theorem, and the connectedness of non-klt centers for F-dlt co-rank one foliations on threefolds. These results bear a resemblance to corresponding outcomes in the minimal model program of algebraic varieties. Furthermore, as an application, under the assumption of non-existence of certain \(\mathbb{A}^1\)-curves, the nefness of the canonical divisor of the foliation is demonstrated. These global results build upon the F-dlt modification technique and rely on a meticulous analysis of the contracted loci.

Reviewer: Zhan Li (Shenzhen)

MSC:

14E30	Minimal model program (Mori theory, extremal rays)
37F75	Dynamical aspects of holomorphic foliations and vector fields
32S65	Singularities of holomorphic vector fields and foliations

Keywords:

foliations; minimal model program; singularities; separatrices; hyperbolicity

Citations:

Zbl 1428.14025; Zbl 1492.14025; Zbl 0355.32013

References:

[1]	Araujo, C., Druel, S.: On Fano foliations. Adv. Math. 238, 70-118 (2013) Zbl 1282.14085 MR 3033631 · Zbl 1282.14085
[2]	Birkar, C.: On connectedness of non-klt loci of singularities of pairs. arXiv:2010.08226 (2020)
[3]	Bogomolov, F., McQuillan, M.: Rational curves on foliated varieties. In: Foliation Theory in Algebraic Geometry, Simons Symp., Springer, Cham, 21-51 (2016) Zbl 1337.14041 MR 3644242 · Zbl 1337.14041
[4]	Brunella, M.: Birational Geometry of Foliations. Monografías de Matemática, Instituto de Matemática Pura e Aplicada, Rio de Janeiro (2000) Zbl 1073.14022 MR 1948251 · Zbl 1073.14022
[5]	Camacho, C.: Quadratic forms and holomorphic foliations on singular surfaces. Math. Ann. 282, 177-184 (1988) Zbl 0657.32007 MR 963011 · Zbl 0657.32007
[6]	Camacho, C., Lins Neto, A.: Geometric Theory of Foliations. Birkhäuser Boston, Boston, MA (1985) Zbl 0568.57002 MR 824240 · Zbl 0568.57002
[7]	Cano, F.: Reduction of the singularities of codimension one singular foliations in dimension three. Ann. of Math. (2) 160, 907-1011 (2004) Zbl 1088.32019 MR 2144971 · Zbl 1088.32019
[8]	Cano, F., Cerveau, D.: Desingularization of nondicritical holomorphic foliations and existence of separatrices. Acta Math. 169, 1-103 (1992) Zbl 0771.32018 MR 1179013 · Zbl 0771.32018
[9]	Cano, F., Mattei, J.-F.: Hypersurfaces intégrales des feuilletages holomorphes. Ann. Inst. Fou-rier (Grenoble) 42, 49-72 (1992) Zbl 0762.32018 MR 1162557 · Zbl 0762.32018
[10]	Cano, F., Molina-Samper, B.: Invariant surfaces for toric type foliations in dimension three. Publ. Mat. 65, 291-307 (2021) Zbl 07405617 MR 4185834 · Zbl 1524.32027
[11]	Cano, F., Ravara-Vago, M.: Local Brunella’s alternative II. Partial separatrices. Int. Math. Res. Notices 2015, 12840-12876 Zbl 1341.32025 MR 3431638 · Zbl 1341.32025
[12]	Cano, F., Ravara-Vago, M., Soares, M.: Local Brunella’s alternative I. RICH foliations. Int. Math. Res. Notices 2015, 2525-2575 (2015) Zbl 1317.32064 MR 3344680 · Zbl 1317.32064
[13]	Cascini, P., Spicer, C.: MMP for co-rank one foliations on threefolds. Invent. Math. 225, 603-690 (2021) Zbl 07387419 MR 4285142 · Zbl 1492.14025
[14]	Cerveau, D., Lins Neto, A.: Frobenius theorem for foliations on singular varieties. Bull. Brazil. Math. Soc. (N.S.) 39, 447-469 (2008) Zbl 1194.32018 MR 2473858 · Zbl 1194.32018
[15]	Druel, S.: Codimension 1 foliations with numerically trivial canonical class on singular spaces. Duke Math. J. 170, 95-203 (2021) Zbl 07322635 MR 4194898 · Zbl 1470.14009
[16]	Druel, S., Ou, W.: Codimension one foliations with trivial canonical class on singular spaces II. arXiv:1912.07727v2 (2019)
[17]	Filipazzi, S., Svaldi, R.: On the connectedness principle and dual complexes for generalized pairs. arXiv:2010.08018v2 (2020)
[18]	Flenner, H.: Divisorenklassengruppen quasihomogener Singularitäten. J. Reine Angew. Math. 328, 128-160 (1981) Zbl 0457.14001 MR 636200 · Zbl 0457.14001
[19]	Gómez-Mont, X., Luengo, I.: Germs of holomorphic vector fields in C 3 without a separatrix. Invent. Math. 109, 211-219 (1992) Zbl 0774.32019 MR 1172688 · Zbl 0774.32019
[20]	Greb, D., Kebekus, S., Kovács, S. J., Peternell, T.: Differential forms on log canonical spaces. Publ. Math. Inst. Hautes Études Sci. 114, 87-169 (2011) Zbl 1258.14021 MR 2854859 · Zbl 1258.14021
[21]	Greb, D., Kebekus, S., Peternell, T.: Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties. Duke Math. J. 165, 1965-2004 (2016) Zbl 1360.14094 MR 3522654 · Zbl 1360.14094
[22]	Hartshorne, R.: Algebraic Geometry. Grad. Texts in Math. 52, Springer, New York (1977) Zbl 0367.14001 MR 0463157 · Zbl 0367.14001
[23]	Kebekus, S.: Pull-back morphisms for reflexive differential forms. Adv. Math. 245, 78-112 (2013) Zbl 1427.14072 MR 3084424 · Zbl 1427.14072
[24]	Kempf, G., Knudsen, F. F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Math. 339, Springer, Berlin (1973) Zbl 0271.14017 MR 0335518 · Zbl 0271.14017
[25]	Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Math. 134, Cambridge Univ. Press, Cambridge (1998) Zbl 0926.14003 MR 1658959 · Zbl 0926.14003
[26]	Loray, F., Pereira, J. V., Touzet, F.: Singular foliations with trivial canonical class. Invent. Math. 213, 1327-1380 (2018) Zbl 1426.32014 MR 3842065 · Zbl 1426.32014
[27]	Malgrange, B.: Frobenius avec singularités. I. Codimension un. Inst. Hautes Études Sci. Publ. Math. 46, 163-173 (1976) Zbl 0355.32013 MR 508169 · Zbl 0355.32013
[28]	Mattei, J.-F., Moussu, R.: Holonomie et intégrales premières. Ann. Sci. École Norm. Sup. (4) 13, 469-523 (1980) Zbl 0458.32005 MR 608290 · Zbl 0458.32005
[29]	McQuillan, M.: Diophantine approximations and foliations. Inst. Hautes Études Sci. Publ. Math. 87, 121-174 (1998) Zbl 1006.32020 MR 1659270 · Zbl 1006.32020
[30]	McQuillan, M.: Canonical models of foliations. Pure Appl. Math. Quart. 4, 877-1012 (2008) Zbl 1166.14010 MR 2435846 · Zbl 1166.14010
[31]	McQuillan, M., Panazzolo, D.: Almost étale resolution of foliations. J. Differential Geom. 95, 279-319 (2013) Zbl 1295.32041 MR 3128985 · Zbl 1295.32041
[32]	Mendes, L. G.: Kodaira dimension of holomorphic singular foliations. Bol. Soc. Brasil. Mat. (N.S.) 31, 127-143 (2000) Zbl 0979.32017 MR 1785264 · Zbl 0979.32017
[33]	Pereira, J. V., Svaldi, R.: Effective algebraic integration in bounded genus. Algebraic Geom. 6, 454-485 (2019) Zbl 1461.37049 MR 3957403 · Zbl 1461.37049
[34]	Siu, Y.-t.: Extending coherent analytic sheaves. Ann. of Math. (2) 90, 108-143 (1969) Zbl 0181.36301 MR 245837 · Zbl 0181.36301
[35]	Spicer, C.: Higher-dimensional foliated Mori theory. Compos. Math. 156, 1-38 (2020) Zbl 1428.14025 MR 4036447 · Zbl 1428.14025
[36]	Svaldi, R.: Hyperbolicity for log canonical pairs and the cone theorem. Selecta Math. (N.S.) 25, art. 67, 23 pp. (2019) Zbl 1442.14057 MR 4030221 · Zbl 1442.14057
[37]	Tian, Z., Xu, C.: Finiteness of fundamental groups. Compos. Math. 153, 257-273 (2017) Zbl 1453.14014 MR 3604863 · Zbl 1453.14014
[38]	Wang, J.: On the Iitaka conjecture C n;m for Kähler fibre spaces. arXiv:1907.06705 (2019)