$ \newcommand\Pic{\mathrm{Pic}} \newcommand\nil{\mathrm{nil}} \newcommand\gfrak{\mathfrak{g}} \newcommand\A{\mathrm{A}} \newcommand\B{\mathrm{B}} \newcommand\C{\mathrm{C}} \newcommand\D{\mathrm{D}} \newcommand\E{\mathrm{E}} \newcommand\F{\mathrm{F}} \newcommand\G{\mathrm{G}} \newcommand\H{\mathrm{H}} \newcommand\h{\mathrm{h}} \newcommand\K{\mathrm{K}} \newcommand\L{\mathrm{L}} \newcommand\M{\mathrm{M}} \newcommand\R{\mathrm{R}} \newcommand\t{\mathrm{t}} \newcommand\T{\mathrm{T}} \newcommand{\bA}{\mathbf{A}} \newcommand{\bF}{\mathbf{F}} \newcommand{\bG}{\mathbf{G}} \newcommand{\bH}{\mathbf{H}} \newcommand{\bP}{\mathbf{P}} \newcommand{\bT}{\mathbf{T}} \newcommand{\bW}{\mathbf{W}} \newcommand{\Gm}{\bG_m} \newcommand\Ascr{\mathcal{A}} \newcommand\Cscr{\mathcal{C}} \newcommand\Dscr{\mathcal{D}} \newcommand\Escr{\mathcal{E}} \newcommand\Fscr{\mathcal{F}} \newcommand\Kscr{\mathcal{K}} \newcommand\Lscr{\mathcal{L}} \newcommand\Oscr{\mathcal{O}} \newcommand\Perf{\mathrm{Perf}} \newcommand\Perfscr{\mathcal{P}\mathrm{erf}} \newcommand\Acscr{\mathcal{A}\mathrm{c}} \newcommand\heart{\heartsuit} \newcommand\cn{\mathrm{cn}} \newcommand\op{\mathrm{op}} \newcommand\gr{\mathrm{gr}} \newcommand\Gr{\mathrm{Gr}} \newcommand\fil{\mathrm{fil}} \newcommand\Ho{\mathrm{Ho}} \newcommand\dR{\mathrm{dR}} \newcommand\dRhat{\widehat{\dR}} \newcommand\we{\simeq} \newcommand\Sym{\mathrm{Sym}} \newcommand\HH{\mathrm{HH}} \newcommand\HC{\mathrm{HC}} \newcommand\HP{\mathrm{HP}} \newcommand\TC{\mathrm{TC}} \newcommand\TR{\mathrm{TR}} \newcommand\THH{\mathrm{THH}} \newcommand{\bMap}{\mathbf{Map}} \newcommand{\End}{\mathrm{End}} \newcommand{\Mod}{\mathrm{Mod}} \newcommand{\coMod}{\mathrm{coMod}} \newcommand{\Fun}{\mathrm{Fun}} \newcommand{\bMap}{\mathbf{Map}} \newcommand\bE{\mathbf{E}} \newcommand\bZ{\mathbf{Z}} \newcommand\bS{\mathbf{S}} \newcommand\bQ{\mathbf{Q}} \newcommand\bC{\mathbf{C}} \newcommand\bN{\mathbf{N}} \newcommand\bAM{\mathbf{AM}} \newcommand\bLM{\mathbf{LM}} \newcommand\Spec{\mathrm{Spec}\,} \newcommand\CAlg{\mathrm{CAlg}} \newcommand\aCAlg{\mathfrak{a}\CAlg} \newcommand\dCAlg{\mathfrak{d}\CAlg} \newcommand{\Cat}{\mathrm{Cat}} \newcommand{\Sscr}{\mathcal{S}} \newcommand{\poly}{\mathrm{poly}} \newcommand{\perf}{\mathrm{perf}} \newcommand\Sp{\mathrm{Sp}} \newcommand\CycSp{\mathrm{CycSp}} \newcommand\TCart{\mathrm{TCart}} \newcommand\Fr{\mathrm{Fr}} \newcommand\Br{\mathrm{Br}} $

The recent paper [3] Hodge numbers are not derived invariants in positive characteristic by Nick Addington and Daniel Bragg answers several open questions about the behavior of Hodge numbers under derived equivalence in positive characteristic. In characteristic $0$, the Hodge numbers are derived invariants up through dimension $3$ by Popa and Schnell [7]. In characteristic $p$, Hodge numbers are derived invariant up through dimension $2$ by my work with Bragg [4] and we showed in that paper that in dimension $3$ the numbers $\chi(\Omega^i)$ are derived invariants.

The authors construct a pair of smooth projective Calabi–Yau $3$-folds $M$ and $X$ over $\overline{\bF}_3$ such that

  • $\D^b(X)\we\D^b(M)$ and
  • $h^{i,j}(X)\neq h^{i,j}(M)$

for some pairs $(i,j)$, where $h^{i,j}(X)=\dim\H^j(X,\Omega^i_X)$.

Important note: Daniel Bragg will be on the job market this year! Just saying.

In fact, their theorem shows that the dimensions $h^j(\Oscr)$ are not derived invariants. In their paper, \(\dim\H^*(X,\Oscr_X)=1\quad 0\quad 0\quad 1\) while \(\dim\H^*(M,\Oscr_M)=1\quad 1\quad 1\quad 1.\) The variety $X$ is a small resolution of a specific intersection of two cubics in $\bP^5$ and it admits an abelian surface fibration $X\rightarrow\bP^1$. The variety $M$ is a compactification of the relative Picard sheaf $\Pic^0_{X/\bP^1}$ at the smooth fibers. Although it takes a lot of work thanks to the presence of reducible fibers, Addington and Bragg show that the usual Mukai-style derived equivalence between an abelian variety and its dual extends to the families $X$ and $M$ over $\bP^1$, so that $\D^b(X)\we\D^b(M)$.

The Hodge numbers of $X$ are determined by a computer algebra system, while those of $M$ are done in several steps. First, the derived invariance of Hochschild homology and the degeneration from [5] of the HKR spectral sequence for $3$-folds in characteristic $3$ shows that certain sums of Hodge numbers are derived invariant. Serre duality and the fact that $\omega_M\cong\Oscr_M$ is enough to resolve everything but the terms $h^{1,0}(M)$ and $h^{1,2}(M)$ which sum to $7$. A Dieudonn'e-module theoretic argument using a theorem of Oda fixes the Hodge number $\dim\H^0(M,\Omega^1_M)=1$ which settles everything.

After this, the paper contains three additional interesting sections.

  • Section 6 gives some information on the crystalline cohomology of $X$ and $M$.
  • Appendix A contains a nice proof of a result of Abuaf [1] that shows that $h^*(\Oscr)$ is a derived invariant in characteristic $0$ up through dimension $4$.
  • Appendix B, written by Sasha Petrov, gives higher-dimensional examples of failure of derived invariants of Hodge numbers in any characteristic.

Petrov’s examples are notable and start from his work in [6] on failure of Hodge symmetry for abeloid varieties.


[1] Abuaf, Homological units, IMRN 22 (2017), 6943-6960. arXiv:1510.01583.

[2] Addington and Bragg, Hodge numbers are not derived invariants in positive characteristic, arXiv:2106.09949.

[4] Antieau and Bragg, Derived invariants from topological Hochschild homology, arXiv:1906.12267.

[5] Antieau and Vezzosi, A remark on the HKR theorem in characteristic $p$, Ann. Sc. Norm. Super. Pisa Cl. Sci (5) 20 (2020), no. 3, 1135-1145. arXiv:1710.06039.

[6] Petrov, Rigid-analytic varieties with projective reduction violating Hodge symmetry, arXiv:2005.02226.

[7] Popa and Schnell, Derived invariance of the number of holomorphic $1$-forms and vector fields, ASENS (4) 44 (2011), no. 3, 527-536. arXiv:0912.4040.