Derived invariance of the space of holomorphic two-forms

Technical Summary: Derived Invariance of the Space of Holomorphic Two-Forms

Problem Statement
The paper addresses a central conjecture in algebraic geometry, generally attributed to Kontsevich, which posits that smooth projective complex varieties $X$ and $Y$ with equivalent bounded derived categories of coherent sheaves ($D^b(X) \simeq D^b(Y)$) must share the same Hodge numbers. While derived invariance of Hochschild homology ensures the invariance of sums $\sum_{p-q=k} h^{p,q}$, and specific cases (dimensions $\le 2$, maximal Albanese dimension) have been resolved, the invariance of individual Hodge numbers $h^{p,q}$ in arbitrary dimensions remains a challenge. Specifically, the paper focuses on the derived invariance of $h^{2,0}(X)$ (the dimension of the space of holomorphic two-forms) and, under specific conditions, all Hodge numbers. This work builds upon foundational ideas and methodologies developed in earlier studies by Căldăraru, Popa and Schnell, Huybrechts and Nieper-Wißkirchen, Taelman, Beckmann, and Leijnse.

Methodology
The author employs a cohomological approach utilizing the Fourier-Mukai transform. Let $\Phi: D^b(X) \to D^b(Y)$ be a derived equivalence with cohomological transform $F$. The core strategy involves constructing a family of homomorphisms between the cohomology rings $H^\bullet(Y, \mathcal{O}_Y)$ and $H^\bullet(X, \mathcal{O}_X)$ parameterized by a complex number $z$.

1. Construction of the Homomorphism $\Theta^\bullet_{F,z}$:
   The paper defines a map $\Theta^\bullet_{F,z}$ by transporting the cup-product action of a class $\beta \in H^\bullet(Y, \mathcal{O}_Y)$ via the equivalence. Specifically, for an ample line bundle $L$ on $X$ with $h = c_1(L)$, the map is defined as:
   $$\Theta^\bullet_{F,z}(\beta) = e^{-zh} \cup F^{-1}\left( \beta \cup F(e^{zh}) \right)$$
   This replaces the categorical transport of line bundles $L^m$ (used in prior work) with a continuous parameter $z$, allowing for the exploitation of orthogonality relations in the limit.

2. Degree One Analysis:
   For $z=i$, the author proves $\Theta^1_{F,i}$ is an isomorphism. The proof utilizes the decomposition of the transported operator $D_\beta = F^{-1} m_\beta F$ into a multiplication part ($m_a$) and a contraction part ($\iota_v$). By analyzing the commutation relations with the complex conjugate and applying the Hodge-Riemann bilinear relations on $H^{0,1}(X)$, the author establishes a norm identity:
   $$\|\Theta^1_{F,i}(\beta)\|^2 = \|a\|^2 + \|\iota_v(h)\|^2$$
   This forces $a=0$ and $\iota_v(h)=0$ if the image is zero, leading to the vanishing of $\beta$ via a Lefschetz argument.

3. Degree Two Analysis:
   The proof for $h^{2,0}$ is more intricate. For $\beta \in H^{0,2}(Y)$, the transported operator $D_\beta$ decomposes into $m_a + K + \Pi$, where $K$ is a Kodaira-Spencer contraction and $\Pi$ is a bivector contraction.

   • Orthogonality: Using the relation $[D_\beta, \overline{D_\beta}] = 0$ and the Hodge-Riemann relations on $H^{0,2}(X)$, the author derives a norm identity involving the terms $a, c = \Pi(h^2)/2$, and $\eta = K(h)$.
   • Positivity via Correspondences: To show $K$ and $\Pi$ vanish, the author lifts the operators to correspondences on $X \times X$. By modifying the correspondence with a sign on the second factor, the commutator identities with $L_h$ translate into primitivity conditions with respect to the ample class $\ell = p_1^*h + p_2^*h$.
   • Trace Vanishing: The Hodge-Riemann relations on $X \times X$ provide non-negative trace expressions for $K$ and $\Pi$. Since $D_\beta$ is nilpotent and commutes with its conjugate, the total trace $\text{Tr}(D_\beta \overline{D_\beta})$ vanishes. Strict positivity of the Hodge-Riemann forms then forces $K = \Pi = 0$, implying $a=c=0$ and thus $\beta=0$.

4. Extension to All Hodge Numbers:
   Under the assumption that $K_X$ is trivial and the generator of $H^n(X, \mathcal{O}_X)$ lies in the subalgebra generated by $H^1$ and $H^2$, the author uses the multiplicativity of $\Theta^\bullet_{F,i}$ to transport a holomorphic symplectic form (or its conjugate) from $X$ to $Y$. By comparing the ranks of cup products by powers of these forms on the full cohomology (using symplectic linear algebra), the individual Hodge numbers are recovered.

Key Contributions and Results

• Theorem 1: For any derived equivalence $\Phi: D^b(X) \to D^b(Y)$ of smooth projective complex varieties, the homomorphism $\Theta^\bullet_{F,i}$ is an isomorphism in degrees one and two. Consequently, $h^{1,0}(X) = h^{1,0}(Y)$ and $h^{2,0}(X) = h^{2,0}(Y)$. This establishes the derived invariance of the space of holomorphic two-forms for all smooth projective complex varieties.
• Theorem 2: If $K_X \simeq \mathcal{O}_X$ and the complex conjugate of a holomorphic volume form belongs to the subalgebra of $H^\bullet(X, \mathcal{O}_X)$ generated by degrees one and two, then all Hodge numbers are derived invariants ($h^{p,q}(X) = h^{p,q}(Y)$).
  • In even dimensions, this condition is equivalent to the existence of a holomorphic symplectic form.
  • In odd dimensions, the result applies to products of symplectic varieties with elliptic curves.
• Proposition 3.2: If $X$ is a smooth projective holomorphic symplectic variety, any derived equivalent partner $Y$ is also holomorphic symplectic, and they share all Hodge numbers.

Significance and Claims
The paper claims to resolve the derived invariance of $h^{2,0}$ in arbitrary dimensions, a result previously known only in low dimensions or under restrictive hypotheses (e.g., specific Fourier-Mukai kernels). The proof introduces a novel analytic technique: replacing discrete tensor powers of line bundles with a continuous complex parameter $z$ to leverage Hodge-Riemann bilinear relations and orthogonality.

Regarding the broader conjecture, the paper notes that while Taelman and Leijnse have established isomorphisms of rational Hodge structures for hyperkähler and specific symplectic varieties, this work provides the numerical conclusion (equality of Hodge numbers) for a broader class of symplectic varieties in arbitrary dimensions, without requiring restrictions on the LLV (Looijenga-Lunts-Verbitsky) algebra. The author explicitly states that the paper does not construct degreewise rational isomorphisms of Hodge structures, focusing instead on the invariance of the dimensions of the cohomology spaces. The results extend the known scope of derived invariance beyond the previously established cases of dimensions $\le 4$ or specific geometric constraints.