Cappell–Shaneson conjecture on characteristic classes

Let $X$ be a compact pure-dimensional complex algebraic variety. The intersection cohomology Hirzebruch class is denoted by ${{\rm IT}_y}_*(X)$, and the Goresky–MacPherson $L$ class by $L_*(X)$.

Cappell–Shaneson conjecture.

$$L_*(X)={{\rm IT}_1}_*(X).$$

This is a characteristic-class version of the Hodge index theorem, concerning the expected commutativity of the characteristic-class constructions associated with intersection cohomology. The paper proves the conjecture for secant varieties when the underlying line bundle is 3-very ample, while the general statement remains open.