Complex Ball Quotients and Line Arrangements in the Projective Plane

Fermat-type configurations of lines in P3 and the containment problem

“…In the book of Barthel, Hirzebruch and Höfer they are called Ceva arrangements, see [1, Section 2.3.I]. The same terminology is kept in the recent book by Tretkoff [20,Chapter V,5.2]. Both these beautiful books are focused on surfaces of general type arising as ball quotients.…”

Bounded negativity and Harbourne constants on ruled surfaces

“…In other words, they are minimal smooth complex projective surfaces Y such that K Y is nef and big and K 2 Y = 3e(Y ), where K Y denotes the canonical divisor and e(Y ) is the topological Euler characteristic. See [24] for more details on ball quotients.…”

The arithmetic and topology of differential equations

“…Hirzebruch made an intensive study of the second class [1], deriving in particular from the Miyaoka-Yau inequality an inequality concerning the combinatorics of line arrangements in the plane that was stronger than anything that had been obtained by elementary methods and was later applied to prove the so-called "bounded negativity conjecture" about such line configurations. There are many links between the three classes, as one can read in detail in [2], in the book [22] of which it is a review, and in the book [67] that, as mentioned in the introduction, was originally an outgrowth of a course that Hirzebruch gave on the subject in 1996. Of particular interest in connection with the present article are the discussions of the classical hypergeometric differential equations in Chapter 2 and of the Appell hypergeometric functions and their associated monodromy groups in Chapter 7 of [67].…”

“…There are many links between the three classes, as one can read in detail in [2], in the book [22] of which it is a review, and in the book [67] that, as mentioned in the introduction, was originally an outgrowth of a course that Hirzebruch gave on the subject in 1996. Of particular interest in connection with the present article are the discussions of the classical hypergeometric differential equations in Chapter 2 and of the Appell hypergeometric functions and their associated monodromy groups in Chapter 7 of [67]. I say no more here except that the whole field is still very active, a very recent example being a new construction by Martin Deraux [23] of non-arithmetic lattices via coverings of line arrangements.…”

“…Three such topics that will appear are: his resolution of the cusp singularities of Hilbert modular surfaces, which serve to explain a puzzling integrality property of a differential equation connected with a certain non-arithmetic modular curve (in Section 4); his theory of multiplicative characteristic classes and genera, which are needed to define the Gamma Class and formulate the Gamma Conjecture in the theory of mirror symmetry (in Section 9); and, very briefly in Section 10, the study of ramified coverings of the projective plane and surfaces with c 2 1 = 3c 2 , which are related via his proportionality principle to quotients of the complex 2-ball but also to the monodromy groups of certain higher-dimensional hypergeometric differential equations. The last topic was of particular interest to Hirzebruch in his later years, and together with Paula Beazley Cohen (later Tretkoff) he wrote a long review of the monograph of Deligne and Mostow on the subject [2] and also originally planned a joint book with her based on a course that he gave in Zürich in 1996, though he eventually abandoned the project and asked her to complete it alone [67].…”