[Paper Review] Counting curves via degeneration

Takeo Nishinou|arXiv (Cornell University)|Aug 13, 2011

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper develops a degeneration technique to count curves in algebraic varieties by reducing the problem to combinatorial calculations on toric degenerations. By analyzing the log normal sheaf and deformation obstructions via tropical and logarithmic geometry, the authors combinatorially prove the existence of 2875 lines on a generic quintic Calabi-Yau threefold and geometrically realize the walls in the Gross-Siebert construction as families of holomorphic disks.

ABSTRACT

We develop a technique to study curves in a variety which has a degeneration into some union of varieties. The class of such varieties is very broad, but the theory becomes particularly useful when the variety has a degeneration into a union of toric varieties. Hypersurfaces are typical examples, and we study lines on K3 surfaces and quintic Calabi-Yau hypersurfaces in detail. In particular, we combinatorially prove the existence of 2875 lines in a generic quintic Calabi-Yau 3-fold. Also, we give a geometric construction of walls in the Gross-Siebert construction of Calabi-Yau varieties.

Motivation & Objective

• To extend tropical and logarithmic geometry techniques beyond toric varieties to study curve counting in more general degenerations.
• To provide a combinatorial method for counting rational curves, particularly lines, in K3 and Calabi-Yau hypersurfaces.
• To geometrically realize the walls in the Gross-Siebert program as families of holomorphic disks in degenerate Calabi-Yau varieties.
• To compute the Kuranishi map for curve deformations when obstructions do not vanish, using degeneration to simplify the calculation.
• To establish a link between combinatorial curve counts and geometric structures in mirror symmetry, particularly in the context of quintic threefolds.

Proposed method

• Use a degeneration of a variety into a union of toric varieties to reduce complex curve counting problems to combinatorial and local analytic calculations.
• Analyze the log normal sheaf of a curve in the central fiber to compute deformation and obstruction spaces.
• Apply the Kuranishi map to study the existence of lifts of curves to higher-order deformations, particularly when obstructions vanish due to degeneration.
• Construct local models for singular points in the degeneration using the equation $XY + tZ = 0$ to simplify the analysis of the normal sheaf and tangent spaces.
• Use a coordinate change and restriction to 2-dimensional surfaces $S_t$ to reduce the $n$-dimensional case to the $n=3$ case, where the deformation problem is solvable.
• Leverage the implicit function theorem in the analytic category to construct a family of Maslov index zero holomorphic disks in the degenerate setting.

Experimental results

Research questions

• RQ1 Can the number of lines on a generic quintic Calabi-Yau threefold be rigorously computed using combinatorial methods?
• RQ2 How can the Kuranishi map be computed explicitly in the presence of non-vanishing obstructions using degeneration techniques?
• RQ3 Can the walls in the Gross-Siebert construction of Calabi-Yau varieties be geometrically realized as families of holomorphic disks?
• RQ4 To what extent can tropical and logarithmic geometry be applied to non-toric varieties via degeneration to toric unions?
• RQ5 What is the geometric structure of the moduli space of holomorphic disks in degenerate Calabi-Yau threefolds?

Key findings

• The paper provides a combinatorial proof of the classical result that a generic quintic Calabi-Yau threefold contains exactly 2875 lines.
• The Kuranishi map for lines on K3 surfaces is computed explicitly using degeneration, enabling the analysis of obstructed deformations.
• The authors construct a family of Maslov index zero holomorphic disks in a degenerate Calabi-Yau threefold, which geometrically realizes the walls in the Gross-Siebert program.
• These disks form an $n-3$ dimensional family in the $n$-dimensional case, with real codimension 1 in the smooth fiber, confirming the expected geometric structure of the walls.
• The method successfully reduces the global curve counting problem to local analytic calculations on toric-like singularities via the $XY + tZ = 0$ model.
• The construction confirms that the walls in the Gross-Siebert program arise as the geometric locus of such holomorphic disk families, as conjectured.