Varieties of complexes and foliations (Q2879063)

This paper offers a description of the set of integrable 1-forms on projective space. A 1-form \(\omega\) being integrable here means \(\omega \wedge d\omega\) -- i.e., \(\omega\) is Frobenius integrable. Letting \(\mathcal{F}(r,d)\) denote the space of integrable 1-forms of degree \(d\) on \(\mathbb{P}^r\), the integrability condition implies that \(\mathcal{F}(r,d)\) is an affine variety defined by homogeneous quadratic equations in the vector space of all 1-forms of degree \(d\) on \(\mathbb{P}^r\). The main result in this paper is that \(\mathcal{F}(r,d)\) is isomorphic to a linear subspace of a certain variety of complexes.NEWLINENEWLINEA variety of complexes is a space of exact sequences of linear maps between vector spaces; like \(\mathcal{F}(r,d)\), a variety of complexes is an affine variety defined by homogeneous quadratic equations. The paper gives a detailed discussion of varieties of complexes, and the upshot of this analysis is a description of the irreducible components of \(\mathcal{F}(r,d)\).