On Hodge polynomials for nonalgebraic complex manifolds.

Area of Science:

• Algebraic Geometry
• Complex Manifold Theory
• Topology

Background:

• Hodge theory is fundamental to understanding algebraic varieties' geometry and topology.
• The Hodge decomposition theorem links variety geometry with cohomology groups.
• Hodge theory is vital in mirror symmetry and studying algebraic cycles and motives.

Purpose of the Study:

• To explore Hodge polynomials and their properties on non-Kähler complex manifolds.
• To investigate the motivic nature of Hodge polynomials in a broader context beyond algebraic varieties.
• To deepen the understanding of complex manifold geometry.

Main Methods:

• Investigating diverse non-Kähler complex manifolds: (quasi-)Hopf, (quasi-)Calabi-Eckmann, and LVM manifolds.
• Analyzing a class of definable complex manifolds including algebraic varieties.
• Performing explicit calculations and thorough analyses.

Main Results:

• Demonstrated the preservation of the motivic nature of Hodge polynomials for investigated manifolds.
• Established that Hodge polynomials retain their motivic properties within a broader class of complex manifolds.
• Provided deeper insights into the geometry of complex manifolds beyond algebraic varieties.

Conclusions:

• Hodge polynomials' motivic nature is preserved in a wider range of complex manifolds.
• This research expands the applicability of Hodge theory beyond traditional algebraic geometry.
• Findings have potential applications in mathematics and physics involving complex manifolds.