The order of finite algebraic fundamental groups of surfaces with K^2<=3\chi-2
We study the structure of the algebraic fundamental group for minimal surfaces of general type S satisfying K_S^2<=3\chi-2$ and not having any irregular etale cover. We show that, if K_S^2<=3\chi-2, then then the algebraic fundamental group of S has order at most 5, and equality only occurs if S is a Godeaux surface. We also show that if K_S^2<= 3\chi-3 and the algebraic fundamental group of S is not trivial, then it is Z_2, or Z_2^2 or Z_3. Furthermore in this last case one has: 2<=\chi<=4, K^2=3\chi-3 and these possibilities do occur.
FACULTAD DE MATEMÁTICAS ON THE GEOGRAPHY OF SURFACES OF GENERAL TYPE WITH FIXED FUNDAMENTAL GROUP POR
In this thesis, we study the geography of complex surfaces of general type with respect to the topological fundamental group. The understanding of this general problem can be coarsely divided into geography of simply-connected surfaces and geography of non-simply-connected surfaces. The geography of simply-connected surfaces was intensively studied in the eighties and nineties by Persson, Chen, and Xiao among others. Due to their works, we know that the set of Chern slopes c1/c2 of simply-connected surfaces of general type is dense in the interval [ 5 , 2]. The last result which closes the density problem for this type of surfaces happened in 2015. Roulleau and Urzúa showed the density of the Chern slopes in the interval [1, 3]. This completes the study since accumulation points of c1/c2 belong to the interval [ 1 5 , 3] by the Noether’s inequality and the Bogomolov-Miyaoka-Yau inequality for complex surfaces. The geography of non-simply-connected surfaces is well understood only for small Chern slopes. Indeed, because of works of Mendes, Pardini, Reid, and Xiao, we know that for c1/c2 ∈ [15 , 1 3 ] the fundamental group is either finite with at most nine elements, or the fundamental (algebraic) group is commensurable with the fundamental (algebraic) group of a curve. Furthermore, a well-known conjecture of Reid states that for minimal surfaces of general type with c1/c2 < 1 2 the topological fundamental group is either finite or it is commensurable with the fundamental group of a curve. Due to Severi-Pardini’s inequality and a theorem of Xiao, Reid’s conjecture is true, at least in the algebraic sense for irregular surfaces or surfaces having an irregular étale cover. Keum showed with an example in his doctoral thesis that Reid’s conjecture cannot be extended over 1 2 . For higher slopes essentially there are no general results. In this thesis, we prove that for any topological fundamental group G of a given non-singular complex projective surface, the Chern slopes c1(S)/c2(S) of minimal non-singular projective surfaces of general type S with π1(S) ' G are dense in the interval [1, 3]. It remains open the question for non-simplyconnected surfaces in the interval [ 2 , 1].
Complex analysis and algebraic geometry : a volume in memory of Michael Schneider
Lucian Badescu, Mauro C. Beltrametti, Paltin Ionescu, almost-lines and quasi-lines on projective manifolds Daniel Barlet, Jon Magnusson, Transfert de metrique Wolf P. Barth, on the classification of K3 surfaces with nine cusps Arnaud Beauville, complex manifolds with split tangent bundle Mauro C. Beltrametti, Alan Howard, Michael Schneider, Andrew J. Sommese, projections from subvarieties Indranil Biswas, Georg Schumacher, generalized Petersson-Weil metric on the Douady space of embedded manifolds Fabrizio Catanese, Roberto Pignatelli, on simply connected Godeaux surfaces Ciro Ciliberto, Angelo Felice Lopez, Rick Miranda, on the Wahl map of plane nodal curves Lawrence Ein, Bo Ilic, Robert Lazarsfeld, a remark on projective embeddings of varieties with non-negative cotangent bundles David Garber, Mina Teicher, the fundamental group's structure of the complement of some configurations of real line arrangements Peter Heinzner, Alan T. Huckleberry, Kahlerian structures on symplectic reductions Klaus Hulek, Nef divisors on moduli spaces of Abelian varieties Klaus Hulek, Kristian Ranestad, Abelian surfaces with two plane cubic curve fibrations and Calabi-Yau threefolds Janos Kollar, real algebraic threefolds IV. Del Pezzo fibrations Christian Okonek, Andrei Teleman, Seiberg-Witten invariants for 4-manifolds with b+ = 0 Jeroen Spandaw, a geometric proof of Ax' Theorem Sheng-Li Tan, Eckart Viehweg, a note on Cayley-Bacharach property for vector bundles Thomas Peternell, the scientific work of Michael Schneider Ulf Persson, Michael Schneider - an alpine vita.