[Paper Review] Some Calabi-Yau coverings over singular varieties and new Calabi-Yau threefolds with Picard rank one

Nam‐Hoon Lee|arXiv (Cornell University)|Oct 2, 2006

Algebraic Geometry and Number Theory3 references2 citations

TL;DR

This paper demonstrates that certain singular varieties, particularly Q-Fano varieties from Takagi's examples, admit Calabi-Yau coverings. By applying degeneration methods to compute invariants, the authors construct at least 22 new Calabi-Yau threefolds with Picard number one, expanding the known list of such manifolds with trivial canonical class and trivial intermediate cohomologies of the structure sheaf.

ABSTRACT

Abstract. This paper is a report on the observation that some singular varieties admit Calabi-Yau coverings. We derive a formula for calculating the invariants of the coverings with degeneration methods. By applying these to Takagi’s Q-Fano examples([Ta1], [Ta2]), we construct several Calabi-Yau threefolds with Picard number one. It turns out that at least 22 of them are new. A Calabi-Yau manifold is a compact Kähler manifold with trivial canonical class such that the intermediate cohomologies of its structure sheaf are all trivial (h i (X, OX) = 0 for 0 < i < dim(X)). One handy way of construction of Calabi-Yau manifolds is by taking coverings of some smooth varieties

Motivation & Objective

• To investigate whether singular varieties can admit Calabi-Yau coverings, extending the known construction methods beyond smooth base varieties.
• To develop a formula for computing topological and geometric invariants of such Calabi-Yau coverings using degeneration techniques.
• To apply the derived formula to Takagi’s Q-Fano examples to construct new Calabi-Yau threefolds with Picard number one.
• To determine the novelty and invariants of the resulting Calabi-Yau manifolds, particularly focusing on those with trivial intermediate cohomologies.

Proposed method

• Utilize degeneration methods to analyze the invariants of Calabi-Yau coverings over singular varieties.
• Apply the derived invariant formula to Takagi’s Q-Fano varieties, which are known singular Fano-type varieties.
• Construct Calabi-Yau threefolds as finite coverings of these Q-Fano varieties, ensuring trivial canonical class and vanishing intermediate cohomologies.
• Verify the Calabi-Yau conditions: trivial canonical bundle and h^i(X, O_X) = 0 for 0 < i < dim(X).
• Employ algebraic geometry techniques, including resolution of singularities and monodromy analysis, to ensure the covering is well-defined and Calabi-Yau.

Experimental results

Research questions

• RQ1 Can Calabi-Yau manifolds be constructed as coverings of singular varieties, particularly Q-Fano varieties?
• RQ2 What formula governs the invariants of Calabi-Yau coverings over singular bases using degeneration methods?
• RQ3 How many new Calabi-Yau threefolds with Picard number one can be constructed from Takagi’s Q-Fano examples?
• RQ4 Do the resulting Calabi-Yau manifolds satisfy the standard cohomological conditions (h^i(X, O_X) = 0 for 0 < i < 3)?

Key findings

• At least 22 new Calabi-Yau threefolds with Picard number one are constructed from coverings of Takagi’s Q-Fano varieties.
• The method successfully produces Calabi-Yau manifolds even when the base variety is singular, extending standard covering constructions.
• The derived formula enables precise computation of invariants for Calabi-Yau coverings via degeneration techniques.
• All constructed threefolds satisfy the Calabi-Yau conditions: trivial canonical bundle and vanishing intermediate cohomologies of the structure sheaf.