Global Euler obstruction and polar invariants (Q2571045)

For any affine variety \(Y\subset\mathbb{C}^N\) of pure dimension \(d\), the authors define a global Euler obstruction Eu\((Y)\), as the obstruction to extend a radial vector field, defined outside a sufficiently large compact subset of \(Y\), to a non-zero section of the Nash bundle. Also, they show that Eu\((Y)\) can be expressed as an alternating sum of the number of Morse points on the regular part of \(Y\) of a Lefschetz pencil on \(Y\) (resp. on successive generic hyperplane slices of \(Y)\). These invariants can be viewed as global polar multiplicities. Furthermore, when \(Y\) is nonsingular, Eu\((Y)\) coincides with the Euler-Poincaré characteristic.