[Paper Review] Stability conditions and Calabi-Yau fibrations

Yukinobu Toda|ArXiv.org|Aug 20, 2006

Algebraic Geometry and Number Theory3 references3 citations

TL;DR

This paper constructs and describes the spaces of stability conditions for triangulated categories associated with three-dimensional Calabi-Yau fibrations, focusing on flat elliptic fibrations and smooth K3 (or Abelian) fibrations. It establishes chamber structures in the stability space analogous to those in birational geometry for elliptic fibrations and identifies a natural correspondence between stability conditions on the total space and those on the special fiber for K3 fibrations, using geometric and categorical techniques.

ABSTRACT

In this paper, we describe the spaces of stability conditions for the triangulated categories associated to three dimensional Calabi-Yau fibrations. We deal with two cases, the flat elliptic fibrations and smooth K3 (Abelian) fibrations. In the first case, we will see there exist chamber structures similar to those of the movable cone used in birational geometry. In the second case, we will compare the space with the space of stability conditions for the closed fiber of the fibration.

Motivation & Objective

• To describe the space of stability conditions on the derived category $ D(X/S) $ for three-dimensional Calabi-Yau fibrations.
• To extend the framework of stability conditions beyond crepant resolutions to include elliptic and K3 fibrations.
• To establish a connection between birational geometry (via movable cone structures) and $ /mathcal{N}=2 $ superconformal field theory through stability conditions.
• To compare the stability space of the total space $ X $ with that of the special fiber $ X_0 $ in the case of $ K3 $ fibrations.
• To address the technical challenges in constructing stability conditions when singular fibers or non-trivial normal bundles (e.g., $ \mathcal{O}_{\mathbb{P}^2}(-3) $) are present.

Proposed method

• Uses the $ \beta, \omega $-twisted Chern character and central charge map $ Z_{(\beta,\omega)} $ to define a t-structure with heart $ \mathcal{A}_{(\beta,\omega)} $, forming a numerical stability condition.
• Applies techniques from Bridgeland’s work on $ K3 $ surfaces, relying on the Bogomolov inequality, under the assumption of smooth or flat fibrations to ensure constructibility.
• For flat elliptic fibrations, constructs a regular covering map from $ \mathop{\rm Stab}^\circ(X/S) $ to a quotient space involving $ \mathop{\rm GL}^+(2,\mathbb{R}) \times V_\mathbb{C} $, modulo hyperplanes indexed by $ (k,w,l) $.
• For $ K3 $ fibrations, compares stability conditions on $ D(X/S) $ with those on the special fiber $ X_0 $, using the known structure of $ \mathop{\rm Stab}^\circ(X_0) $ as a covering over $ \mathcal{P}_0^+(X_0) \subset \mathbb{C} \oplus N^1(X_0)_\mathbb{C} \oplus \mathbb{C} $.
• Employs autoequivalence groups and commutative diagrams of functors to identify generators of $ \mathop{\rm Auteq}(\mathrsfs{X}/\mathrsfs{Y}) $, particularly using twist functors $ T_{\mathcal{E}_{a,b}} $ and line bundle tensoring.
• Uses Chen’s lemma and properties of spherical objects to show that certain autoequivalences are trivial or tensor products, enabling identification of the full autoequivalence group.

Experimental results

Research questions

• RQ1 How do the spaces of stability conditions on $ D(X/S) $ for flat elliptic fibrations compare to those in birational geometry, particularly in terms of chamber structures?
• RQ2 Can the space of stability conditions on a $ K3 $ fibration be related to the space of stability conditions on its special fiber?
• RQ3 What role do autoequivalences, especially twist functors and line bundle actions, play in the structure of the stability space for Calabi-Yau fibrations?
• RQ4 How do the geometric properties of the singular fiber $ f^{-1}(0) $, such as the presence of $ \mathbb{P}^2 $ with normal bundle $ \mathcal{O}(-3) $, obstruct the construction of stability conditions?
• RQ5 What is the precise structure of the autoequivalence group $ \mathop{\rm Auteq}(\mathrsfs{X}/\mathrsfs{Y}) $ for a Calabi-Yau fibration over a base with a degenerate fiber?

Key findings

• For flat elliptic fibrations, the space $ \mathop{\rm Stab}^\circ(X/S) $ admits a regular covering map to $ (\mathop{\rm GL}^+(2,\mathbb{R}) \times V_\mathbb{C}) \setminus \bigcup \widetilde{H}_{k,w(l)} $, with chamber structure analogous to the movable cone in birational geometry.
• The space $ \mathop{\rm Stab}^\circ(X/S) $ for elliptic fibrations is shown to be a regular covering over a quotient space involving reflections in $ W_{\rm ref} $ and Dynkin-type rational curves.
• For $ K3 $ (or Abelian) fibrations, the space $ \mathop{\rm Stab}^\circ(X/S) $ is naturally related to $ \mathop{\rm Stab}^\circ(X_0) $, the stability space of the special fiber, via a comparison of central charges and t-structures.
• The autoequivalence group $ \mathop{\rm Auteq}(\mathrsfs{X}/\mathrsfs{Y}) $ is generated by twist functors $ T_{\mathcal{E}_{a,b}} $, line bundle tensorings, and the shift functor $[1]$, with $ 1 \leq a \leq b \leq n $.
• The generators $ T_{\mathcal{E}_{a,b}} $ correspond to loops around codimension-two hyperplanes in the stability space, and their action is shown to be non-trivial and independent of line bundle actions.
• The proof identifies $ \Phi_{a,b} = T_{\mathcal{E}_{a,b}} $ via commutative diagrams and the fact that the composition $ \Phi_{a,b}^{-1} \circ T_{\mathcal{E}_{a,b}} $ acts as identity on spherical objects, implying triviality up to line bundle twist.