Differential topology of complex surfaces. Elliptic surfaces with \(p_ g=1\): smooth classification. With the collaboration of Millie Miss (Q689071)

This lecture notes show that two diffeomorphic elliptic surfaces \(S\) and \(S'\) with \(q=0\) and \(p_ g=1\) are deformations from one another (a result also due to \textit{S. Bauer} in the simply connected case). In particular \(S'\) is \(K3\) if so is \(S\). Known arguments reduce the problem to the case where there are at most two multiple fibers of multiplicities \((m_ 1,m_ 2)\) on each. When \(S\) is minimal, this pair is a deformation invariant. It is shown here that it is also a diffeomorphism invariant. A second main result is that any diffeomorphism between such minimal surfaces preserves the canonical class up to sign and modulo torsion. -- The pair \((m_ 1,m_ 2)\) is obtained from the computation of the two coefficients \(c_ 0=15\) \(m_ 1m_ 2\) and \(c_ 1=15\) \(m_ 1m_ 2\) \((2(m_ 1m_ 2)^ 2-(m^ 2_ 1+ m^ 2_ 2))\) of the Donaldson polynomial \(\gamma_ 3(S)=\sum^ 3_{i=0}c_ i q_ s^{3- i} k_ s^{2i} \), where \(q_ S\) is the intersection form and \(k_ S={c_ 1(S) \over m_ 1m_ 2}\). This computation is done in several steps, each of which is very involved and technical. A brief description is this (simplifying somehow): 1. Stable algebro-geometric analogues \(\delta_ c (S,H)\) of the Donaldson polynomials \(\gamma_ c(s)\) are introduced. By a blowing-up procedure, unstable polynomials \(\delta_ c(S,H)\) are defined for any \(c>0\) (and in particular \(c=3\), which is in the unstable range) (\S2). 2. For any pair \((m_ 1,m_ 2)\), a family of regular elliptic surfaces with \(p_ g=1\) and given pair of multiplicities is constructed, by means of relative logarithmic transformations, from an appropriate family of elliptic \(K3\)-surfaces with a section. For the generic member \(S\) of this family and sufficiently generic polarisation, one has: \(\gamma_ 3(S) =\delta_ 3(S,H)\) (\S3). 3. The remaining chapters are devoted to the computation of \(\delta_ 3(S,H)\), which is written (after restriction to \(H^ +_ 2(S,\mathbb{Q}))\) in \S6 as a sum of 3 more manageable terms, two of which corresponding to boundary or degenerate components of the moduli space of sheaves involved.