[Paper Review] Construction of Integral Cohomology in Some Degenerations and its Application to Smoothing of Degenerate Calabi-Yau

Nam‐Hoon Lee|arXiv (Cornell University)|Apr 27, 2006

Algebraic Geometry and Number Theory1 citations

TL;DR

This paper establishes a general formula for constructing integral cohomology classes—specifically Picard groups and Chern classes—of Calabi-Yau manifolds arising from smoothing normal crossings varieties. By proving the constructibility of these invariants from the degenerate central fiber, the authors provide a systematic method to generate new examples, including Calabi-Yau 3-folds with Picard number one, demonstrating the utility of smoothing degenerations as a construction tool.

ABSTRACT

Abstract. A smoothing theorem for normal crossings to Calabi-Yau manifolds was proved by Y. Kawamata and Y. Namikawa ([KaNa]). This paper is a study of the observation that the Picard groups and Chern classes of these Calabi-Yau manifolds are constructible from the normal crossings in such smoothings. We provide and prove the formula for the construction in its full generality and various applications are discussed, including the construction of many new examples of Calabi-Yau 3-folds with Picard number one. With this construction as a starting point, we hope to convince readers that smoothing normal crossings is a promising method of constructing Calabi-Yau manifolds. 1.

Motivation & Objective

• To investigate the constructibility of integral cohomology invariants—particularly Picard groups and Chern classes—of Calabi-Yau manifolds obtained by smoothing normal crossings varieties.
• To generalize the smoothing theorem of Kawamata and Namikawa to provide a full formula for these invariants in terms of the degenerate central fiber.
• To demonstrate the method's power by constructing new examples of Calabi-Yau 3-folds with Picard number one.
• To establish that smoothing normal crossings is a viable and promising method for constructing Calabi-Yau manifolds with desired topological properties.

Proposed method

• Utilizes the smoothing theorem of Kawamata and Namikawa for normal crossings to Calabi-Yau manifolds as a foundational framework.
• Applies intersection theory and cohomological techniques to relate the integral cohomology of the smooth Calabi-Yau to the geometry of the degenerate central fiber.
• Derives a general formula expressing the Picard group and Chern classes of the smooth Calabi-Yau in terms of the components and their intersections in the normal crossings degeneration.
• Employs the long exact sequence in cohomology associated with the degeneration to track the behavior of characteristic classes and line bundles.
• Applies the formula to specific configurations of normal crossings to construct explicit examples of Calabi-Yau 3-folds.
• Validates the construction by verifying that the resulting manifolds satisfy the Calabi-Yau condition, including trivial canonical bundle and Hodge numbers.

Experimental results

Research questions

• RQ1 Can the Picard group and Chern classes of a Calabi-Yau manifold obtained by smoothing a normal crossings variety be reconstructed from the degenerate central fiber?
• RQ2 What is the general formula governing the construction of integral cohomology classes in such smoothings?
• RQ3 How can this formula be applied to generate new examples of Calabi-Yau 3-folds with specific topological invariants, such as Picard number one?
• RQ4 To what extent does the smoothing of normal crossings provide a systematic method for constructing Calabi-Yau manifolds with desired geometric and cohomological properties?

Key findings

• A general formula is established that constructs the Picard group and Chern classes of a Calabi-Yau manifold from the components and their intersections in a normal crossings degeneration.
• The formula confirms that the integral cohomology invariants of the smooth Calabi-Yau are fully determined by the degenerate central fiber.
• The method successfully produces new examples of Calabi-Yau 3-folds with Picard number one, demonstrating its effectiveness in constructing rare and interesting cases.
• The construction shows that smoothing normal crossings is a powerful and systematic approach to building Calabi-Yau manifolds with controlled topological invariants.
• The results validate the theoretical framework of Kawamata and Namikawa’s smoothing theorem by providing explicit cohomological invariants for the resulting smooth Calabi-Yau varieties.
• The paper establishes that the Chern classes and Picard group of the smooth Calabi-Yau are not arbitrary but are constrained and computable from the degenerate data.