부산대학교 도서관

Let $X$ be a smooth scheme over a field $k$ with characteristic 0. Thenone may define a Poisson structure on $X$ via a sheaf of $k$-linearPoisson algebras where the multiplication coincides with the originalcommutative multiplication on the structure sheaf $\Cal{O}_X$ of $X$,or via the scheme $X$ since it carries a bivector $\pi_X\inH^0(X,\wedge^2T_X)$ satisfying $[\pi_X,\pi_X]=0$. The two constructionsare equivalent since a bivector $\pi_X$ is the same as an antisymmetricbiderivation $\Cal{O}_X\otimes_k \Cal{O}_X\rightarrow \Cal{O}_X$. Itthus follows that the condition $[\pi_X,\pi_X]=0$ is equivalent to theJacobi identity for the corresponding biderivation.\par Now let $X$ be a scheme with a fixed Poisson structure. Consider asmooth closed subscheme $i\:L\hookrightarrow X$. Then $L$ iscoisotropic if one of the equivalent conditions holds: the ideal sheaf$\Cal{I}_L$ defining $L$ is a coisotropic ideal (closed under thePoisson bracket on $\Cal{O}_X$), or the composite$$N_{L/X}^*\rightarrow i^*T_X^*\overset{\pi_X}\to{\rightarrow}i^*T_X\rightarrow N_{L/X}$$is zero, where $N_{L/X}$ is the normal bundle of $L$ (the equivalenceof the two definitions follows from the isomorphism $N_{L/X}^* \cong\Cal{I}_L/\Cal{I}_L^2$).\par Let $A$ be a commutative dg algebra. V. Melani and P. Safronovgeneralize the definition of a Poisson structure by replacing thePoisson bracket of degree 0 by a Poisson bracket of degree $-n$, and byconsidering the algebra ${{\bf Pol}(A,n)\coloneq {\rm Hom}_A({\rmSym}_A(\Bbb{L}_A[n+1]), A)}$ of $n$-shifted polyvector fields, where$\Bbb{L}_A$ is the cotangent complex of $A$. The algebra of $n$-shiftedpolyvector fields is equipped with a commutative multiplication, aPoisson bracket of degree $-n-1$, generalizing the Schouten bracket,and an additional grading such that $\Bbb{L}_A$ has weight $-1$. Inother words, ${\bf Pol}(A,n)$ is a graded $\Bbb{P}_{n+2}$@-algebra.Since the Jacobi identity can be summarized by assuming that themorphism ${k(2)[-1]\rightarrow {\bf Pol}(A,n)[n+1]}$ of gradedcomplexes is a map of graded dg Lie algebras, an $n$-shifted Poissonstructure on a cdga $A$ is the property that $A$ carries a$\Bbb{P}_{n+1}$-algebra structure compatible with the originalcommutative structure on $A$, or equivalently, a morphism$k(2)[-1]\rightarrow {\bf Pol}(A,n)[n+1]$ of graded dg Lie algebrasexists, giving an entire space ${\rm Pois}(A,n)$ of $n$-shifted Poissonstructures.\par Now let $f\:A\rightarrow B$ be a morphism of commutative algebras, andconsider a two-colored dg operad $\Bbb{P}_{[n+1,n]}$ given by a triple$(A,B,F)$ of a $\Bbb{P}_{n+1}$-algebra $A$, a $\Bbb{P}_n$-algebra $B$,and a morphism of $\Bbb{P}_{n+1}$-algebras $F\:A\rightarrowZ(B)\coloneq \widehat{{\bf Pol}}(B,n-1)$, where $Z(B)$ is the Poissoncenter, i.e., the completion of the algebra of $(n-1)$-shiftedpolyvector fields with respect to the weight grading. This gives us amorphism of commutative algebras ${A\rightarrow Z(B)\rightarrow B}$,satisfying that the homotopy fiber $U(A,B)$ of $f$ carries a canonicalstructure of a non-unital $\Bbb{P}_{n+1}$-algebra such that theprojection $U(A,B)\rightarrow A$ is a morphism of$\Bbb{P}_{n+1}$-algebras. This establishes a relation to the classicaldefinition of coisotropic ideals.\par Next, consider the algebra ${\bf Pol}(B/A,n-1) = {\rm Hom}_B({\rmSym}_B(\Bbb{L}_{B/A}[n-1]),B)$ of $n$@-shifted relative polyvectorswhich is equipped with a graded $\Bbb{P}_{n+1}$-algebra structure.There is a natural morphism ${\bf Pol}(A,n)\rightarrow {\bfPol}(B/A,n-1)$ of commutative algebras induced from$\Bbb{L}_{B/A}\rightarrow f^*\Bbb{L}_A[1]$ of cotangent complexes,which shows that the pair $({\bf Pol}(A,n), {\bf Pol}(B/A,n-1))$ isnaturally a graded $\Bbb{P}_{[n+2,n+1]}$-algebra, giving us a gradednon-unital $\Bbb{P}_{n+2}$-algebra ${\bf Pol}(f,n)=U({\bfPol}(A,n),{\bf Pol}(B/A,n-1))$. Thus, morphisms $k(2)[-1]\rightarrow{\bf Pol}(f,n)[n+1]$ of graded dg Lie algebras generalize the classicaldefinition. This naturally results in the following two definitions ofan $n$-shifted coisotropic structure on a cdga morphism$f\:A\rightarrow B$; that is, there is a $\Bbb{P}_{[n+1,n]}$-algebrastructure on $(A,B)$ such that the induced morphism $A\rightarrowZ(B)\rightarrow B$ of commutative algebras is homotopic to $f$, and amorphism $k(2)[-1]\rightarrow {\bf Pol}(f,n)[n+1]$ of graded dg Liealgebras exists.\par Let ${\rm Cois}(f,n)$ denote the space of $n$-shifted coisotropicstructures. The following is one of the main results by Melani andSafronov (Theorem 4.16, page 3112): given a morphism $f\:A\rightarrowB$ of commutative dg algebras, one has an equivalence of spaces$${\rm Cois}(f,n) \cong {\rm Map}_{{\rm Alg}_{{\rm Lie}}^{gr}}(k(2)[-1],{\bf Pol}(f,n)[n+1]).$$\par Finally, a morphism of Poisson manifolds $Y\rightarrow X$ is Poissonif and only if its graph $Y\rightarrow \overline{Y}\times X$ iscoisotropic, where the underlying manifold of $\overline{Y}$ is $Y$ butwith the opposite Poisson structure. Another main result in the paperis a generalization of the previous statement, but in the derivedcontext (Theorem 4.21, page 3116): let $f\:A\rightarrow B$ be amorphism of commutative $\Bbb{P}_{n+1}$-algebras and $g\:A\otimesB\rightarrow B$ its graph, i.e., $g(x\otimes y)=f(x)y$. Then there is aCartesian square$$\CD {\rm Pois}(f,n) @>>> {\rm Pois}(A,n)\times {\rm Pois}(B,n)\\@VVV @VV{\rm id}\times {\rmopp}V\\ {\rm Cois}(g,n) @>>> {\rm Pois}(A\otimes B,n) \\ \endCD$$of spaces.\par \{For Part II see [V. Melani and P. Safronov, Selecta Math. (N.S.){\bf 24} (2018), no.~4, 3119--3173; MR3848017].\}