Quadratic functions and smoothing surface singularities
This paper is, in the reviewers opinion, a major contribution to singularity theory. Based on the remarkable paper by \textit{V. V. Nikulin} [Math. USSR Izv. 14, 103-167 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 111- 177 (1979; Zbl 0408.10011)] the authors attack the problem of characterizing the smoothing components and the intersection forms of the corresponding Milnor fiber, for isolated complex-analytic surface singularities. Let (X,0) be any such singularity, i.e. a Stein space of dimension \(2\) with a smooth \((C^{\infty})\) boundary \(\partial X=L\), and a unique singular point 0. L is usually called the link of the singularity. Let \(\pi: \tilde X\to X\) be a good resolution, i.e. one with an exceptional curve \(E=\pi^{-1}(0)\) consisting of nonsingular irreducible components intersecting transversally. Consider a smoothing \(f: {\mathcal X}\to \Delta,\) where \(\Delta\) is an open disc in \({\mathbb{C}}\) containing 0. Here \({\mathcal X}\) is a Stein space with a partial \(C^{\infty}\)-boundary \(\partial {\mathcal X}\). There is an isomorphism of analytic spaces with boundary \(i: f^{- 1}(0)\overset \sim \rightarrow X\) and \(f| int({\mathcal X}-\{0\})\) and \(f| \partial {\mathcal X}\) are both submersions. The Milnor fiber M, i.e. the generic fiber of f, has a boundary \(\partial M\) which is diffeomorphic to L. Therefore the 3-dimensional manifold L bounds both of the two oriented 4- dimensional manifolds \(\tilde X\) and M. The main result of this paper, (3.7), shows that the classical linking pairing of the link L, \(b: H_ 1(L)_ t\times H_ 1(L)_ t\to Q/Z\) where the subscript t denotes the torsion part, is in an obvious sense the link between the intersection pairings of \(H_ 2(\tilde X)\) and \(H_ 2(M)\). In fact, the complex structures of \(\tilde X\) and M induces an, up to homotopy, unique complex structure on the sum of the trivial bundle R and the tangent bundle \(\tau_ L\) of L. Using the first Chern class of \(-\tau_ L\oplus R\) the authors construct a quadratic function \(q: H_ 1(L)_ t\to Q/Z\) such that the corresponding bilinear form \(b(x,y)=-(q(x+y)-q(x)-q(y))\) is the linking pairing. Now, let \(K_{\tilde X}:=-c_ 1(\tau_{\tilde X})\in H^ 2(\tilde X)=Hom(H_ 2(\tilde X),Z)\) then the bilinear form of the quadratic function \(Q_{\tilde X}: H_ 2(\tilde X)\to Z\) defined by \(Q_{\tilde X}(x)=(x\cdot x+K_{\tilde X}(x)),\) is the intersection form of \(H_ 2(\tilde X)\). Notice that since this form is negative definite, there is an element \(k\in H_ 2(\tilde X):=\{x\in H_ 2(\tilde X)\oplus Q| \forall y\in H_ 2(\tilde X), x\cdot y\in Z\}\) such that \(K_{\tilde X}(x)=k\cdot X\) for all \(x\in H_ 2(\tilde X).\) To \(Q_{\tilde X}\) there is associated a discriminant quadratic function (DQF) \(q_{\tilde X}: H_ 2(\tilde X)/H_ 2(\tilde X)\to Q/Z\) and there is a natural isomorphism \(H_ 2(\tilde X)/H_ 2(\tilde X)\simeq H_ 1(L)_ t\) identifying \(q_{\tilde X}\) and q. Moreover, see (3.7) and (4.5), let \(K_ M\in Hom(H_ 2(M),Z)\) be the image of \(-c_ 1(\tau_ M)\), then the corresponding bilinear form of the quadratic function \(Q_ M: H_ 2(M)\to Z\) defined by \(Q_ M(x)=(x\cdot x+K_ M(x))\) is the intersection form of \(H_ 2(M)\), and the associated DQF is isomorphic to the quadratic function \(q_ I: I^{\perp}/I\to Q/Z\) induced by q, where \(I:=im(H_ 2(M,L)_ t\to H_ 1(L)_ t)\subseteq H_ 1(L)_ t\) is q-isotropic. Specializing to Gorenstein singularities, the authors observe, see (4.8), that \(k\in H_ 2(\tilde X)\) and that q is a quadratic form. Moreover, \(K_ M=0\) implying that \(\bar H{}_ 2(M):=H_ 2(M)\) modulo torsion and the radical of the intersection form, is an even lattice with DQF canonically isomorphic to \(q_ I.\) For Gorenstein singularities the formulas of \textit{J. H. M. Steenbrink} [in Singularities, Summer Inst., Arcata/Calif. 1981, Proc. Symp. Pure Math. 40, Part 2, 513-536 (1983; Zbl 0515.14003)] express the Sylvester invariants \((\mu_ 0,\mu_+,\mu_-)\) of the intersection form of \(H_ 2(M)\) as linear functions in the Betti number \(b_ 1(\tilde X)\), the genus p(X) and \(k\cdot k.\) Consider now a smoothing component of the base space of a versal deformation. Any two smoothings contained in the same component will determine the same q-isotropic subgroup \(I\subseteq H_ 1(L)_ t\) and the same quadratic function \(q_ I\). Therefore \((\mu_ 0,\mu_+,\mu_-)\), I and \(q_ I\) are invariants of the smoothing component. The authors refine these invariants to what they call a smoothing datum. The last part of the paper is concerned with the problem of characterizing the smoothing components, together with the intersection forms of the corresponding Milnor fibers, studying the subset of ''permissible'', see {\S} 6, smoothing data, and applying the classification of Nikulin, loc. cit. The discussion is confined to the minimally elliptic singularities, i.e. Gorenstein and \(p(X)=1\). For simple elliptic and triangle singularities this analyses gives a rather complete answer to the problems posed.
isolated complex- analytic surface singularities
scientific article; zbMATH DE number 3996104