compatible almost complex structure and proceeding with a complex blow-up. This makes sense on a purely topological level; however, endowing the blow-up with a symplectic form requires some care, because one cannot arbitrarily extend the symplectic form across the exceptional divisor E. One must alter the symplectic form in a neighborhood of E, or perform the blow-up by cutting out a neighborhood of Z and collapsing the boundary in a well-defined way. This is best understood using the formalism of symplectic cutting, of which symplectic blow-up is a special case. Symplectic cutting, together with the inverse operation of symplectic summation, is the symplectic analogue of deformation to the normal cone along a smooth divisor. See also Infinitely near point Resolution of singularities References Birational geometry