Lecture 10- Moment maps in the algebraic setting

Lecture 10- Moment maps in the algebraic setting - LECTURES...

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LECTURES ON SYMPLECTIC REFLECTION ALGEBRAS IVAN LOSEV 10. Moment maps in algebraic setting 10.1. Symplectic algebraic varieties. An affine algebraic variety X is said to be Poisson if C [ X ] is equipped with a Poisson bracket. Exercise 10.1. Let A be a commutative algebra and B be a localization of A . Let A be equipped with a bracket. Show that there is a unique bracket on B such that the natural homomorphism A B respects the bracket. Thanks to this exercise, the sheaf O X of regular functions on X acquires a bracket (i.e., we have brackets on all algebras of sections and the restriction homomorphisms are compatible with the bracket). We say that an arbitrary (=not necessarily affine) variety X is Poisson if the sheaf O X comes equipped with a Poisson bracket. Recall that on a variety X such that O X is equipped with a bracket we have a bivector (=a bivector field) P Γ( X reg , 2 TX reg ). This gives rise to a map v x : T x X T x X for x X reg , α 7→ P x ( α, · ). We say that P is nondegenerate in x if this map is an isomorphism. In this case, we can use this map to get a 2-form ω x 2 T x X : ω x ( v x ( α ) , v x ( β )) = P x ( α, β ) = α, v x ( β ) = −⟨ v x ( α ) , β . Now suppose X is smooth. Suppose that P is non-degenerate (=non-degenerate at all points). So we have a non-degenerate form ω on X . The condition that P is Poisson is equivalent to = 0. A non-degenerate closed form ω is called symplectic (and X is called a symplectic variety ). The most important for us class of symplectic varieties is cotangent bundles. Let X 0 be a smooth algebraic variety, set X := T X 0 . A symplectic form ω on X is introduced as follows. First, let us introduce a canonical 1-form α . We need to say how α x pairs with a tangent vector for any x X . A point X can be thought as a pair ( x 0 , β ), where x 0 X 0 and β T x 0 X 0 . Consider the projection π : X X 0 (defined by π ( x ) = x 0 ). For x = ( x 0 , β ) we define α x by α x , v = β, d x π ( v ) . We can write α in “coordinates”. If we worked in the C - or analytic setting, we could use the usual coordinates. However, we cannot do this because we want to show that α is an algebraic form. So we will use an algebro-geometric substitute for coordinate charts: ´ etale coordinates. Namely, we can introduce ´ etale coordinates in a neighborhood of each point x 0 X 0 . Let us choose functions x 1 , . . . , x n with a property that d x 0 x 1 , . . . d x 0 x n form a basis in T x 0 X 0 . Then dx 1 , . . . , dx n are linearly independent at any point from some neighborhood X 0 0 of x 0 . So the map φ : X 0 0 C n given by ( x 1 , . . . , x n ) is ´ etale and we call x 1 , . . . , x n ´ etale coordinates. Then we can get ´ etale coordinates y 1 , . . . , y n on T X 0 0 as follows: by definition y i ( x 0 , β ) is the coefficient of d x 0 x i in β , i.e., β = n i =1 y i ( x 0 , β ) d x 0 x i (and we view x 1 , . . . , x n as functions on T X 0 0 via pull-back). Then, on T X 0 0 , α is given by n i =1 y i dx i .
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