On the smoothability problem with rational coefficients

TL;DR

This paper investigates the problem of smoothing algebraic cycles with rational coefficients on complex varieties, revealing its deep connection to the Hartshorne conjecture and providing an unconditional solution to a symplectic variant.

Contribution

It establishes a link between the smoothability problem and the Hartshorne conjecture, and solves a symplectic version unconditionally.

Findings

Smoothing algebraic cycles with rational coefficients relates to the Hartshorne conjecture.

A symplectic variant of the problem is solved unconditionally.

The compatibility of smoothing solutions with the Hartshorne conjecture is demonstrated.

Abstract

We consider the problem of smoothing algebraic cycles with rational coefficients on smooth projective complex varieties up to homological equivalence. We show that a solution to this problem would be incompatible with the validity of the Hartshorne conjecture on complete intersections in projective space. We also solve unconditionally a symplectic variant of this problem.