On the cohomology of a tautological fibre bundle on the Hilbert scheme of a surface (Q2710516)

Let \(X\) be a smooth projective surface on \({\mathbb C}\). Let \(S^m(X)\) be the \(m\)-th symmetric product and \(\omega_X\) the canonical bundle of \(X\). Let \(X^{[m]}\) denote the Hilbert scheme of subschemes of \(X\) of length \(m\). Let \(L\) and \(A\) be two line bundles on \(X\). The symmetrization of the \(m\)-fold tensor product of \(A\) gives a line bundle on \(S^m(X)\). Let \({\delta}^A\) (the determinant bundle for \(A\)) denote the pull back of this line bundle to \(X^{[m]}\) by the canonical (Hilbert-Chow) morphism from \(X^{[m]}\) to \(S^m(X)\). Let \(S \subset X^{[m]} \times X\) be the universal scheme of couples \((Z,x)\) with \(x\in Z\). Let \(p_1\) and \(p_2\) be the projections from \(S\) to \(X^{[m]}\) and \(X\) respectively and \(L^{[m]} = {p_1}_* {p_2}^*L\). The main result is the following. NEWLINENEWLINENEWLINETheorem: If \({\omega_X}^{-1} \otimes A\) and \({\omega _X}^{-1} \otimes A \otimes L\) are ample, thenNEWLINENEWLINENEWLINE(i) \(H^q(X^{[m]}, L^{[m]}\otimes {\delta ^A}) = 0\) for \(q > 0\);NEWLINENEWLINENEWLINE(ii) \(H^0(X^{[m]}, L^{[m]} \otimes {\delta ^A}) \approx S^{m-1}(H^0(A)) \otimes H^0(L\otimes A)\).NEWLINENEWLINENEWLINEAs a corollary, it is shown that the conclusion of the theorem holds under the hypothesis \(H^q(X, A) = H^q(X, A\otimes L)= 0\) for \(q>0\). These results can be used to give examples supporting Le Potier's strange duality conjecture about moduli of semistable rank \(2\) torsionfree sheaves on a projective plane.