A Miyaoka-Yau Type Inequality for Threefolds with Nef Anti-canonical Divisors in Positive Characteristic

Abstract

We prove a Miyaoka–Yau type inequality for threefolds such that −K_X is nef and of numerical dimension ≥ 2

$A \cdot \left(c_{2}(X)+\lambda{c}_{1}^{2}(X)\right) >0,$

where A is an ample line bundle and λ > 0 is a constant dependent on the characteristic p and the Cartier index n₀ of K_X. This is an analogue of the Miyaoka–Yau type inequality presented in [Algebra Number Theory, 2022, 16(10): 2339–2384], which treats minimal threefolds of general type. We attain this inequality by following the strategy of [Algebra Number Theory, 2022, 16(10): 2339–2384, Duke Math. J., 2019, 168(7): 1269–1301]. Applying a similar argument, we also show that in characteristic ≥ 5, if −K_X is nef and −K_X · c₂(X) < 0, then the nonvanishing theorem for anti-canonical divisor holds.

Keywords

Miyaoka–Yau inequality / threefold / positive characteristic / 14C17 / 14E99 / 14F99

References

[1] Bauer T, Campana F, Eckl T, Kebekus S, Peternell T, Rams S, Szemberg T, Wotzlaw L. A Reduction Map for Nef Line Bundles, 2002. Berlin, Springer.

[2] Birkar C, Waldron J. Existence of Mori fibre spaces for 3-folds in char p. Adv. Math., 2017, 313: 62-101.

[3] Cascini P, Tanaka H, Xu C. On base point freeness in positive characteristic. Ann. Sci. Éc. Norm. Supér. (4), 2015, 48(5): 1239-1272.

[4] Ekedahl T. Foliations and Inseparable Morphisms, 1987. Providence, RI, Amer. Math. Soc.. Part 2. 46

[5] Ekedahl T. Canonical models of surfaces of general type in positive characteristic. Inst. Hautes Études Sci. Publ. Math., 1988, 67: 97-144.

[6] Gu Y, Sun X, Zhou M. Slope inequalities and a Miyaoka–Yau type inequality. J. Eur. Math. Soc. (JEMS), 2023, 25(2): 611-632.

[7] Hartshorne R. Algebraic Geometry, 1977. New York-Heidelberg, Springer-Verlag. 52

[8] Huybrechts D, Lehn M. The Geometry of Moduli Spaces of Sheaves, 2010, Second Edition, Cambridge, Cambridge University Press.

[9] Iwai M, Jiang C, Liu H. Miyaoka-type inequalities for terminal threefolds with nef anti-canonical divisors. Sci. China Math., 2025, 68(1): 1-18.

[10] Katz NM. Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Études Sci. Publ. Math., 1970, 39: 175-232.

[11] Kawamata Y. Boundedness of Q-Fano threefolds, 1992. Providence, RI, Amer. Math. Soc.. Part 3. 131

[12] Keeler DS. Fujita’s conjecture and Frobenius amplitude. Amer. J. Math., 2008, 130(5): 1327-1336.

[13] Langer A. Semistable sheaves in positive characteristic. Ann. of Math. (2), 2004, 159(1): 251-276.

[14] Langer A. Generic positivity and foliations in positive characteristic. Adv. Math., 2015, 277: 1-23.

[15] Liu H, Liu J. Kawamata–Miyaoka type inequality for ℚ-Fano varieties with canonical singularities. J. Reine Angew. Math., 2025, 819: 265-281

[16] Liu H., Liu J., Kawamata–Miyaoka-type inequality for ℚ-Fano varieties with canonical singularities II: terminal ℚ-Fano threefolds. Épijournal Géom. Algébrique, 2025, 9: Art. 12, 21 pp.

[17] Mehta VB, Ramanathan A. Homogeneous Bundles in Characteristic p, 1983. Berlin, Springer. 997

[18] Miyaoka Y. On the Chern numbers of surfaces of general type. Invent. Math., 1977, 42: 225-237.

[19] Miyaoka Y. The Chern Classes and Kodaira Dimension of a Minimal Variety, 1987. Amsterdam, North-Holland. 10

[20] Mukai S. Counterexamples to Kodaira’s vanishing and Yau’s inequality in positive characteristics. Kyoto J. Math., 2013, 53(2): 515-532.

[21] Ou W., On generic nefness of tangent sheaves. Math. Z., 2023, 304(4): Paper No. 58, 23pp.

[22] Patakfalvi Z, Waldron J. Singularities of general fibers and the LMMP. Amer. J. Math., 2022, 144(2): 505-540.

[23] Raynaud M. Contre-exemple au “vanishing theorem” en caractéristique p > 0. C. P. Ramanujam—a tribute, 1978. Berlin–New York, Springer: 273-2788

[24] Shepherd-Barron NI. Fano threefolds in positive characteristic. Compositio Math., 1997, 105(3): 237-265.

[25] Tanaka H. Abundance theorem for surfaces over imperfect fields. Math. Z., 2020, 195(1–2): 595-622.

[26] Tate J. Genus change in inseparable extensions of function fields. Proc. Amer. Math. Soc., 1952, 3: 400-406.

[27] Xu C, Zhang L. Nonvanishing for 3-folds in characteristic p > 5. Duke Math. J., 2019, 168(7): 1269-1301.

[28] Zhang L. Abundance for 3-folds with non-trivial Albanese maps in positive characteristic. J. Eur. Math. Soc. (JEMS), 2020, 22(9): 2777-2820.

[29] Zhang L. Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic. Algebra Number Theory, 2022, 16(10): 2339-2384.

RIGHTS & PERMISSIONS

Peking University