Topological Radon transforms and the local Euler obstruction
Let \(X \subseteq P^N\) be a complex projective variety, and denote by \(X^{<k}\) the \(k\)-dual variety of \(X\) living in the Grassmannian of linear subspaces of \(P^N\) of dimension \(k\). In the paper under review, the author proves that the \(k\)-Radon transform of the local Euler obstruction of \(X\) coincides with the local Euler obstruction of \(X^{< k}\), up to multiplication by sign and addition of a constant function on the Grassmannian [see \textit{J. L. Brylinski}, Astérisque 140-141, 3-134 (1986; Zbl 0624.32009) for the definition of the Radon transform at the level of constructible sheaves]. In the case where \(k = N - 1\), this gives an affirmative answer to a conjecture in [\textit{O. Y. Viro}, Lect. Notes Math. 1346, 127-138 (1987; Zbl 0686.14019)]. As an application, the author obtains generalized Zeuthen-Klein formulas for complex projective varieties with real structure, like in Viro's paper. The proofs depend on the theory of MacPherson-Chern classes [\textit{R. MacPherson}, Ann. Math., II. Ser. 100, 423-432 (1974; Zbl 0311.14001)], and its reformulation in terms of Lagrangian cycles [\textit{C. Sabbah}, Astérisque 130, 161-192 (1985; Zbl 0598.32011)].
scientific article; zbMATH DE number 727817
scientific article