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Fourier-Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes. (English) Zbl 1473.14020

Given a smooth projective variety \(X\), the Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) is again a smooth projective variety of dimension \(\dim X^{[n]} = n\, \dim X\). Each vector bundle \(E\) on \(X\) defines the vector bundle \(E^{[n]} = p_{2,*}p_1^*E\). Here \(p_k\) is the projection to the \(k\)-factor from the universal subscheme \(\Pi_n \subset X \times X^{[n]}\). The bundle \(E^{[n]}\) is called the Fourier-Mukai transform of \(E\) (with respect to \(\Pi_n\)). By work of Lehn and Lehn-Sorger, these transforms are important tools to study the topology and geometry of Hilbert schemes. Conversely, they are useful to study bundles on \(X\) itself e.g. by work of Voison, Ein-Lazarsfeld and Agostini.
The present article enhances the Fourier-Mukai transform to so-called V-cotwisted Hitchin pairs \((E, \theta)\). Here \(E\) and \(V\) are vector bundles on \(X\) and \(\theta\colon E \otimes V \to E\) is a section. The outcome of the enhanced Fourier-Mukai transform are \(V^{[n]}\)-cotwisted Hitchin pairs \((E^{[n]}, \theta^{[n]})\) on \(X^{[n]}\). Note here that if \(V = T_X\), then \(V^{[n]}\cong T_{X^{[n]}}(-\log B_n)\) (by a result of Stapleton) where \(B_n\subset X^{[n]}\) is the locus of non-reduced sub-schemes of \(X\). In particular, the enhanced Fourier-Mukai transforms of Higgs bundles (i.e. \(T_X\)-cotwisted Hitchin pairs) are logarithmic Higgs bundles on \(X^{[n]}\).
After establishing basic results on the enhanced Fourier-Mukai transform, which are of independent interest, the authors prove various interesting results on the relationship between Hitchin pairs on \(X\) and their enhanced Fourier-Mukai transforms (similar results for vector bundles were already obtained by the second author), for example:
If \((E, \theta)^{[n]} \cong (F, \eta)^{[n]}\) on \(X^{[n]}\), then \((E,\theta) \cong (F, \eta)\) on \(X\) where \(X\) is any smooth projective curve of genus \(\geq 1\) or any smooth quasi-projective variety of \(\dim X \geq 2\);
relationship between the stability conditions for \((E, \theta)\) on \(X\) and \((E, \theta)^{[n]}\) on \(X^{[n]}\) for any smooth projective curve \(X\).

MSC:

14D23 Stacks and moduli problems
14D20 Algebraic moduli problems, moduli of vector bundles
14H30 Coverings of curves, fundamental group
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
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