Autoequivalences of Derived Categories of Bielliptic Surfaces

Abstract: We determine the generators of the autoequivalence group of the derived category of coherent sheaves on a bielliptic surface over an algebraically closed field of arbitrary characteristic. As a consequence, we prove that any algebraic variety derived equivalent to such a surface is isomorphic to the surface itself.

• The paper determines that the autoequivalence group of a bielliptic surface’s derived category is generated by standard autoequivalences and relative Fourier–Mukai transforms along its two elliptic fibrations.
• It rigorously proves that any derived equivalence between a bielliptic surface and another variety forces an isomorphism, confirming the rigidity of these surfaces.
• The analysis leverages explicit computations on the numerical Chow group and even lattice structures, providing clear classification-theoretic implications and derived Torelli insights.

Autoequivalences of Derived Categories of Bielliptic Surfaces

Introduction and Context

The study of autoequivalences of the bounded derived category of coherent sheaves, $\Auteq D(X)$, on algebraic varieties is central in modern algebraic geometry, revealing deep structural and categorically intrinsic properties. The subgroup generated by automorphisms, twists by line bundles, and shifts of the derived category (the "standard autoequivalences") often encapsulates significant symmetries of $X$. However, for a variety with trivial or numerically trivial canonical bundle, $\Auteq D(X)$ can be much richer, admitting non-standard autoequivalences of Fourier–Mukai type. The present work determines $\Auteq D(X)$ for bielliptic surfaces over an algebraically closed field, in arbitrary characteristic, providing both an explicit description of generators and a classification-theoretic consequence: rigidity of bielliptic surfaces up to derived equivalence.

A bielliptic surface is a minimal projective surface with $K_X \equiv 0$, $b_2=2$, and Albanese fibration with smooth elliptic fibers. These surfaces have two intrinsic elliptic fibrations, and possess canonical covers that are abelian or, in positive characteristic, bielliptic surfaces themselves. Their derived categories display a tapestry of standard and non-standard autoequivalences, with the latter associated to relative Fourier–Mukai transforms along the elliptic structures.

Main Results

Generators of the Autoequivalence Group

The principal theorem establishes that for any bielliptic surface $X$ over an algebraically closed field of arbitrary characteristic, the autoequivalence group $\Auteq D(X)$ is generated by the standard autoequivalences together with the relative Fourier–Mukai transforms associated to the two elliptic fibrations:

$\Auteq D(X) = \langle \, \Pic(X),\ \Aut(X),\ [1],\ \text{relative FM transforms along } f_1, f_2 \, \rangle.$

Explicitly, the non-standard generators are realized via relative Fourier–Mukai transforms, constructed along the elliptic fibrations $f_1$ and $X$0, modulating torsion data tailored to the fibration structure and respecting the lattice of numerical classes specific to bielliptic surfaces. This comprehensive description unifies the cyclic and non-cyclic types, and transcends the restriction to characteristic zero previously given in the literature [potter2017derived].

Moreover, the action on the numerical Chow group $X$1 is described via an explicit short exact sequence: $X$2 where $X$3 are explicit congruence subgroups of $X$4 determined by intersection-theoretic data, and where the tensor structure encodes the bielliptic fibration product.

Rigidity Under Derived Equivalence

A second major consequence is a strong rigidity statement: If $X$5 is an algebraic variety over an algebraically closed field (possibly of positive characteristic) such that $X$6 as triangulated categories, then $X$7 as varieties; that is, bielliptic surfaces are rigid in their derived equivalence class. No non-trivial Fourier–Mukai partners exist for $X$8: $X$9 This holds regardless of characteristic. In particular, this extends previous results in characteristic zero and characteristic $\Auteq D(X)$0 to all characteristics (including the difficult cases of $\Auteq D(X)$1) by explicit intrinsic analysis of the autoequivalence group and the structure of their actions [MR4247995].

Methods and Technical Contributions

Structure of $\Auteq D(X)$2 and Intersection Theory

A substantial technical component involves the explicit computation of the numerical equivalence group $\Auteq D(X)$3 for bielliptic surfaces, producing a basis associated to the fibrations and their fiber multiplicities. The work gives precise formulas for the generators in terms of intersection numbers, and proves that $\Auteq D(X)$4 is always an even lattice.

Relative Fourier–Mukai Transforms

The construction and analysis of the relative Fourier–Mukai transforms leverages the classification of moduli spaces $\Auteq D(X)$5 of stable vector bundles of fixed rank and fiber degree along the elliptic fibrations. These moduli spaces are identified, up to isomorphism, with $\Auteq D(X)$6, allowing the construction of autoequivalences by universal families and matrix parametrization. The explicit computation of their induced action on numerical invariants is derived, and their matrix representations are obtained.

Canonical Covers and Characteristic Considerations

The proof addresses complications arising from the behavior of canonical covers in arbitrary characteristic (including situations where the cover is itself bielliptic), and systematically generalizes the lifting properties and the action of functors between derived categories under finite covers, extending Bridgeland-Maciocia’s and Orlov’s theoretical frameworks [MR3713877, MR1998775]. The approach employs homological algebra, the structure of semi-homogeneous sheaves, and the non-existence of rigid or exceptional sheaves (especially those supported on effective divisors with negative self-intersection), facilitated by results on positivity (e.g., the Bogomolov-Gieseker inequality in positive characteristic).

The No Non-Trivial FM Partners Theorem

By showing that any triangulated equivalence $\Auteq D(X)$7 is, up to composition with appropriate relative Fourier–Mukai transforms, equivalent to an isomorphism followed by twist, the work essentially categorically reconstructs $\Auteq D(X)$8 from $\Auteq D(X)$9. The analysis exploits the standard criterion for when a derived equivalence maps skyscraper sheaves to sheaves associated to closed points, and applies rigidification results to conclude the isomorphism type [MR2244106].

Numerical Results and Contradictory Claims

• Explicit Identification: The paper describes, in terms of explicit congruence conditions and matrix multiplication, all autoequivalences of $\Auteq D(X)$0, with all non-standard ones being accounted for by relative Fourier–Mukai transforms.
• Rigidity: It is explicitly proved that no non-trivial derived equivalence exists (in any characteristic), strengthening and generalizing previously known results.
• Even Lattice Structure: The numerical groups $\Auteq D(X)$1 are always even, and "evenness" is essential for the argument that eliminates the possibility of rigid torsion sheaves and exceptional objects, which would otherwise complicate the structure of the group.

Implications and Future Directions

The characterization of $\Auteq D(X)$2 for bielliptic surfaces enriches the broader understanding of the relationship between surface geometry and categorical symmetries. The explicit description of generators contributes to the classification of varieties by their derived categories, illuminating the "derived Torelli" phenomena for surfaces with $\Auteq D(X)$3. The rigidity of bielliptic surfaces among their derived equivalence classes parallels results for abelian and K3 surfaces, but with an even stricter outcome: bielliptic surfaces have no FM partners, trivial or otherwise, across all characteristics.

Possible extensions include generalizations to families of $\Auteq D(X)$4-trivial surfaces in positive characteristic with more complicated canonical covers, analysis of autoequivalence groups for quasi-bielliptic or generalized hyperelliptic surfaces, and understanding derived invariants under nontrivial group actions in mixed characteristic settings. The computational methodology and use of intersection theory, Mukai lattices, and integral transforms may be adapted to higher-dimensional analogues, enriched moduli, or in the framework of derived categories with enhancements (dg categories and stability conditions).

Conclusion

This work offers a complete, explicit description of the group of autoequivalences of the derived category of a bielliptic surface in arbitrary characteristic and demonstrates that such surfaces are rigid in their derived category: any derived equivalent variety is isomorphic to the original. The analysis manages torsion phenomena, fiber structures, canonical covers, and the interplay of algebraic and categorical symmetries, reinforcing the role of the derived category as an invariant of high fidelity for surfaces with trivial canonical class.

Reference: "Autoequivalences of Derived Categories of Bielliptic Surfaces" (2603.29471).