Geodesic
Coherent sheaves, Chern classes, and superconnections
Izvestiya. Mathematics , Tome 87 (2023) no. 3, pp. 439-468

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A twist-closed enhancement of the bounded derived category $\mathcal{D}^b_{\mathrm{coh}} (X)$ of complexes of $\mathcal{O}_X$-modules with coherent cohomology is constructed by means of the DG-category of $\overline\partial$-superconnections. The machinery of $\overline\partial$-superconnections is applied to define Chern classes and Bott–Chern classes of objects in the category, in particular, of coherent sheaves.
Keywords: coherent sheaves, derived category, DG-category, Dolbeault operator, superconnection. 
@article{IM2_2023_87_3_a2,
     author = {A. I. Bondal and A. A. Roslyi},
     title = {Coherent sheaves, {Chern} classes, and superconnections},
     journal = {Izvestiya. Mathematics },
     pages = {439--468},
     publisher = {mathdoc},
     volume = {87},
     number = {3},
     year = {2023},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2023_87_3_a2/}
}
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A. I. Bondal; A. A. Roslyi. Coherent sheaves, Chern classes, and superconnections. Izvestiya. Mathematics , Tome 87 (2023) no. 3, pp. 439-468. http://geodesic.mathdoc.fr/item/IM2_2023_87_3_a2/