Geometry of the Donaldson-Friedman Pushout: Twistor degenerations and instanton charge

Abstract

We study the Donaldson-Friedman semistable twistor degeneration by combining the scheme-theoretic geometry of its Ferrand pushout with the logarithmic topology of its Kato-Nakayama realisation. For the central fibre $Z_0={\tilde{Z}}_1{\cup }_Q{\tilde{Z}}_2,$ the Ferrand description yields an explicit equaliser presentation of the operational Chow ring and a componentwise specialisation formula, with sharp restrictions on surfaces that glue across the exceptional quadric. The logarithmic structure of the same normal-crossing fibre retains data not visible in ordinary intersection theory: after fixing the phase of the smoothing parameter, the Kato-Nakayama space over $Q$ is the unit circle bundle of the normal line bundle, and over a ruling fibre its anti-diagonal quotient is diffeomorphic to ${\mathbb{RP}}^3$. Restriction to curves in $Q$ gives a log-topological refinement of the algebraic intersection data. For bundles $E_0$ on $Z_0$ obtained by gluing Ward or Hartshorne-Serre data from the two components, we prove additivity of the second Chern cycle $c_2(E_0)\cap [Z_0]$. If $E_0$ extends over the semistable smoothing, its polarised charge is the sum of the component charges; under the usual reality and triviality conditions of the Ward correspondence, the bundle on a smooth fibre determines an anti-self-dual $SU(2)$-instanton on the connected sum of the underlying four-manifolds.

24 pages, 1 figure