[Paper Review] Stability conditions and extremal contractions

Yukinobu Toda|arXiv (Cornell University)|Apr 3, 2012

Algebraic Geometry and Number Theory28 references4 citations

TL;DR

This paper establishes that extremal contractions from smooth projective threefolds and surfaces arise as moduli spaces of Bridgeland (semi)stable objects in the derived category of coherent sheaves. Using a one-parameter family of stability conditions, it shows that the original variety $X$ and the target $Y$ of the extremal contraction are realized as fine or coarse moduli spaces of stable objects depending on the stability parameter, with the contraction occurring via wall-crossing at a critical stability condition corresponding to the pullback of an ample class from $Y$. The key contribution is a categorical realization of the first step of the Minimal Model Program via stability conditions.

ABSTRACT

We show that any extremal contraction from a smooth projective variety with dimension less than or equal to three appears as a moduli space of (semi)stable objects in the derived category of coherent sheaves.

Motivation & Objective

• To investigate whether extremal contractions in the Minimal Model Program (MMP) can be realized as moduli spaces of Bridgeland (semi)stable objects in the derived category of coherent sheaves.
• To establish a categorical interpretation of the first step of MMP for varieties of dimension ≤3 using stability conditions.
• To construct a one-parameter family of stability conditions such that the moduli space of stable objects transitions from $X$ to $Y$ under wall-crossing.
• To verify that the target variety $Y$ of an extremal contraction is corepresented by the moduli functor of semistable objects in the derived category.

Proposed method

• Construct a one-parameter family of Bridgeland stability conditions $\sigma_t = (Z_{f^*\omega + \varepsilon t D}, \mathcal{P}_t)$ for $t \in (-1,1)$, where $f: X \to Y$ is an extremal contraction.
• Define the central charge $Z_{f^*\omega}(E) = -\int_X e^{-i f^*\omega} \mathop{\rm ch}\nolimits(E)$, linking the stability condition to the pullback of an ample divisor on $Y$.
• Use tilting of the perverse heart $\mathop{\rm Per}\nolimits(X/Y)$ to define the heart $\mathcal{B}_{f^*\omega}$ at $t=0$, corresponding to the boundary of the geometric chamber.
• Show that for $t < 0$, $X$ is the fine moduli space of $\sigma_t$-stable objects with $\mathop{\rm ch}\nolimits(E) = \mathop{\rm ch}\nolimits(\mathcal{O}_x)$, while for $t > 0$, $Y$ becomes the fine moduli space.
• For $t = 0$, $Y$ is the coarse moduli space of $S$-equivalence classes of semistable objects.
• Use corepresentation of functors via $\mathop{\rm Hom}(\ast, Z)$ to show that $Y$ (or $\widetilde{Y}$ when $f(D)$ is a curve) corepresents the moduli functor of semistable objects.

Experimental results

Research questions

• RQ1 Can every extremal contraction $f: X \to Y$ of a smooth projective variety of dimension ≤3 be realized as a moduli space of Bridgeland-stable objects in $D^b\mathop{\rm Coh}\nolimits(X)$?
• RQ2 Does wall-crossing in the space of stability conditions correspond to the geometric transition from $X$ to $Y$ in the MMP?
• RQ3 Is the target variety $Y$ of an extremal contraction corepresented by the moduli functor of semistable objects in the derived category?
• RQ4 What is the role of the geometric chamber in the space of stability conditions, and how does its boundary relate to the pullback of the ample cone of $Y$?
• RQ5 How does the moduli space of stable objects evolve as the stability condition crosses the wall at $t=0$?

Key findings

• For surfaces, the one-parameter family $\sigma_t$ with $t \in (-1,1)$ realizes $X$ as the fine moduli space for $t < 0$, $Y$ as the coarse moduli space for $t = 0$, and $Y$ as the fine moduli space for $t > 0$, with the transition occurring at $t=0$.
• The stability condition $\sigma_0 = (Z_{f^*\omega}, \mathcal{B}_{f^*\omega})$ lies at the boundary of the geometric chamber and corresponds to the pullback of the ample cone of $Y$ to $X$, linking the nef cone to the space of stability conditions.
• When $f(D)$ is a point, $Y$ corepresents the moduli functor $\mathcal{M}^{\sigma_{B,f^*\omega}}([\mathcal{O}_x])$, and the morphism $Y \to Z$ is induced via the corepresentation property.
• When $f(D)$ is a curve, the moduli space is realized as $\widetilde{Y} = Y \cup (f(D) \times f(D))$, glued along the diagonal, and $\widetilde{Y}$ corepresents the moduli functor.
• The moduli space for $t > 0$ is expected to be isomorphic to $Y$, though the explicit description of the stability conditions $\sigma_t$ for $t > 0$ remains conjectural.
• The construction suggests that the entire MMP step can be understood as wall-crossing in the space of stability conditions, with the derived category encoding the birational geometry.