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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].

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.)
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