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

TL;DR

This paper studies special symplectic instanton bundles on odd-dimensional projective spaces, analyzing their cohomology, moduli space dimension, and singularity properties, revealing linear growth in moduli dimension and quadratic growth in certain cohomology groups.

Contribution

It extends the theory of instanton bundles to higher dimensions, providing explicit cohomology calculations and analyzing the structure of their moduli space, including singularities.

Findings

The dimension of the moduli space grows linearly with the second Chern class k.

The second cohomology group h^2End(E) grows quadratically with k.

Special symplectic instanton bundles are singular points in their moduli space.

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.