2000 character limit reached
On the Miyaoka-Yau inequality for manifolds with nef anti-canonical line bundle
Published 14 Mar 2024 in math.DG and math.AG | (2403.09120v2)
Abstract: Based on the recent work of K.~Zhang, we discuss the Miyaoka-Yau type inequality for projective manifolds with nef anti-canonical line bundle, assuming the lower bound of the delta-invariant introduced by Fujita and Odaka.
- R. J. Berman: A thermodynamical formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler-Einstein metrics. Adv. Math. 248 (2013), 1254–1297.
- Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties. to appear in J. Reine Angew. Math., arXiv:1111.7158.
- A variational approach to the Yau-Tian-Donaldson conjecture. J. Amer. Math. Soc. 34 (2021), 605–652
- Convexity of the extended K-energy and the large time behavior of the weak Calabi flow. Geometry and Topology, 24, 1907–1967 (2020)
- S. Boucksom T. Hisamoto and M. Jonsson: Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs. Ann. Inst. Fourier (Grenoble) 67 no. 2 (2017), 743–841.
- S. Boucksom T. Hisamoto and M. Jonsson: Uniform K-stability and asymptotics of energy functionals in Kähler geometry. J. Eur. Math. Soc.. 21 no. 9 (2019), 2905–2944
- H. Blum and M. Jonsson: Thresholds, valuations, and K-stability. arXiv:1706.04548v1.
- I. Cheltsov: Log canonical thresholds of del Pezzo surfaces. Geometric and Functional Analysis 11 (2008), 1118–1144.
- X. Chen: On the lower bound of the Mabuchi energy and its application. Int. Math. Res. Not. 4 (2000), no. 12, 607–623.
- X. Chen and J. Cheng: On the constant scalar curvature Kähler metrics, a priori estimates. J. Am. Math. Soc., 34 (2021), 909–936.
- X. Chen and J. Cheng: On the constant scalar curvature Kähler metrics, existence results. J. Am. Math. Soc., 34 (2021), 937–1009.
- B. Y. Chen and K. Ogiue: Some characterization of complex space forms in terms of Chern classes. Quart. J. Math., 26 (1975), 119–121.
- G. Chen: The J-equation and the supercritical deformed Hermitian-Yang-Mills equation. Invent. Math. 225 (2021), Pages 529–602.
- T. Darvas and K. Zhang: Twisted Kähler-Einstein metrics in big classes. arXiv:2208.08324.
- R. Dervan: Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds. Int. Math. Res. Notices 26 (2015), 7162–7189.
- S. K. Donaldson: Moment maps and diffeomorphisms. Asian J. Math., 3 (1999) no. 1, 1–15.
- Z. S. Dyrefelt: Existence of cscK metrics on smooth minimal models. arXiv:2004.02832.
- On the K-stability of Fano varieties and anticanonical divisors. Tohoku Math. J. 70(4) 70(4), 511–521 (2018).
- Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor. J. Algebraic Geom. 31 (2022), 467–496.
- Degenerate complex Monge-Ampère equations. EMS Tracts in Mathematics 26 (2017).
- M. Hattori: A decomposition formula for J-stability and its applications. arXiv:2103.04603.
- S. Kobayashi: Differential geometry of complex vector bundles. Princeton University Press, 1987.
- W. Liu: The Miyaoka-Yau inequality on smooth minimal models. Bulletin of the London Mathematical Society 55 (3), 1196–1202.
- T. Mabuchi: K-energy maps integrating Futaki invariants. Tohoku Math. J. (2) 38 (1986), no. 4, 575–593.
- Y. Miyaoka: On the Chern numbers of surfaces of general type. Invent. Math. 42 (1977), no. 1, 225–237.
- A. M. Nadel: Multiplier Ideal Sheaves and Kähler-Einstein Metrics of Positive Scalar Curvature. Ann. of Math. 132 (1990), no. 3, 549–596.
- J. Song and B. Weinkove: On the convergence and singularities of the J-flow with applications to the Mabuchi energy. Commun. Pure Appl. Math. 61 (2008), 210–229.
- J. Song: Nakai-Moischezon criterions for complex Hessian equations. arXiv:2012.07956.
- G. Tian: emphOn stability of the tangent bundles of Fano varieties. Internat. J. Math. 3 (1992), no. 3, 401–413.
- G. Tian: Kähler-Einstein metrics with positive scalar curvature. Inv. Math. 130 (1997), 239–265.
- S. T. Yau: Calabi’s conjecture and some new results in algebraic geometry. Proc. Nat. Acad. Sci. U.S.A 74 (1977), no. 5, 1798–1799.
- S. T. Yau: On the Ricci curvature of a com pact Kähler manifold and the complex Monge-Ampére equation I. Com. Pure and Appl. Math., 31 (1978), 339–411.
- K. Zhang: Continuity of delta invariants and twisted Kähler-Einstein metrics. Adv. Math. 388 (2021)
- K. Zhang: A quantization proof of the uniform Yau-Tian-Donaldson conjecture. to appear in the Journal of the European Mathematical Society.
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.