Algebraic Geometry Santa Cruz 1995, Part 1 by Kollar J., Lazarsfeld R., Morrison D. (eds.)

By Kollar J., Lazarsfeld R., Morrison D. (eds.)

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G+ from i) and L+ - J from ii) are clearly inverse to one another. A Lagrangian subspace L. of a complex symplectic space VV is called a real Lagrangian subspace, if it is the complexification of a Lagrangian subspace L C V; that is, L, = L OR C. This is satisfied precisely when Lc is carried to itself by complex conjugation. that is. when Lc = L. In V c. all real Lagrangian subspaces Lc are transversal to every F E C+. This is because for 00v=a+vr 1bELcnF witha,bEL we have 0<- / u; (v, v)_-2w(a,b)=0.

4 With this, we clearly have ytat =dsAa(x, s + t, t)dxk+b(x, s+t, t)dxk+i and so (1) d (iP at) = ds A T (x, s + t, t)dxk + -(x, s + t, t) dxk+i dt 8s as (x,a+t,t)dxk+ +dsA 5i(x, a+t, t)dxk+i We immediately deduce that (2) ti ( j- J = ds A (x, s + t, t) dxk + (x, s + t, t) dxk+l 2. Symplectic Manifolds 38 The vector field Xt, in this special situation, can be written as Xt = $ And so we then have a ) of = a(x, s, t) d? )(xIs t ) dx, A ... A (') _ dii) (8am (x, s, t) ds A dxt, ... ik) ax 7- (x, s, t) dxj Adxt, A...

25. 1) A subspace Q of V with ";IQ = 0 is called an isotropic subspace of (V, w). 2) A subspace W C V with w'N, non-degenerate is called a symplectic subspace of V. 3) A subspace W C V with W1 isotropic is called coisotropic. 4) A subspace L C V which is both isotropic and coisotropic (thus with Ll = L) is called a Lagrangian subspace. Before we discuss Lagrangian subspaces further, we prove the earlier statement that the dimension and rank are the only two independent symplectic invariants of a subspace.

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