Derived isogenies and isogenies for abelian surfaces

Technical Summary: Derived Isogenies and Isogenies for Abelian Surfaces

Problem Statement
The paper addresses the classification of Fourier–Mukai partners for abelian surfaces, specifically focusing on the concept of "derived isogeny." Two abelian varieties are defined as derived isogenous if they can be connected by a sequence of derived equivalences between twisted abelian varieties (Definition 1.1.1). While Stellari [66] established that derived isogenous complex abelian surfaces are isogenous, the converse is false: isogenous surfaces are not necessarily derived isogenous. The central problem is to provide a cohomological and geometric classification of derived isogenies between abelian surfaces over algebraically closed fields of arbitrary characteristic, distinguishing them from standard isogenies.

Methodology
The authors employ a multi-faceted approach combining Hodge theory, $p$-adic Hodge theory, and the geometry of moduli spaces of twisted sheaves:

1. Generalization of Shioda's Trick: A core methodological innovation is the extension of Shioda's trick (originally for integral Hodge structures on complex abelian surfaces) to rational Hodge structures, $\ell$-adic Tate modules, and $F$-crystals. This involves defining "admissible bases" and "admissible isomorphisms" for cohomology groups across different characteristics. The authors utilize the spinor norm and the Cartan–Dieudonné decomposition to relate isometries of second cohomology groups ($H^2$) to isogenies of the underlying abelian surfaces.
2. Twisted Derived Equivalences: The paper utilizes the theory of twisted derived categories $D^b(X, \alpha)$ and their realization via Fourier–Mukai transforms. It establishes that derived equivalences between twisted abelian surfaces correspond to isomorphisms of twisted Mukai lattices.
3. Lifting and Specialization: For positive characteristic, the authors employ a lifting-specialization technique. They lift twisted abelian surfaces and derived equivalences from characteristic $p$ to characteristic 0 (using discrete valuation rings and the Kummer construction to relate abelian surfaces to K3 surfaces). They then apply results from characteristic 0 and specialize them back to characteristic $p$.
4. Moduli Spaces of Twisted Sheaves: The paper relies heavily on the correspondence between twisted derived equivalences and moduli spaces of stable twisted sheaves (Theorem 3.5.3), extending results by Minamide–Yanagida–Yoshioka and Bragg–Lieblich to include supersingular cases via Ogus's crystalline Torelli theorem.

Key Contributions and Results

• Twisted Derived Torelli Theorem (Characteristic 0):
  The primary result for characteristic 0 (Theorem 1.2.1) establishes that two abelian surfaces $X$ and $Y$ over an algebraically closed field of characteristic 0 are derived isogenous if and only if they are principally isogenous. A principal isogeny is defined as an isogeny of square degree. This equivalence is further linked to:

  • The derived isogeny of their associated Kummer surfaces.
  • The isomorphism of their Chow motives (specifically the even-degree parts) as Frobenius exterior algebras.
  • The existence of a rational Hodge isometry between their second cohomology groups (or Mukai lattices).

• Generalized Shioda's Trick and Tate Conjecture:
  The authors prove a generalized Shioda's trick (Theorem 1.3.1): for any admissible Hodge isometry $\psi: H^2(X, \mathbb{Q}) \to H^2(Y, \mathbb{Q})$, there exists an isogeny $f: Y \to X$ of square degree such that $\psi = f^*/d$. This result is extended to $\ell$-adic and $p$-adic cohomology, providing a proof of the Tate conjecture for isometries between the second cohomology groups of abelian surfaces over finitely generated fields (Corollary 4.6.3).

• Results in Positive Characteristic:
  For fields of characteristic $p > 3$, the paper proves a "prime-to-$p$" version of the Torelli theorem (Theorem 1.4.1). Two abelian surfaces are prime-to-$p$ derived isogenous if and only if they are prime-to-$p$ principally isogenous.

  • The proof utilizes the lifting of prime-to-$p$ derived equivalences to characteristic 0 and the specialization of moduli spaces of twisted sheaves.
  • The paper addresses the supersingular case separately, showing that if $X$ is supersingular, $Y$ is derived isogenous to $X$ if and only if $Y$ is also supersingular.
  • The authors note that the equivalence for non-prime-to-$p$ derived isogenies remains open due to obstructions in the specialization of Brauer classes of order $p$ (Remark 6.2.2).

• Kummer Surfaces:
  The paper investigates the relationship between derived isogenies of abelian surfaces and their associated Kummer surfaces. It proves that if $X$ and $Y$ are prime-to-$p$ derived isogenous, then $Km(X)$ and $Km(Y)$ are also prime-to-$p$ derived isogenous (Theorem 6.5.1). The converse is partially established under specific conditions regarding the orders of the twisting Brauer classes.

Significance and Claims
The paper claims to provide a complete cohomological and geometric classification of derived isogenies for abelian surfaces in characteristic 0 and a partial classification (restricted to prime-to-$p$ cases) in positive characteristic.

• Unification: It unifies the geometric notion of "principal isogeny" (isogenies of square degree) with the categorical notion of "derived isogeny," showing they are equivalent conditions for abelian surfaces.
• Extension of Classical Results: The work generalizes Mukai and Orlov's derived Torelli theorem (which relates derived equivalence to Hodge isometry of $H^2$) to the twisted setting and to positive characteristic.
• Methodological Advance: The generalization of Shioda's trick to rational and $p$-adic contexts is presented as a key technical tool that allows the authors to bypass the lack of a global Torelli theorem in positive characteristic by linking $H^2$ isometries directly to isogenies.
• Modesty on Open Problems: The authors explicitly state that the equivalence between derived isogeny and isogeny for K3 surfaces in positive characteristic remains a conjecture (Conjecture 1.4.2), and the full equivalence for non-prime-to-$p$ derived isogenies of abelian surfaces is obstructed by the specialization of Brauer classes (Conjecture 6.4.2). They frame their results on Kummer surfaces as providing evidence for the K3 conjecture rather than a full proof.

The paper does not propose new experimental applications but rather establishes foundational theoretical connections between derived categories, Hodge theory, and arithmetic geometry for abelian surfaces.