Holomorphy and semiholomorphy (Q5960206)

The aim of this note is to review some of the recent advances in Gauge theory, Donaldson theory and Calabi--Yau geometry [cf.: \textit{S. K. Donaldson} and \textit{R. Thomas}, Gauge theory in higher dimensions; The geometric universe: Science, geometry, and the work of Roger Penrose (Oxford: University Press), 31--47 (1998; Zbl 0926.58003)], and to propose to use a ``semiholomorphic analogue'' (i.e., replacing holomorphic connections by ``semiholomorphic connections'') to obtain results like the Hodge conjecture for products of \(K3\) surfaces. The setup and language is close to the one of physicist's which makes the paper sometimes hard to follow. Comparing the results and methods of the paper with classical calculations and approaches [cf. \textit{S. Mukai}, in: Vector bundles on algebraic varieties, Pap. Colloq., Bombay 1984, Stud. Math., Tata Inst. Fundam. Res. 11, 341--413 (1987; Zbl 0674.14023)], it seems that it is not necessary to deviate from holomorphic connections, since there are stronger results obtained without the need to exploit semiholomorphic connections.