Ran, Ziv On the local geometry of moduli spaces of locally free sheaves. (English) Zbl 0884.14005 Maruyama, Masaki (ed.), Moduli of vector bundles. Papers of the 35th Taniguchi symposium, Sanda, Japan, and a symposium held in Kyoto, Japan, 1994. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 179, 213-219 (1996). Let \(X\) be a paracompact locally-ringed space over a field \(C\) of sufficiently large or zero characteristic. Let \(E\) be a simple locally free sheaf over \(X\). The author gives a canonical construction for the universal \(m\)-th order deformation space of \(E\) and for the Poincaré bundle over it. The construction involves the Jacobi complexes \(J^\bullet_m(sl (E))\) whose definition is briefly recalled. The above deformation space turns out to be \(\text{Spec}(C\oplus H^0(J^\bullet_m(sl (E)))\). The proof uses the Kodaira-Spencer bi-complexes associated to a deformation of \(E\) and it is only sketched. The author claims that the same construction can be applied to deformation of manifolds, yielding a simplification of the author’s preprint “Canonical infinitesimal deformations”. An interesting application, stated in full generality, is the closedness of the trace forms, that are \(H^2({\mathcal O}_X)\)-valued \(2\)-forms on a germ parametrizing a deformation of a simple locally free sheaf over \(X\). A striking consequence of this closedness theorem is a unified proof of the following two known results: (i) the existence of a symplectic structure on the moduli space of bundles over a compact complex surface \(S\) with \(K_S={\mathcal O}_S\) [S. Mukai, Invent. Math. 77, 101-116 (1984; Zbl 0565.14002)], (ii) the existence of a symplectic structure on the moduli space of local systems over a compact Riemann surface [N. J. Hitchin, Common trends in mathematics and quantum field theories, Kyoto and Tokyo 1990, Prog. Theor. Phys., Suppl. 102, 159-174 (1990; Zbl 0793.53033)].For the entire collection see [Zbl 0842.00034]. Reviewer: G.Ottaviani (L’Aquila) Cited in 2 Documents MSC: 14D20 Algebraic moduli problems, moduli of vector bundles 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14D15 Formal methods and deformations in algebraic geometry 14J10 Families, moduli, classification: algebraic theory 14H55 Riemann surfaces; Weierstrass points; gap sequences 32J15 Compact complex surfaces 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:deformation space; Poincaré bundle; deformation of a simple locally free sheaf; symplectic structure on the moduli space bundles; compact complex surface; Riemann surface Citations:Zbl 0565.14002; Zbl 0793.53033 PDFBibTeX XMLCite \textit{Z. Ran}, Lect. Notes Pure Appl. Math. 179, 213--219 (1996; Zbl 0884.14005)