The Atiyah-Hizrebruch theorem suggests that this cannot hold integrally.
The Atiyah-Hizrebruch theorem suggests that this cannot hold integrally. It is also remarked that it might be useful to express the namesake classes in terms of Chern classes, which is possible because vector bundles have plenty of meromorphic sections. A naïve version of this assumes that relevant projective space cannot be weakened to be merely Kahler. It deals with schemes of finite type over a field, and the geometric interpretation in R-space is defined as the set of points satisfying a system of polynomial equations, which according the Hilbert basis theorem-a finite number of equations is sufficient. Futhermore, it asserts that these can be represented by linear combination of homology classes, otherwise known as its namesake cycles. For 10 points, name this Clay Institute Millennium Prize problem, which asserts that for projective algebraic varieties, these eponymous cycles are rational linear combinations of algebraic cycles.
The Atiyah-Hizrebruch theorem suggests that this cannot hold integrally.
Hodge Conjecture