Decomposition into pairs-of-pants for complex algebraic hypersurfaces
  It is well-known that a Riemann surface can be decomposed into the so-called pairs-of-pants. Each pair-of-pants is diffeomorphic to a Riemann sphere minus 3 points. We show that a smooth complex projective hypersurface of arbitrary dimension admits a similar decomposition. The n-dimensional pair-of-pants is diffeomorphic to the complex projective n-space minus n+2 hyperplanes.   Alternatively, these decompositions can be treated as certain fibrations on the hypersurfaces. We show that there exists a singular fibration on the hypersurface with an n-dimensional polyhedral complex as its base and a real n-torus as its fiber. The base accomodates the geometric genus of a hypersurface V. Its homotopy type is a wedge of h^{n,0}(V) spheres S^n. 
Grigory Mikhalkin