[Paper Review] On nonexistence of semi-orthogonal decompositions in algebraic geometry

Xun Lin|arXiv (Cornell University)|Jul 20, 2021

Algebraic Geometry and Number Theory22 references4 citations

TL;DR

This paper establishes a new criterion for the nonexistence of nontrivial semi-orthogonal decompositions in derived categories of smooth projective varieties by introducing the intersection of base loci of line bundles numerically equivalent to the canonical bundle. It proves that if this intersection locus $Z = \mathsf{PBs}|\omega_X|$ is empty or zero-dimensional, then the derived category $D(X)$ is indecomposable. The result is applied to symmetric products of curves of genus $g \geq 2$ with $i \leq g-1$, and to certain elliptic surfaces and surfaces of general type, confirming indecomposability. An inequality on phases of skyscraper sheaves under Bridgeland stability is also derived.

ABSTRACT

The nonexistence of semi-orthogonal decompositions in algebraic geometry is known to be governed by the base locus of the canonical bundle. We study another locus, namely the intersection of the base loci of line bundles that are isomorphic to the canonical bundle in the Néron-Severi group, and show that it also governs the nonexistence of semi-orthogonal decompositions. As an application by using algebraically moving techniques, we prove that the bounded derived category of the $i$-th symmetric product of a smooth projective curve $C$ has no nontrivial semi-orthogonal decompositions when the genus $g(C)\geq 2$ and $i\leq g(C)-1$. We prove indecomposability of derived categories of some examples of elliptic surfaces with $P_{g}(X)=0$, and some natural examples of minimal surfaces of general type. Finally, an inequality involving phases of skyscraper sheaves for any Bridgeland stability condition is obtained.

Motivation & Objective

• To develop a new numerical criterion for the nonexistence of nontrivial semi-orthogonal decompositions in derived categories of smooth projective varieties.
• To generalize previous results by focusing on the intersection of base loci of line bundles in the Néron-Severi group rather than the full Picard group.
• To prove indecomposability of derived categories for symmetric products $S^iC$ of curves with $g(C) \geq 2$ and $i \leq g-1$, and for certain elliptic surfaces and surfaces of general type.
• To establish a universal inequality on the phase differences of HN factors of skyscraper sheaves under any Bridgeland stability condition.

Proposed method

• Introduce the locus $Z = \mathsf{PBs}|\omega_X| := \bigcap_{L \in \mathrm{Pic}^0(X)} \mathsf{Bs}|\omega_X \otimes L|$, the intersection of base loci of all line bundles numerically equivalent to $\omega_X$.
• Prove that any semi-orthogonal decomposition $D(X) = \langle \mathcal{A}, \mathcal{B} \rangle$ must have all objects in one component supported in $Z$, while skyscraper sheaves $k(x)$ for $x \notin Z$ must lie entirely in one component.
• Use algebraic moving techniques to analyze the base locus behavior under deformation in families, particularly for symmetric products and elliptic fibrations.
• Apply the theory to specific classes: symmetric products of curves, elliptic surfaces with $P_g = 0$, and minimal surfaces of general type with $P_g = q = 1$, $K^2 = 2$ or $3$, showing $Z$ is empty or zero-dimensional.
• Use the preservation of HN filtrations under tensoring with $\mathrm{Pic}^0$-line bundles to derive phase inequalities in Bridgeland stability conditions.
• Establish that for any $x \in X \setminus Z$, the phase difference between consecutive HN factors of $k(x)$ satisfies $\phi_i - \phi_{i+1} \leq \dim X - 1$.

Experimental results

Research questions

• RQ1 Under what conditions on the base locus of the canonical bundle does the derived category $D(X)$ admit no nontrivial semi-orthogonal decompositions?
• RQ2 Can the nonexistence criterion be extended beyond the base locus of $|\omega_X|$ to the intersection of base loci of all $\omega_X \otimes L$ for $L \in \mathrm{Pic}^0(X)$?
• RQ3 Does the derived category of the $i$-th symmetric product $S^iC$ of a curve $C$ of genus $g(C) \geq 2$ with $i \leq g(C)-1$ admit nontrivial semi-orthogonal decompositions?
• RQ4 Are the derived categories of minimal elliptic surfaces with $P_g = 0$ and certain surfaces of general type with $P_g = q = 1$ indecomposable?
• RQ5 What universal bounds exist on the phase differences of HN factors of skyscraper sheaves under any Bridgeland stability condition?

Key findings

• The derived category $D(S^iC)$ for a smooth projective curve $C$ with $g(C) \geq 2$ and $i \leq g(C)-1$ admits no nontrivial semi-orthogonal decompositions.
• For minimal elliptic surfaces $X$ of Kodaira dimension 1 with $P_g(X) = 0$, $q(X) = 1$, $g(C) = 1$, and $\deg L = 0$, the derived category $D(X)$ is indecomposable.
• The derived category of minimal surfaces of general type with $P_g = q = 1$ and $K^2 = 2$ or $3$ is indecomposable.
• The intersection base locus $Z = \mathsf{PBs}|\omega_X|$ governs indecomposability: if $Z$ is empty or zero-dimensional, then $D(X)$ has no nontrivial semi-orthogonal decompositions.
• For any Bridgeland stability condition on $D(X)$, the phase difference between consecutive HN factors of a skyscraper sheaf $k(x)$ with $x \notin Z$ satisfies $\phi_i - \phi_{i+1} \leq \dim X - 1$.
• The result shows that the numerical class of $\omega_X$ in $NS(X)$, rather than the full $Pic(X)$, controls derived category indecomposability, providing a bridge from numerical geometry to homological algebra.