The Tangent Space at a Special Symplectic Instanton Bundle on $P_{2n+1}$

Abstract

Mathematical instanton bundles on $ P_3$ have their analogues in rank--$2n$ instanton bundles on odd dimensional projective spaces $ P_{2n+1}$. The families of special instanton bundles on these spaces generalize the special 'tHooft bundles on $ P_3$. We prove that for a special symplectic instanton bundle $ E$ on $ P_{2n+1}$ with $c_2=k$ $h^1End( E) = 4(3n-1) k + (2n-5)(2n-1)$. Therefore the dimension of the moduli space of instanton bundles grows linearly in $k$. The main difference with the well known case of $ P_3$ is that $h^2End( E)$ is nonzero, in fact we prove that it grows quadratically in $k$. Special symplectic instanton bundles turn out to be singular points of the moduli space. Such bundles $ E$ are $SL(2)$--invariant and the result is obtained regarding the cohomology groups of $ E$ as $SL(2)$--representations.