Manchester Geometry Seminar 2000/2001

In late 1970's Miyaoka [8] and Yau [12] proved inequality between Chern numbers of surfaces of general type. This inequality was later generalized by many people including Sakai [9], Kobayashi [2], Wahl [11], Tian [10], Miyaoka-Lu [3] and Megyesi [7].

The talk will be devoted to an inequality generalizing all the previous inequalities and extending them to much larger class of surfaces with Q-divisors.

The talk will begin with studying Chern numbers of reflexive sheaves on singular algebraic surfaces (see [4]). Using them one can define logaritmic orbifold Euler numbers (see [6]). Then one can give the promised inequality between such numbers.

Finally I will mention a few applications of the obtained inequality (see [6]) to singularities of plane curves, line arrangements and effective versions of Bogomolov's results [1] bounding curves in surfaces of general type.

References :

1. F. A. Bogomolov, Families of curves on a surface of general type, Soviet Math. Dokl. 18 (1977), 1294-1297
2. R. Kobayashi, Uniformization of complex surfaces, Adv. Stud. Pure Math. 18, (1990) 313--394
3. S. Lu, Y. Miyaoka, Bounding curves in algebraic surfaces by genus and Chern numbers, Math. Research Letters 2 (1995), 663--676
4. A. Langer, Chern classes of reflexive sheaves on normal surfaces, to appear in Math. Z., electronic version: DOI 10.1007/s002090000149
5. A.Langer, The Bogomolov-Miyaoka-Yau inequality for log canonical surfaces, preprint (1999), electronic version available at http://www.uni-math.gwdg.de/jarekw/EAGER/alan2.ps,
6. A.Langer, Logaritmic orbifold Euler numbers of surfaces with applications, in preparation
7. G. Megyesi, Generalisation of the Bogomolov--Miyaoka--Yau inequality to singular surfaces, Proc. London Math. Soc. (3) 78, (1999) 241--282
8. Y. Miyaoka, On the Chern numbers of surfaces of general type, Invent. Math. 42 (1977), 225--237
9. F. Sakai, Semi-stable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980), 89--120
10. G. Tian, K\"ahler--Einstein metrics on algebraic manifolds, in Transcendental Methods in Algebraic Geometry, Lecture Notes in Math. 1646 (1996), 143--185
11. J.Wahl, Miyaoka--Yau inequality for normal surfaces and local analogues, Contemporary Mathematics 162 (1994), 381--402
12. S.-T. Yau, On Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Sci. USA 74 (1977), 1798-1800