Comparison of the Donaldson polynomial invariants with their algebro- geometric analogues
When \(M\) is a closed simply connected smooth 4-manifold with \(b_ 2^ + > 1\) and odd, and \(\beta\) is an orientation for \(H^ 2_ +(M,\mathbb{R})\), Donaldson has defined polynomial invariants \(\gamma_ c(M,\beta)\). These are diffeomorphism type invariants and provide a major technique for understanding differential structures on 4-manifolds; much current research is devoted to their structure and computation. Their definition uses the Uhlenbeck compactification \(\overline{\mathcal M}(P_ c,g)\) of the moduli space \({\mathcal M}(P_ c,g)\) of \(g\)-ASD connections of a principal \(SU(2)\)-bundle \(P_ c\) over \(M\) with \(c_ 2(P_ c) = c\) and a map \(\mu: H_ 2(M) \to H^ 2(\overline{\mathcal M}(P_ c,g))\). For \(S\) a simply connected algebraic surface with \(p_ g(S) > 0\) with polarization \(\mathcal L\), \textit{K. G. O'Grady} [Invent. Math. 107, No. 2, 351-395 (1992; Zbl 0769.14008)] has defined analogues \(\delta_ c(S,{\mathcal L})\) of these polynomials using the Gieseker compactification \(\overline{\mathcal M}^ G_ c(S,{\mathcal L})\) of the moduli space of \(\mathcal L\)-slope stable vector bundles within the Gieseker moduli space \(\widehat{\mathcal M}^ G_ c(S,{\mathcal L})\) and a map \(\nu: H_ 2(S) \to H^ 2(\widehat{\mathcal M}^ G_ c(S,{\mathcal L}))\) similar to \(\mu\). Since the algebro-geometric polynomials \(\delta_ c(S,{\mathcal L})\) depend on the complex structure and the polarization, it is unclear whether they are in fact diffeomorphism type invariants. However, their definition allows their computation via algebraic geometric techniques, which have provided the most powerful computational techniques in the gauge theory of 4-manifolds until fairly recently. \textit{S. K. Donaldson} [Proc. Lond. Math. Soc., III. Ser. 50, 1- 26 (1985; Zbl 0547.53019)] had shown there is a homeomorphism between the moduli space of \(\mathcal L\)-slope stable holomorphic vector bundles with \(c_ 1 = 0\) and \(c_ 2 = c\) and the moduli space \({\mathcal M}(P_ c,g_{\mathcal L})\), where \(g_{\mathcal L}\) is the Kähler metric induced by the polarization \(\mathcal L\). For this reason it had been conjectured that there is a close relationship between these two polynomials, and this had been verified in special cases. However, there were difficulties in establishing this since the homeomorphism did not extend to the compactifications as well as problems with the lack of genericity of the Kähler metric. The main theorem of this paper establishes the appropriate correct formulation of the relationship. Theorem 1: Let \(S\) be a simply connected algebraic surface with \(p_ g(S) > 0\). For any compact subset \(K\) in the ample cone of \(S\) there is \(c_ 0(K)\) such that for any \(c \geq c_ 0(K)\) with \(c\) odd and for any \(c\)-generic polarization \(\mathcal L\) for \(S\) whose Kähler class lies in the subcone \(\mathbb{R}^ + \cdot K\) we have \(\delta_ c(S,{\mathcal L}) = \gamma_ c(S,\beta_ S)\) as multilinear functions on \(H_ 2(S)\). The first main ingredient of the proof is to establish the existence of a continuous map from the Gieseker moduli space to the Uhlenbeck compactification extending the known homeomorphism before compactifying. Theorem 2: Let \(S\) be a simply connected algebraic surface and let \(\mathcal L\) be a polarization. Let \(\widehat{\mathcal M}^ G_ c(S,\mathcal L)\) be the moduli space of Gieseker semi-stable rank-two sheaves with \(c_ 1 = 0\) and \(c_ 2 = c > 0\). Let \(g_{\mathcal L}\) be the Kähler metric associated to \(\mathcal L\). Let \(\overline{\mathcal M}(P_ c,g_{\mathcal L})\) be the Uhlenbeck compactification of the moduli space of gauge equivalence classes of \(g_{\mathcal L}\)-ASD connections on \(P_ c\) where \(P_ c\) is a principal \(SU(2)\)-bundle over \(S\) with \(c_ 2(P_ c) = c\). Then there is a continuous map \(\Phi: {\mathcal M}^ G_ c(S,{\mathcal L}) \to \overline{\mathcal M}(P_ c,g_{\mathcal L})\) from the Gieseker moduli space to the Uhlenbeck compactification which extends the Donaldson homeomorphism between the moduli space of \(\mathcal L\)-slope stable vector bundles with \(c_ 1 = 0\) and \(c_ 2 = c\) and \({\mathcal M}(P_ c,g_{\mathcal L})\). Theorem 1 is established by proving Theorem 3: Let \(S\) be a simply connected algebraic surface with \(p_ g(S) > 0\). For any compact subset \(K\) in the ample cone of \(S\) there is \(c_ 0(K)\) such that for \(c \geq c_ 0(K)\) with \(c\) odd and for any \(c\)-generic polarization \(\mathcal L\) for \(S\) whose Kähler class lies in the subcone \(\mathbb{R}^ + \cdot K\) the following holds. There is a map \(\mu: H_ 2(S) \to \overline M(P_ c,g_{\mathcal L})\) such that for any classes \(\alpha_ 1,\dots,\alpha_{d(c)} \in H_ 2(S)\) we have \[ \gamma_ c(S,\beta_ S)(\alpha_ 1,\dots,\alpha_{d(c)}) = \langle \mu(\alpha_ 1)\cup \dots \cup \mu(\alpha_{d(c)}),[\overline {\mathcal M}(P_ c, g_{\mathcal L})]\rangle. \] Furthermore, we have \[ \Phi^*\circ\mu=\nu:H_ 2(S)\to H^ 2(\overline{\mathcal M}^ G_ c(S,{\mathcal L})). \] As a corollary of this theorem, it is shown that the canonical class \(k_ S \in H^ 2(S;\mathbb{Z})\) is a diffeomorphism invariant up to sign for \(S\) a complete intersection in \(\mathbb{C} \mathbb{P}^ N\) that is of general type with \(p_ g(S)\) odd; the case with \(p_ g(S)\) even had been established earlier. Generalizations of Theorem 1 are given to the case when \(b_ 2^ + = 1\), evaluation on the 4-dimensional class, the \(SO(3)\) case, and the nonsimply connected case.
closed simply connected smooth 4- manifold
scientific article; zbMATH DE number 488107