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A Survey on the Weighted Log Canonical Threshold and the Weighted Multiplier Ideal Sheaf

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Geometric Complex Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 246))

Abstract

We present some recent results and propose a list of questions on the weighted log canonical threshold and the weighted multiplier ideal sheaf. This survey is dedicated to Prof. Kang-Tae Kim on the Occasion of His 60th Birthday.

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References

  1. Berndtsson, B.: The openness conjecture and complex Brunn-Minkowski inequalities. Comp. Geom. Dyn. 10, 29–44 (2015)

    Article  MathSciNet  Google Scholar 

  2. De Fernex, T., Ein, L., Mustata, M.: Shokurov’s ACC conjecture for log canonical thresholds on smooth varieties. Duke Math. J. 152, 93–114 (2010)

    Article  MathSciNet  Google Scholar 

  3. Demailly, J.-P.: Monge-Ampère operators, Lelong numbers and intersection theory. In: Ancona, V., Silva, A. (eds.) Complex Analysis and Geometry, University Series in Math. Plenum Press, New-York (1993)

    Google Scholar 

  4. Demailly, J.-P.: A numerical criterion for very ample line bundles. J. Diff. Geom. 37, 323–374 (1993)

    Article  MathSciNet  Google Scholar 

  5. Demailly, J.-P., Hiep, Pham Hoang: A sharp lower bound for the log canonical threshold. Acta Math. 212, 1–9 (2014)

    Article  MathSciNet  Google Scholar 

  6. Demailly, J.-P., Kollár, J.: Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. Ecole Norm. Sup. 34, 525–556 (2001)

    Article  MathSciNet  Google Scholar 

  7. Favre, C., Jonsson, M.: Valuations and multiplier ideals. J. Am. Math. Soc. 18, 655–684 (2005)

    Article  MathSciNet  Google Scholar 

  8. Guan, Q.A., Zhou, X.Y.: Effectiveness of Demaillys strong openness conjecture and related problems. Invent. Math. 202(2), 635–676 (2015)

    Article  MathSciNet  Google Scholar 

  9. Guan, Q.A., Zhou, X.Y.: A proof of Demailly’s strong openness conjecture. Ann. Math. 182, 605–616 (2015)

    Article  MathSciNet  Google Scholar 

  10. Hacon, C., McKernan, J., Xu, C.: ACC for log canonical thresholds. Ann. Math. 180, 523–571 (2014)

    Article  MathSciNet  Google Scholar 

  11. Hiep, P.H.: Continuity properties of certain weighted log canonical thresholds. C. R. Acad. Sci. Paris. 355, 34–39 (2017)

    Google Scholar 

  12. Hiep, P.H.: The weighted log canonical threshold. C. R. Acad. Sci. Paris 352, 283–288 (2014)

    Google Scholar 

  13. Kiselman, C.O.: Attenuating the singularities of plurisubharmonic functions. Ann. Polon. Math. 60, 173–197 (1994)

    Article  MathSciNet  Google Scholar 

  14. Kollár, J. (with 14 coauthors): Flips and abundance for algebraic threefolds, Astérisque No. 211, 1992

    Google Scholar 

  15. Nadel, A.M.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. Math. 132, 549–596 (1990)

    Article  MathSciNet  Google Scholar 

  16. Ohsawa, T., Takegoshi, K.: On the extension of \(L^2\) holomorphic functions. Math. Z. 195, 197–204 (1987)

    Article  MathSciNet  Google Scholar 

  17. Phong, D.H., Sturm, J.: Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions. Ann. Math. 152, 277–329 (2000)

    Google Scholar 

  18. Shokurov, V.V.: Three-dimensional log perestroikas. Izv. Ross. Akad. Nauk Ser. Mat. 56(1), 105–203 (1992)

    MathSciNet  Google Scholar 

  19. Skoda, H.: Sous-ensembles analytiques d’ordre fini ou infini dans \(\mathbb{C}^n\). Bull. Soc. Math. Fr. 100, 353–408 (1972)

    Article  Google Scholar 

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Correspondence to Pham Hoang Hiep .

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Hoang Hiep, P. (2018). A Survey on the Weighted Log Canonical Threshold and the Weighted Multiplier Ideal Sheaf. In: Byun, J., Cho, H., Kim, S., Lee, KH., Park, JD. (eds) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol 246. Springer, Singapore. https://doi.org/10.1007/978-981-13-1672-2_13

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