Complex Surfaces of General Type: Some Recent Progress

The classification of isotrivially fibred surfaces with p g = q = 2

“…, 6. Then G = Z 2 , and since it is (0 | 2 6 )-generated, we recover the first case in Table 1, i.e., g(F) = 2 g(C) = 3 and m = (2 6 ).…”

“…If this is the case one has 1 = χ(O S ) = 1 − q + p g and it follows that p g = q. If we also assume that the surface is irregular (i.e., q > 0) then the Bogomolov-Miyaoka-Yau and Debarre inequalities, K 2 S ≤ 9, K 2 S ≥ 2 p g , imply 1 ≤ p g ≤ 4, see [6]. If p g = q = 4 we have a product of curves of genus 2, as shown by Beauville in the appendix to [21], while the case p g = q = 3 was understood through the work of several authors [18,26,32].…”

The fundamental group of a quotient of a product of curves

“…They were used successfully to address various questions (see e.g. the survey paper [6]) and, in particular, to obtain substantial information about various moduli spaces of surfaces of general type (see e.g. [3][4][5]).…”

Generic vanishing index and the birationality of the bicanonical map of irregular varieties

“…In fact, the surfaces with irregularity q(S) ≤ 1 and χ(S, ω S ) = 1 are not completely understood and there is no classification about which ones have birational ϕ 2 . For a modern review of the state of the art in the surface case, we refer to [BCP,Thm. 8].…”