K-theory, local cohomology and tangent spaces to Hilbert schemes

Abstract

Using K-theory, we construct a map $\pi :T_Y{Hilb}^p\left(X\right)\to H_y^p\left({\mathrm{\Omega }}_{X∕\mathbb{Q}}^{p-1}\right)$ from the tangent space to the Hilbert scheme at a point $Y$ to the local cohomology group. We use this map $\pi$ to answer (after slight modification) a question by Mark Green and Phillip Griffiths on constructing a map from the tangent space $T_Y{Hilb}^p\left(X\right)$ to the Hilbert scheme at a point $Y$ to the tangent space to the cycle group $T{Z}^p\left(X\right)$.