Viro-Zvonilov inequalities for flexible curves on an almost complex four-dimensional manifold

TL;DR

This paper extends Viro-Zvonilov inequalities to flexible curves on almost complex four-manifolds, providing new topological restrictions and examples, and explores lifting Lie group actions to branched coverings with conditions on homology.

Contribution

It introduces generalized inequalities for flexible curves on almost complex four-manifolds and analyzes the lifting of Lie group actions to branched coverings.

Findings

Extended Viro-Zvonilov inequalities to new settings

Provided examples demonstrating sharpness of inequalities

Established conditions for trivial first homology in branched covers

Abstract

The restrictions on the topology of nonsingular plane projective real algebraic curves of odd degree, obtained by O. Viro and the author in the paper published in the early 90s, are extended to flexible curves lying on an almost complex four-dimensional manifold. Some examples of real algebraic surfaces and real curves on them prove the sharpness of the obtained inequalities. In addition, it is proved that a compact Lie group smooth action can be lifted to a cyclic branched covering space $ \tilde{X} $ over a closed four-dimensional manifold, and a sufficient condition for $ H_1(\tilde{X})=0 $ was found.