[Paper Review] Examples of non-Kähler Calabi-Yau 3-folds with arbitrarily large $b_2$

Kenji Hashimoto, Taro Sano|arXiv (Cornell University)|Feb 4, 2019

Algebraic Geometry and Number Theory29 references4 citations

TL;DR

This paper constructs the first known examples of non-Kähler Calabi-Yau 3-folds with arbitrarily large second Betti numbers $b_2$, using log deformation theory to smooth a simple normal crossing variety composed of two Kähler components. The resulting manifolds are simply connected, non-projective, have algebraic dimension 1, and exhibit unobstructed deformations with degenerating Hodge-to-de Rham spectral sequence at $E_1$. The key contribution is bimeromorphic unboundedness of non-Kähler Calabi-Yau 3-folds.

ABSTRACT

We construct non-Kähler simply connected Calabi-Yau 3-folds with arbitrarily large 2nd Betti numbers by smoothing normal crossing varieties with trivial dualizing sheaves.

Motivation & Objective

• To construct non-Kähler Calabi-Yau 3-folds with arbitrarily large second Betti numbers $b_2$, addressing the open question of topological unboundedness in this class.
• To demonstrate bimeromorphic unboundedness of non-Kähler Calabi-Yau 3-folds by showing that the examples $X(a)$ and $X(a')$ are not bimeromorphic for $a \neq a'$.
• To establish that such Calabi-Yau 3-folds can be simply connected, non-projective, and have algebraic dimension 1, despite trivial canonical bundle.
• To extend the applicability of log deformation theory to non-projective SNC varieties with Kähler components, enabling smoothing to Calabi-Yau manifolds.
• To verify that the Hodge-to-de Rham spectral sequence degenerates at $E_1$ and deformations are unobstructed, confirming Calabi-Yau structure preservation.

Proposed method

• Construct a simple normal crossing (SNC) variety $X_0(a) = X_1 \cup X_2$, where $X_1$ is the blow-up of $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ along $a$ distinct smooth fibers of an elliptic fibration and a curve $C$, and $X_2 = \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$.
• Glue $X_1$ and $X_2$ along their isomorphic intersection $S$, an anticanonical divisor, using a twist by an automorphism of infinite order on $S$ to allow arbitrarily many blow-ups.
• Apply Kawamata–Namikawa log deformation theory to smooth $X_0(a)$ into a smooth Calabi-Yau 3-fold $X(a)$, ensuring trivial canonical bundle and vanishing $H^i(X, \mathcal{O}_X)$ for $0 < i < 3$.
• Use the upper semicontinuity theorem and cohomological vanishing to lift a big line bundle from the central fiber to the general fiber, preserving morphism to $\mathbb{P}^1$.
• Prove non-projectivity and algebraic dimension 1 by contradiction: assuming $a(X) \geq 2$ leads to a contradiction with the existence of a big line bundle on the central fiber.
• Verify simply connectedness via the Lefschetz hyperplane theorem and fundamental group comparison, and confirm unobstructed deformations and $E_1$-degeneration of the Hodge-to-de Rham spectral sequence.

Experimental results

Research questions

• RQ1 Can non-Kähler Calabi-Yau 3-folds with arbitrarily large $b_2$ be constructed using smoothing techniques?
• RQ2 Do such examples exist that are simply connected, non-projective, and have algebraic dimension 1?
• RQ3 Is bimeromorphic unboundedness possible for non-Kähler Calabi-Yau 3-folds, i.e., are there infinitely many non-bimeromorphic such manifolds?
• RQ4 Can log deformation theory be applied to non-projective SNC varieties with Kähler components to produce Calabi-Yau manifolds?
• RQ5 Do the Hodge-to-de Rham spectral sequence and deformation theory behave well on these non-Kähler Calabi-Yau 3-folds?

Key findings

• For any positive integer $a$, there exists a simply connected non-Kähler Calabi-Yau 3-fold $X(a)$ with $b_2(X(a)) = a + 3$.
• $X(a)$ has topological Euler characteristic $e(X(a)) = -256a^2 + 32a - 224$, which can be arbitrarily negative.
• The algebraic dimension of $X(a)$ is exactly 1, and $X(a)$ is not bimeromorphic to any Kähler manifold.
• The Hodge-to-de Rham spectral sequence degenerates at $E_1$ for $X(a)$, and $X(a)$ admits unobstructed deformations.
• For $a \neq a'$, the manifolds $X(a)$ and $X(a')$ are not bimeromorphic, establishing bimeromorphic unboundedness in the non-Kähler Calabi-Yau 3-fold category.
• The construction uses a non-trivial automorphism of infinite order on the intersection surface $S$, enabling arbitrarily many blow-ups and thus arbitrarily large $b_2$.