Main references

Alexander Polishchuk. Algebraic geometry of Poisson brackets. J. Math. Sci. (New York) 84 (1997), no. 5, 1413–1444.
Mathscinet        zbMATH             Journal               
 
Brent Pym. Constructions and classifications of projective Poisson varieties. Lett. Math. Phys. 108 (2018), no. 3, 573–632.
Mathscinet        zbMATH             Journal                arXiv
 
Textbooks
​
Jean-Paul Dufour and Nguyen Zung. Poisson structures and their normal forms. Progress in Mathematics, 242. Birkhäuser Verlag, Basel, 2005. xvi+321 pp.
Mathscinet        zbMATH             
 
Camile Laurent-Gengoux; Anne Pichereau; Pol Vanhaecke. Poisson structures. Grundlehren der Mathematischen Wissenschaften, 347. Springer, Heidelberg, 2013. xxiv+461 pp. 
Mathscinet        zbMATH