[Paper Review] Bridgeland Stability conditions on threefolds II: An application to Fujita's conjecture

Arend Bayer, Aaron Bertram+2 more|arXiv (Cornell University)|Jun 17, 2011

Algebraic Geometry and Number Theory9 references11 citations

TL;DR

This paper applies a conjectured Bogomolov-Gieseker-type inequality for tilt-stable objects in the derived category of a threefold to prove a Reider-type theorem in dimension three, which implies Fujita's conjecture for threefolds: $K_X + 6L$ is very ample when $L$ is ample, and $5L$ is very ample when $K_X$ is trivial, under the assumption of the conjectured inequality on third Chern classes.

ABSTRACT

We apply a conjectured inequality on third chern classes of stable two-term complexes on threefolds to Fujita's conjecture. More precisely, the inequality is shown to imply a Reider-type theorem in dimension three which in turn implies that K_X + 6L is very ample when L is ample, and that 5L is very ample when K_X is trivial.

Motivation & Objective

• To extend Reider’s method for adjoint line bundles from surfaces to threefolds using derived categories and Bridgeland stability.
• To establish a Reider-type criterion for global generation and very ampleness of adjoint line bundles on smooth projective threefolds.
• To show that the conjectured Bogomolov-Gieseker inequality on third Chern classes implies Fujita’s conjecture in dimension three.
• To provide effective numerical bounds on the adjoint line bundle $K_X \otimes L^m$ for global generation and very ampleness.
• To demonstrate that the conjecture on third Chern classes is consistent with known results in the literature, such as those by Ein-Lazarsfeld and Kawamata.

Proposed method

• Use of tilt-stability in the derived category $\mathrm{D}^b(X)$ for threefolds, defined via a pair of parameters $\omega$ and $B$ in $\mathrm{NS}_{\mathbb{Q}}(X)$.
• Reduction of the vanishing of $H^1(X, K_X \otimes L \otimes I_Z)$ to the non-existence of certain stable objects in the heart $\mathcal{B}_{\omega,B}$.
• Application of a conjectured Bogomolov-Gieseker inequality for third Chern classes of stable two-term complexes in $\mathcal{B}_{\omega,B}$.
• Construction of a non-trivial extension $\mathcal{O}_X[1] \to E \to L \otimes I_Z \to \mathcal{O}_X[2]$ to derive a contradiction under the conjecture.
• Use of Serre duality and cohomological vanishing to relate the non-vanishing of $H^1(X, K_X \otimes L \otimes I_Z)$ to the existence of a destabilizing subobject in $\mathcal{B}_{\omega,B}$.
• Computation of Chern characters and genus formulas via Hirzebruch-Riemann-Roch to derive constraints on curve classes and normal bundles.

Experimental results

Research questions

• RQ1 Can Reider’s method for proving Fujita’s conjecture on surfaces be generalized to threefolds using derived categories and stability conditions?
• RQ2 Does the conjectured Bogomolov-Gieseker inequality on third Chern classes of tilt-stable objects imply Fujita’s conjecture in dimension three?
• RQ3 What numerical bounds on $L^3$, $L^2.D$, and $L.C$ ensure that $K_X \otimes L^m$ is globally generated or very ample?
• RQ4 How does the conjectured inequality relate to known results such as those by Ein-Lazarsfeld and Kawamata on adjoint line bundles?
• RQ5 Can the conjecture be verified in special cases, such as when $K_X$ is numerically trivial or when $Z$ has length one?

Key findings

• Assuming the conjectured Bogomolov-Gieseker inequality on third Chern classes, $H^1(X, K_X \otimes L \otimes I_Z) = 0$ holds for all zero-dimensional subschemes $Z$ of length $\alpha$ under conditions (A), (B), and (C) on $L$.
• The conjecture implies that $K_X \otimes L^m$ is globally generated for $m \geq 4$, and for $m \geq 3$ if $L^3 \geq 2$, under the same assumptions.
• The conjecture implies that $K_X \otimes L^m$ is very ample for $m \geq 6$, and for $m \geq 5$ if $K_X$ is trivial or $K_X.C$ is even for all curves $C \subset X$.
• For $\alpha = 2$, the conditions (A), (B), and (C) yield a Reider-type criterion ensuring that $K_X \otimes L^2$ is very ample, provided the restriction to degree one curves is very ample.
• The paper shows that the conjecture is consistent with known results: when $H^1(X, K_X \otimes L \otimes I_x) \neq 0$, the conjecture holds for the corresponding extension objects, providing evidence for its validity.
• In the case of a rational curve $C$ with $L.C = 1$, the conjecture leads to a contradiction unless $K_X.C$ is even, supporting the necessity of the parity condition in the main theorem.