# Two results about (shifted) symplectic structures

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.

• "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.
– YCor
May 8, 2020 at 11:52
• 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. May 8, 2020 at 12:18

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).