부산대학교 도서관

Let $X$ be a smooth scheme over a field $k$ with characteristic 0. Then one may define a Poisson structure on $X$ via a sheaf of $k$-linear Poisson algebras where the multiplication coincides with the original commutative multiplication on the structure sheaf $\Cal{O}_X$ of $X$, or via the scheme $X$ since it carries a bivector $\pi_X\in H^0(X,\wedge^2T_X)$ satisfying $[\pi_X,\pi_X]=0$. The two constructions are equivalent since a bivector $\pi_X$ is the same as an antisymmetric biderivation $\Cal{O}_X\otimes_k \Cal{O}_X\rightarrow \Cal{O}_X$. It thus follows that the condition $[\pi_X,\pi_X]=0$ is equivalent to the Jacobi identity for the corresponding biderivation. \par Now let $X$ be a scheme with a fixed Poisson structure. Consider a smooth closed subscheme $i\:L\hookrightarrow X$. Then $L$ is coisotropic if one of the equivalent conditions holds: the ideal sheaf $\Cal{I}_L$ defining $L$ is a coisotropic ideal (closed under the Poisson 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 equivalence of 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. Safronov generalize the definition of a Poisson structure by replacing the Poisson bracket of degree 0 by a Poisson bracket of degree $-n$, and by considering the algebra ${{\bf Pol}(A,n)\coloneq {\rm Hom}_A({\rm Sym}_A(\Bbb{L}_A[n+1]), A)}$ of $n$-shifted polyvector fields, where $\Bbb{L}_A$ is the cotangent complex of $A$. The algebra of $n$-shifted polyvector fields is equipped with a commutative multiplication, a Poisson bracket of degree $-n-1$, generalizing the Schouten bracket, and an additional grading such that $\Bbb{L}_A$ has weight $-1$. In other words, ${\bf Pol}(A,n)$ is a graded $\Bbb{P}_{n+2}$@-algebra. Since the Jacobi identity can be summarized by assuming that the morphism ${k(2)[-1]\rightarrow {\bf Pol}(A,n)[n+1]}$ of graded complexes is a map of graded dg Lie algebras, an $n$-shifted Poisson structure on a cdga $A$ is the property that $A$ carries a $\Bbb{P}_{n+1}$-algebra structure compatible with the original commutative structure on $A$, or equivalently, a morphism $k(2)[-1]\rightarrow {\bf Pol}(A,n)[n+1]$ of graded dg Lie algebras exists, giving an entire space ${\rm Pois}(A,n)$ of $n$-shifted Poisson structures. \par Now let $f\:A\rightarrow B$ be a morphism of commutative algebras, and consider 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 $F\:A\rightarrow Z(B)\coloneq \widehat{{\bf Pol}}(B,n-1)$, where $Z(B)$ is the Poisson center, i.e., the completion of the algebra of $(n-1)$-shifted polyvector fields with respect to the weight grading. This gives us a morphism of commutative algebras ${A\rightarrow Z(B)\rightarrow B}$, satisfying that the homotopy fiber $U(A,B)$ of $f$ carries a canonical structure of a non-unital $\Bbb{P}_{n+1}$-algebra such that the projection $U(A,B)\rightarrow A$ is a morphism of $\Bbb{P}_{n+1}$-algebras. This establishes a relation to the classical definition of coisotropic ideals. \par Next, consider the algebra ${\bf Pol}(B/A,n-1) = {\rm Hom}_B({\rm Sym}_B(\Bbb{L}_{B/A}[n-1]),B)$ of $n$@-shifted relative polyvectors which is equipped with a graded $\Bbb{P}_{n+1}$-algebra structure. There is a natural morphism ${\bf Pol}(A,n)\rightarrow {\bf Pol}(B/A,n-1)$ of commutative algebras induced from $\Bbb{L}_{B/A}\rightarrow f^*\Bbb{L}_A[1]$ of cotangent complexes, which shows that the pair $({\bf Pol}(A,n), {\bf Pol}(B/A,n-1))$ is naturally a graded $\Bbb{P}_{[n+2,n+1]}$-algebra, giving us a graded non-unital $\Bbb{P}_{n+2}$-algebra ${\bf Pol}(f,n)=U({\bf Pol}(A,n),{\bf Pol}(B/A,n-1))$. Thus, morphisms $k(2)[-1]\rightarrow {\bf Pol}(f,n)[n+1]$ of graded dg Lie algebras generalize the classical definition. This naturally results in the following two definitions of an $n$-shifted coisotropic structure on a cdga morphism $f\:A\rightarrow B$; that is, there is a $\Bbb{P}_{[n+1,n]}$-algebra structure on $(A,B)$ such that the induced morphism $A\rightarrow Z(B)\rightarrow B$ of commutative algebras is homotopic to $f$, and a morphism $k(2)[-1]\rightarrow {\bf Pol}(f,n)[n+1]$ of graded dg Lie algebras exists. \par Let ${\rm Cois}(f,n)$ denote the space of $n$-shifted coisotropic structures. The following is one of the main results by Melani and Safronov (Theorem 4.16, page 3112): given a morphism $f\:A\rightarrow B$ 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 Poisson if and only if its graph $Y\rightarrow \overline{Y}\times X$ is coisotropic, where the underlying manifold of $\overline{Y}$ is $Y$ but with the opposite Poisson structure. Another main result in the paper is a generalization of the previous statement, but in the derived context (Theorem 4.21, page 3116): let $f\:A\rightarrow B$ be a morphism of commutative $\Bbb{P}_{n+1}$-algebras and $g\:A\otimes B\rightarrow B$ its graph, i.e., $g(x\otimes y)=f(x)y$. Then there is a Cartesian square $$ \CD {\rm Pois}(f,n) @>>> {\rm Pois}(A,n)\times {\rm Pois}(B,n)\\ @VVV @VV{\rm id}\times {\rm opp}V\\ {\rm Cois}(g,n) @>>> {\rm Pois}(A\otimes B,n) \\ \endCD $$ of spaces. \par \{For Part II see [V. Melani and P. Safronov, Selecta Math. (N.S.) {\bf 24} (2018), no.~4, 3119--3173; MR3848017].\}