Hodge numbers are derived invariants for complex fourfolds

In July 2026 several members of my FRG grant met at UC Berkeley to work. Several participants were coming directly from the ICM and AI was an impossible-to-avoid topic. Akhil Mathew and I had been playing around extensively with these tools for quite a while, but not everyone had, so we decided to brainstorm a list of problems related to our grant and pose them as problems to Codex and Claude Code. The most surprising result is a Codex proof that Hodge numbers are derived invariants in dimension $4$, the proof of which we present below.

The Leiden Declaration asserts the primacy of human authors. To that end, and since this is my blog, I am the author of this blog post. Full credit to humans and models is given toward the end of this post.

Background and statement of theorem

Let $X$ and $Y$ be smooth projective varieties over $\mathbf C$. Orlov’s conjecture [1] asserts that an equivalence

\[D^b(X)\simeq D^b(Y)\]

implies that the rational Chow motives of $X$ and $Y$ are isomorphic, which in turn implies $h^{p,q}(X)=h^{p,q}(Y)$ for all $p,q$.

Popa and Schnell proved that Hodge numbers are derived invariants for complex threefolds [2]. The characteristic-zero hypothesis is essential: Addington and Bragg constructed derived-equivalent Calabi–Yau threefolds in characteristic $3$ with different Hodge numbers [3].

Theorem A. If $X$ and $Y$ are derived-equivalent smooth projective complex fourfolds, then

\[h^{p,q}(X)=h^{p,q}(Y)\]

for every $p,q$.

Reductions

We can assume that $X$ and $Y$ are connected and of dimension $4$. We have Serre duality, which implies that $h^{p,q}=h^{4-p,4-q}$, and Hodge symmetry, which implies that $h^{p,q}=h^{q,p}$.

Popa and Schnell in [2] prove that $h^{0,1}$ is a derived invariant (for $X$ and $Y$ of any dimension, in characteristic $0$).

In their appendix, Addington and Bragg prove that, in dimensions at most four and in characteristic $0$, derived equivalence preserves the Hodge numbers $h^{0,q}$. In fact, their argument gives an alternative proof of the Popa–Schnell result in this case.

A final classical argument is to use the derived invariants of Hochschild homology together with HKR degeneration. Specifically, $\dim_k\HH_i(X/k)$ is a derived invariant and we have by HKR that

\[\dim_k\HH_i(X/k)=\sum_j h^{i+j,j}.\]

Taken together, the facts above imply that for fourfolds in characteristic $0$ the only ambiguity is what happens in $\HH_0$. Of course, $h^{0,0}$ and $h^{4,4}$ are derived invariants and we have $h^{1,1}=h^{3,3}$. By Hochschild homology, $2h^{1,1}+h^{2,2}$ is a derived invariant.

It remains to find an additional relation that lets us separate $h^{1,1}$ from $h^{2,2}$.

Topological signature

The main insight due to GPT 5.6 Sol is that the topological signature is a derived invariant of even-dimensional smooth projective complex varieties. If $X$ is a $2d$-dimensional smooth projective complex variety, then the cup product pairing on $\H^{2d}(X;\mathbf{R})$ is symmetric and non-degenerate. The topological signature of $X$, denoted here by $\sigma(X)$, is the signature of this pairing. Hirzebruch’s signature formula gives \(\sigma(X)=\sum_{p,q}(-1)^qh^{p,q}.\) See Voisin [6], Theorem 6.33.

Theorem B. Let $X$ and $Y$ be derived equivalent smooth projective complex varieties of even dimension. Then, $\sigma(X)=\sigma(Y)$.

Theorem B implies Theorem A. By our previous reductions, we have

\[0=\sigma(X)-\sigma(Y)=(h^{2,2}(X)-2h^{1,1}(X))-(h^{2,2}(Y)-2h^{1,1}(Y)).\]

Thus, the quantity $h^{2,2}-2h^{1,1}$ is a derived invariant. Together with the derived invariance of $2h^{1,1}+h^{2,2}$, this implies the derived invariance of $h^{1,1}$ and $h^{2,2}$, which completes the proof of Theorem A.

The Mukai pairing and the proof of Theorem B.

The Mukai pairing on $\HH_0(X/\bC)$ is induced by the functor $D^b(X)^\op\times D^b(X)\rightarrow D^b(\bC)$ given by taking mapping complexes together with the fact that $\HH(D^b(X)^\op/\bC)\we\HH(D^b(X)/\bC)$. Using topological $K$-theory one can also obtain a version of this pairing on the direct sum of even rational or complex cohomology groups of $X$ using the Chern character. The Mukai pairing is a derived invariant, by construction.

In this form, if $\alpha=\sum\alpha_j$ is an element of $\H^\mathrm{ev}(X;\bC)=\oplus_j\H^{2j}(X;\bC)$, then let

\[\alpha^\vee=\sum_j (-1)^j \alpha_j.\]

In terms of $\H^\mathrm{ev}(X;\bC)$, the Mukai pairing is

\[P_X(\alpha,\beta) =\int_X e^{c_1(X)/2}\alpha^\vee\beta.\]

see Căldăraru and Willerton [5] or Huybrechts [4], Propositions 5.39 and 5.44.

We can symmetrize the Mukai pairing to obtain

\[P_X(\alpha,\beta)+P_X(\beta,\alpha)=\int_X e^{c_1(X)/2}\alpha^\vee\beta+e^{c_1(X)/2}\beta^\vee\alpha.\]

Since the involution $(-)^\vee$ acts by $(-1)^{2d}=1$ on top cohomology,

\[\int_X e^{c_1(X)/2}\beta^\vee\alpha=\int_X e^{-c_1(X)/2}\alpha^\vee\beta,\]

so that

\[S_X(\alpha,\beta):=P_X(\alpha,\beta)+P_X(\beta,\alpha) =2\int_X\mathrm{cosh}(c_1(X)/2)\alpha^\vee\beta.\]

Put

\[Q_X(\alpha,\beta)=\int_X\alpha^\vee\beta \qquad\text{and}\qquad B_X=\sqrt{\mathrm{cosh}(c_1(X)/2)}.\]

Since $B_X^\vee=B_X$, we have

\[S_X(\alpha,\beta)=2Q_X(B_X\alpha,B_X\beta).\]

Multiplication by $B_X$ is a real automorphism, and the factor $2$ is positive, so $S_X$ and $Q_X$ have the same signature. For $j<d$, the restriction of $Q_X$ to

\[\H^{2j}(X;\mathbf{R})\oplus\H^{4d-2j}(X;\mathbf{R})\]

is hyperbolic by Poincaré duality and therefore has signature zero. On the middle cohomology $\H^{2d}(X;\mathbf{R})$, the pairing $Q_X$ is $(-1)^d$ times the ordinary cup product pairing. Consequently,

\[\operatorname{sig}(S_X)=(-1)^d\sigma(X).\]

A derived equivalence preserves $P_X$, hence also $S_X$, and it preserves the dimension. It therefore preserves $\sigma(X)$. This completes the proof of Theorem B.

Comments on the process

I provided the following prompt.

There is a conjecture that if X and Y are derived equivalent smooth projective varieties over the complex numbers, then they have the same Hodge numbers. Can you find a counterexample to this conjecture (or prove it)? If so, please write this up as a TeX/PDF file for me to read.

This resulted in a summary after 20 minutes of the state of the art. I asked the model to keep pushing. After an additional 1 hour and 50 minutes it had solved dimension $4$, but hadn’t stopped. It was working on the general case. I paused it and asked for the proof in dimension $4$, which it provided here, after some back-and-forth on the correctness of some results in the literature (a paper it cited on the arXiv had been withdrawn).

As you can see from the prompt, I expect this to be false in general. I was pleasantly surprised to see a proof in dimension $4$.

The original output included a formalization, but it was very incomplete, just formalizing Theorem A from Theorem B and all of the easy reduction steps. Theorem B itself was not formalized. I have no doubt this could have been done; however, the humans were able to check the proof directly.

Thanks to Bryna Kra for suggesting I include the exact details here.

Contributions and support

The following uses the CRediT contributor-role taxonomy.

• GPT-5.6 Sol Extra High (OpenAI): Conceptualization, Formal analysis, Investigation, Methodology, Validation, Writing – original draft.
• Opus 4.8 (Anthropic): Validation.
• Benjamin Antieau (Northwestern): Formal analysis, Funding acquisition, Validation, Writing – review & editing.
• Andrei Căldăraru (Wisconsin): Formal analysis, Funding acquisition, Validation, Writing – review & editing.
• Akhil Mathew (Chicago): Formal analysis, Funding acquisition, Validation, Writing – review & editing.
• Martin Olsson (Berkeley): Formal analysis, Funding acquisition, Validation, Writing – review & editing.
• Ruoxi Li (Berkeley): Formal analysis, Validation, Writing – review & editing.
• Noah Olander (Berkeley): Formal analysis, Validation, Writing – review & editing.
• Joshua Mundinger (Berkeley): Formal analysis, Validation, Writing – review & editing.

All of the named contributors checked the proof carefully. Benjamin Antieau provided the subscription used to access GPT-5.6 Sol Extra High and Akhil Mathew provided the subscription used to access Opus 4.8. This work was supported by NSF grant DMS-2152235, FRG: Higher categorical structures in algebraic geometry (Benjamin Antieau, Andrei Căldăraru, Akhil Mathew, and Martin Olsson).

References

[1] Dmitri Orlov, Derived categories of coherent sheaves and motives, Russian Math. Surveys 60 (2005), 1242–1244. doi:10.1070/RM2005v060n06ABEH004292.

[2] Mihnea Popa and Christian Schnell, Derived invariance of the number of holomorphic $1$-forms and vector fields, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 527–536. doi:10.24033/asens.2149.

[3] Nicolas Addington and Daniel Bragg, Hodge numbers are not derived invariants in positive characteristic, Math. Ann. 387 (2023), 847–878. arXiv:2106.09949.

[4] Daniel Huybrechts, Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, Oxford University Press, 2006.

[5] Andrei Căldăraru and Simon Willerton, The Mukai pairing, I: a categorical approach, New York J. Math. 16 (2010), 61–98. arXiv:0707.2052.

[6] Claire Voisin, Hodge theory and complex algebraic geometry, I, Cambridge Studies in Advanced Mathematics 76, Cambridge University Press, 2002.