American Institute of Mathematical Sciences

Advanced
  2017, 24: 21-27. doi: 10.3934/era.2017.24.003

The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities

Piotr Pokora

1. 

Instytut Matematyki, Pedagogical University of Cracow, Podchorążych 2, PL-30-084 Kraków, Poland
2. 

Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany

To Professor Kamil Rusek, on the occasion of his 70th birthday.

Received  December 26, 2016 Published  April 2017

Fund Project: I am very grateful to Adrian Langer for stimulating conversations, useful comments about the content of the note, and for pointing out [2]. Finally, I would like to thank the anonymous referees for valuable comments that allowed to improve the note. The author is partially supported by National Science Centre Poland Grant 2014/15/N/ST1/02102.

Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality, we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.

Keywords: orbifold Miyaoka-Yau inequality, Hirzebruch-type inequalities, curve configurations.
Mathematics Subject Classification: 52C35, 14C20, 32S22.
Citation: Piotr Pokora. The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities. Electronic Research Announcements, 2017, 24: 21-27. doi: 10.3934/era.2017.24.003
References:
[1]

G. Barthel, F. Hirzebruch and Th. Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4, Vieweg, Braunschweig, 1987. doi: 10.1007/978-3-322-92886-3.  Google Scholar

[2]

R. Bojanowski, Zastosowania Uogólnionej Nierówno÷ci Bogomolova-Miyaoka-Yau, Master Thesis (in Polish), 2003. Available from: http://www.mimuw.edu.pl/%7Ealan/postscript/bojanowski.ps. Google Scholar

[3]

E. Brieskorn and H. Knörrer, Ebene Algebraische Kurven, Birkhäuser Verlag, Basel u. a., 1981.  Google Scholar

[4]

P. Cassou-Noguè and A. Płoski, Invariants of plane curve singularities and Newton diagrams, Univ. Iagel. Acta Math., 49 (2011), 9-34.   Google Scholar

[5]

F. Hirzebruch, Arrangements of lines and algebraic surfaces, in Arithmetic and Geometry, Vol. II, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983,113-140.  Google Scholar

[6]

F. Hirzebruch, Singularities of algebraic surfaces and characteristic numbers, in The Lefschetz Centennial Conference, Part I (Mexico City, 1984), Contemp. Math., 58, Amer. Math. Soc., Providence, RI, 1986,141-155. doi: 10.1090/conm/058.1/860410.  Google Scholar

[7]

V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, The Mathematical Association of America, 1991.  Google Scholar

[8]

A. Langer, Logarithmic orbifold Euler numbers with applications, Proc. London Math. Soc., 86 (2003), 358-396.  doi: 10.1112/S0024611502013874.  Google Scholar

[9]

Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math., 42 (1977), 225-237.  doi: 10.1007/BF01389789.  Google Scholar

[10]

P. Pokora, X. Roulleau and T. Szemberg, Bounded negativity, Harbourne constants and transversal arrangements of curves, to appear in Ann. Inst. Fourier Grenoble, arXiv: 1602.02379. Google Scholar

[11]

H. Schenck and S. Tohaneanu, Freeness of Conic-Line arrangements in ℙ2, Commentarii Mathematici Helvetici, 84 (2009), 235-258.  doi: 10.4171/CMH/161.  Google Scholar

[12]

L. Tang, Algebraic surfaces associated to arrangements of conics, Soochow Journal of Mathematics, 21 (1995), 427-440.   Google Scholar

[13]

Z. Han, A note on the weak Dirac conjecture, The Electronic Journal of Combinatorics, 24 (2017). Available from: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p63. Google Scholar

show all references

References:
[1]

G. Barthel, F. Hirzebruch and Th. Höfer, Geradenkonfigurationen und Algebraische Flächen, Aspects of Mathematics, D4, Vieweg, Braunschweig, 1987. doi: 10.1007/978-3-322-92886-3.  Google Scholar

[2]

R. Bojanowski, Zastosowania Uogólnionej Nierówno÷ci Bogomolova-Miyaoka-Yau, Master Thesis (in Polish), 2003. Available from: http://www.mimuw.edu.pl/%7Ealan/postscript/bojanowski.ps. Google Scholar

[3]

E. Brieskorn and H. Knörrer, Ebene Algebraische Kurven, Birkhäuser Verlag, Basel u. a., 1981.  Google Scholar

[4]

P. Cassou-Noguè and A. Płoski, Invariants of plane curve singularities and Newton diagrams, Univ. Iagel. Acta Math., 49 (2011), 9-34.   Google Scholar

[5]

F. Hirzebruch, Arrangements of lines and algebraic surfaces, in Arithmetic and Geometry, Vol. II, Progr. Math., 36, Birkhäuser, Boston, Mass., 1983,113-140.  Google Scholar

[6]

F. Hirzebruch, Singularities of algebraic surfaces and characteristic numbers, in The Lefschetz Centennial Conference, Part I (Mexico City, 1984), Contemp. Math., 58, Amer. Math. Soc., Providence, RI, 1986,141-155. doi: 10.1090/conm/058.1/860410.  Google Scholar

[7]

V. Klee and S. Wagon, Old and New Unsolved Problems in Plane Geometry and Number Theory, The Mathematical Association of America, 1991.  Google Scholar

[8]

A. Langer, Logarithmic orbifold Euler numbers with applications, Proc. London Math. Soc., 86 (2003), 358-396.  doi: 10.1112/S0024611502013874.  Google Scholar

[9]

Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math., 42 (1977), 225-237.  doi: 10.1007/BF01389789.  Google Scholar

[10]

P. Pokora, X. Roulleau and T. Szemberg, Bounded negativity, Harbourne constants and transversal arrangements of curves, to appear in Ann. Inst. Fourier Grenoble, arXiv: 1602.02379. Google Scholar

[11]

H. Schenck and S. Tohaneanu, Freeness of Conic-Line arrangements in ℙ2, Commentarii Mathematici Helvetici, 84 (2009), 235-258.  doi: 10.4171/CMH/161.  Google Scholar

[12]

L. Tang, Algebraic surfaces associated to arrangements of conics, Soochow Journal of Mathematics, 21 (1995), 427-440.   Google Scholar

[13]

Z. Han, A note on the weak Dirac conjecture, The Electronic Journal of Combinatorics, 24 (2017). Available from: http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p63. Google Scholar

[1]

Marek Janasz, Piotr Pokora. On Seshadri constants and point-curve configurations. Electronic Research Archive, 2020, 28 (2) : 795-805. doi: 10.3934/era.2020040
[2]

James Scott, Tadele Mengesha. A fractional Korn-type inequality. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3315-3343. doi: 10.3934/dcds.2019137
[3]

Barbara Brandolini, Francesco Chiacchio, Cristina Trombetti. Hardy type inequalities and Gaussian measure. Communications on Pure & Applied Analysis, 2007, 6 (2) : 411-428. doi: 10.3934/cpaa.2007.6.411
[4]

Friedemann Brock, Francesco Chiacchio, Giuseppina di Blasio. Optimal Szegö-Weinberger type inequalities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 367-383. doi: 10.3934/cpaa.2016.15.367
[5]

Juan Luis Vázquez, Nikolaos B. Zographopoulos. Hardy type inequalities and hidden energies. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5457-5491. doi: 10.3934/dcds.2013.33.5457
[6]

Shu-Yu Hsu. Some results for the Perelman LYH-type inequality. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3535-3554. doi: 10.3934/dcds.2014.34.3535
[7]

Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad, Saed F. Mallak, Hussam Alrabaiah. Lyapunov type inequality in the frame of generalized Caputo derivatives. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2335-2355. doi: 10.3934/dcdss.2020212
[8]

Angelo Alvino, Roberta Volpicelli, Bruno Volzone. A remark on Hardy type inequalities with remainder terms. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 801-807. doi: 10.3934/dcdss.2011.4.801
[9]

Xiaobao Zhu. Remarks on singular trudinger-moser type inequalities. Communications on Pure & Applied Analysis, 2020, 19 (1) : 103-112. doi: 10.3934/cpaa.2020006
[10]

Rong Hu, Ya-Ping Fang, Nan-Jing Huang. Levitin-Polyak well-posedness for variational inequalities and for optimization problems with variational inequality constraints. Journal of Industrial & Management Optimization, 2010, 6 (3) : 465-481. doi: 10.3934/jimo.2010.6.465
[11]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[12]

Jun Zhou, Jun Shen, Weinian Zhang. A powered Gronwall-type inequality and applications to stochastic differential equations. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7207-7234. doi: 10.3934/dcds.2016114
[13]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320
[14]

Giovanni Cupini, Eugenio Vecchi. Faber-Krahn and Lieb-type inequalities for the composite membrane problem. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2679-2691. doi: 10.3934/cpaa.2019119
[15]

Andrea Cianchi, Adele Ferone. Improving sharp Sobolev type inequalities by optimal remainder gradient norms. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1363-1386. doi: 10.3934/cpaa.2012.11.1363
[16]

Kunquan Lan, Wei Lin. Lyapunov type inequalities for Hammerstein integral equations and applications to population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1943-1960. doi: 10.3934/dcdsb.2018256
[17]

Bernhard Kawohl. Symmetry results for functions yielding best constants in Sobolev-type inequalities. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 683-690. doi: 10.3934/dcds.2000.6.683
[18]

István Győri, László Horváth. Sharp estimation for the solutions of delay differential and Halanay type inequalities. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3211-3242. doi: 10.3934/dcds.2017137
[19]

Xiaofei He, X. H. Tang. Lyapunov-type inequalities for even order differential equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 465-473. doi: 10.3934/cpaa.2012.11.465
[20]

Simona Fornaro, Maria Sosio, Vincenzo Vespri. Harnack type inequalities for some doubly nonlinear singular parabolic equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5909-5926. doi: 10.3934/dcds.2015.35.5909

2020 Impact Factor: 0.929

Tools

Metrics

  • PDF downloads (193)
  • HTML views (1567)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy