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I am now interested in shifted symplectic structures.

I found Zhang's results about symplectic structures (2011, p.3-4, arXiv link, Comm. Anal. Geom. 2017) and Pantev–Toen–Vaquié–Vezzosi's results on shifted symplectic structures (2011, see p.1-2, arXiv link, Publ. IHES 2013).

I thought Zhang's results were completely included in the results of Pantev–Toen–Vaquié–Vezzosi. Is this idea correct? (About the reason of asking the question, see my comment below)

Any comment welcome! Thank you.

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    $\begingroup$ "I thought were completely included": you mean "Zhang's results were completely included"? Zhang's paper was posted on arXiv 2 weeks after the other one. Possibly the comparison is relevant but I'm not sure this is the right place to compare papers and discuss their merits. $\endgroup$
    – YCor
    May 8, 2020 at 11:52
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    $\begingroup$ That's right, I'm sorry. The reason I posted was that I first found Zhang's results and thought that similar results would be worth doing for moduli stacks of other kinds of sheaves (torsion-free sheaves and more generally coherent sheaves). I found the result of Pantev–Toen–Vaquié–Vezzosi and thought it was obvious to them. $\endgroup$ May 8, 2020 at 12:18

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First of all, let me say that it is not always easy to recover non-derived results from derived ones.

Second of all, it seems to me that Zhang's definition of a symplectic stack may not be accurate, in the sense that it doesn't generalize the notion of a smooth symplectic algebraic variety (or, homomophic symplectic manifold). Indeed, their definition only requires to have a non necessarily closed non-degenerate 2-form. This is manifest in Example 6.7 of Zhang's paper.

Now, can one deduce the result of Zhang from the one of Pantev-Toën-Vaquié-Vezzosi (PTVV, for short)? I do think so.

Indeed, as a special case of PTVV's general existence result of shifted symplectic structures on derived mapping stacks, one gets a 0-shifted symplectic structure on the derived moduli stack of perfect sheaves on a K3 surface. PTVV explain in their paper how one can recover the usual symplectic structure on the coarse moduli of simple sheaves (this is in subsection 3.1). The comparison with Zhang's non-degenerate pairing on the tangent complex of semi-stable sheaves on a K3 shall be very similar.

More precisely, in PTVV, if one forgets about the closedness of the 2-form (which is really the hardest part of the story) and only cares about the induced pairing on the tangent complex, it is just the standard construction (also used in Zhang's paper) with the Atiyah class and Serre duality. Then for the comparison, one shall just compare the cotangent complexes of the derived stack from PTVV and of the underived stack from Zhang, over the semi-stable locus. They do not exactly match, but I suspect that the discrepancy between the two is something like $\mathcal O[-1]\oplus\mathcal O[1]$ (see PTVV for more details).

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  • $\begingroup$ Thank you very much for your comment ! $\endgroup$ Jan 27, 2021 at 10:03

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