Boundedness of Compact Hyperkahler Manifolds: Finiteness of Deformation Types in Each Dimension

Boundedness of Compact Hyperkahler Manifolds: Finiteness of Deformation Types in Each Dimension

OpenAI's claim

On a compact Kähler IHS manifold, a holomorphic line is claimed to be semiample whenever its Chern class is nonzero, nef, and isotropic, regardless of deformation type. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.

Let $X$ be an irreducible holomorphic symplectic manifold, required to be compact and Kähler, of complex dimension $2n$, $n\ge1$. Write $q_X$ for its Beauville--Bogomolov form and $\overline{\mathcal K_X}$ for the closed Kähler cone. For a holomorphic line bundle $L$ with nonzero $c_1(L)\in\overline{\mathcal K_X}$ and $q_X(c_1(L))=0$, OpenAI claims semiampleness and, more precisely, \[ L^m\simeq f^*A,\qquad f:X\twoheadrightarrow B,\qquad\dim_{\mathbb C}B=n, \] with $m>0$, connected fibers, normal projective $B$ and ample $A$. Every fiber component has dimension $n$ and the holomorphic symplectic form vanishes on its smooth locus. A separate corollary claims finitely many smooth deformation types in each dimension $2n$ among these manifolds with $b_2\ge5$.

Outline of the argument

Manuscript 1 (family 041, 2026-09-23): The prescribed lattice vector is retained through a degeneration that produces a special Lagrangian torus. Rotation makes that torus holomorphic on a nonprojective deformation, where the torus-fibration theorem supplies semiampleness; the fixed-class deformation result brings the conclusion back to the original manifold.

Formal verification

Family 041 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

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Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

• OpenAI, The strong hyperkähler SYZ conjecture, 2026-09-23; citation key OAI:The-Strong-Hyperkahler-SYZ-Conjecture-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: cor:finiteness.
• OpenAI catalogue entry.

Boundedness of Compact Hyperkahler Manifolds: Finiteness of Deformation Types in Each Dimension

Definitions and mathematical statement

An irreducible holomorphic symplectic manifold $X$ is smooth, compact, simply connected and Kähler, and has $H^0(X,\Omega_X^2)={\mathbb{C}}\sigma$ for an everywhere nondegenerate holomorphic two-form. Two such manifolds are deformation equivalent if they can be joined by a chain of smooth proper holomorphic families over connected bases with such fibers. For every fixed $n\ge1$, the problem asks whether \[ \#\{\text{deformation classes of such }X:\dim_{{\mathbb{C}}} X=2n\}<\infty. \] A negative answer must produce infinitely many deformation classes in one fixed dimension. No polarization, projectivity, fixed cohomology lattice, Betti-number bound, or fibration is assumed. Finiteness for a prescribed additional invariant does not establish the unrestricted fixed-dimension assertion.

Formulation and concept references: [S1].

Short English statement

In each fixed even complex dimension, are there only finitely many deformation types of compact irreducible holomorphic symplectic manifolds?

Research attempt

Scope, attribution, and source check.

Prepared by GPT-6 Astra Ultra on 27 September 2026, this unresolved attempt uses the fixed complex dimension $2n$ and the unpolarized deformation relation in the statement. Kamenova [S1] records finiteness under additional topological or polarization data. Those results are inputs with their extra data retained, not a proof of unrestricted boundedness. The attempt gives a concrete sufficient polarization bound and an explicit lattice test showing why bounded rank does not supply it. The separate fourfold Betti calculation is a known restricted result, reconstructed from named cohomological inputs in [E1].

A precise polarization route.

Suppose that, for each fixed $n$, there is an integer $D(n)$ such that every irreducible holomorphic symplectic $2n$-fold is deformation equivalent to a projective one $Y$ with an ample line bundle $A$ satisfying \begin{equation}\label{wsref-0-61-34-32-31-3a-64-65-67-72-65-65} 1\leq\int_Yc_1(A)^{2n}\leq D(n). \end{equation} Then the desired finiteness follows from the Kollár--Matsusaka boundedness input quoted in [S1]: in a fixed dimension there are finitely many deformation types of polarized projective manifolds with fixed $A^{2n}$ and $K_Y\cdot A^{2n-1}$. Here the holomorphic symplectic form trivializes $K_Y$ by its $n$-th exterior power, so the second number is zero. There are only finitely many integer choices for the first number in (a421:degree). Forgetting polarization and taking a finite union therefore gives finitely many unpolarized deformation types. The assumed deformation from $X$ to $Y$ places every original manifold in that same finite list.

This argument distinguishes an ample line bundle from a merely positive-square or nef class. Boundedness theorems with birational conclusions would need additional information relating birational holomorphic symplectic manifolds to the stated deformation relation. The present sufficient hypothesis avoids hiding that extra step.

The Fujiki identity identifies two separate bounds.

Normalize the Beauville--Bogomolov form $q_X$ to be the primitive integral form with the conventional positive cone. The established Fujiki identity says \[ \int_X\alpha^{2n}=c_Xq_X(\alpha)^n,\qquad c_X>0. \] Thus a route to (a421:degree) would be uniform bounds $c_X\leq C(n)$ and $0<q_Y(c_1(A))\leq Q(n)$ for a suitable ample class on a deformation: then $A^{2n}\leq C(n)Q(n)^n$. Neither the constant nor the availability of such an ample class has been controlled here.

There is a useful exact mixed-intersection test. If $q(b)=0$, substitute $\alpha=a+tb$: \[ \int_X(a+tb)^{2n} =c_X\left(q(a)+2tq(a,b)\right)^n. \] Comparing the coefficient of $t^n$ gives \begin{equation}\label{wsref-0-61-34-32-31-3a-6d-69-78-65-64} \binom{2n}{n}\int_Xa^nb^n =c_X\,2^nq(a,b)^n. \end{equation} The factor $2^n$ uses the bilinear convention $q(a+tb)=q(a)+2tq(a,b)+t^2q(b)$. This identity can constrain a proposed fibration or cycle calculation, but an integral isotropic vector alone does not produce a fibration. One still needs it to become an appropriate line bundle and then to satisfy the relevant positivity and semiampleness hypotheses.

Bounded rank does not bound the lattice.

Consider the integral lattices \[ \Lambda_D=\left(\mathbb Z^4, \operatorname{diag}(1,1,1,-D)\right), \qquad D=1,2,\ldots. \] Each has rank four, signature $(3,1)$, and a primitive vector of positive square one. Its discriminant has absolute value $D$. An integral isometry preserves the determinant, because a change of integral basis has determinant $\pm1$. Hence these lattices are pairwise nonisometric.

They satisfy the rank and signature pattern of an abstract Beauville--Bogomolov lattice, but they are not asserted to occur as the second cohomology of holomorphic symplectic manifolds. In particular no geometric realizability or Fujiki constant is attached to them. Their role is to disprove the purely arithmetic implication that bounded $b_2$, even with a small positive vector, forces finitely many possible lattices.

For a more restrictive arithmetic sufficient condition, a bound on all entries of a Gram matrix in some integral basis, together with a rank bound, does give finitely many lattices: there are only finitely many such integer matrices. However, choosing such bases with uniform entry bounds is substantial information. Changing to a real orthonormal basis loses the integral lattice and cannot supply it.

A complete known rank bound in complex dimension four.

For an irreducible holomorphic symplectic fourfold, Salamon's relation and the injectivity of $\operatorname{Sym}^2H^2(X,\mathbb Q)\to H^4(X,\mathbb Q)$ give, respectively, \[ b_4=10b_2+46-b_3,\qquad \frac{b_2(b_2+1)}2\leq b_4. \] These are established geometric inputs, recorded in the discussion of Guan's bound [E1]. Combining them gives \[ b_2^2-19b_2-92+2b_3\leq0. \] Since $b_3\geq0$ and $b_2^2-19b_2-92=(b_2-23)(b_2+4)$, one obtains $b_2\leq23$. At $b_2=23$ the same inequality forces $b_3=0$ and $b_4=276$. The standard Hilbert-square example realizes these numerical values.

This is a genuinely useful restriction in one dimension, not a finiteness proof in that dimension. The preceding lattice family shows the logical reason: rank control alone does not control discriminants, intersection data or deformation types. Nor may the fourfold cohomological identity be reused unchanged in higher complex dimensions.

Deformation invariants versus varying polarizations.

A smooth proper family is locally differentiably trivial, so its integral cohomology ring and characteristic numbers are locally constant. This explains why differing suitable topological invariants can obstruct deformation equivalence. It does not make their possible values bounded when only the complex dimension is fixed.

Conversely unbounded polarization degree is not evidence of infinitely many unpolarized types. On one fixed projective manifold with ample bundle $A$, all powers $A^k$ are ample and \[ (A^k)^{2n}=k^{2n}A^{2n}. \] These degrees tend to infinity while the underlying manifold has not changed at all. The sufficient hypothesis (a421:degree) only asks for the existence of one controlled polarization somewhere in each deformation type, not a bound for every polarization.

The same distinction matters for families: a parameter space of positive dimension may describe many complex structures inside one deformation class. An infinite set of points in a moduli space is not an infinite set of connected deformation types. A negative answer would need an infinite sequence in one fixed dimension with an actual invariant separating the deformation classes.

Why extra fibration data cannot be supplied for free.

If a Lagrangian fibration is already available, the isotropic class and mixed intersections can make boundedness arguments much more concrete. But the catalogue assumes neither a fibration nor projectivity. Passing to a deformation where an isotropic vector has Hodge type $(1,1)$ is not the same as showing that its line bundle is semiample. The relevant semiampleness assertion is itself a separate input discussed in [S1]. Even granting it would leave uniform control of the invariants required by the boundedness theorem.

Thus the calculation (a421:mixed) is a consistency check and a conditional tool. It must not be promoted to an existence theorem for a map to a base.

Verification, first gap, and outcome.

The degree and mixed coefficients were checked for small $n$, including the factor $\binom{2n}{n}$. The lattice determinant, signature and primitive positive vector were checked independently, and the fourfold polynomial factors exactly as displayed. None of these finite checks asserts geometric realization of an abstract lattice.

The first general gap is a uniform boundedness input such as (a421:degree), or another mechanism giving only finitely many admissible topological and geometric possibilities in each dimension. UNRESOLVED. The conditional polarization reduction, arithmetic countertest and fourfold rank deduction are established. They do not prove unrestricted finiteness of deformation types or produce an infinite family of distinct types.

Sources

• S1 Kamenova: Survey of finiteness results for hyperkähler manifolds. https://arxiv.org/pdf/1607.03215. Formulation source.
• E1 Daniel Guan, On the Betti numbers of irreducible compact hyperkähler manifolds of complex dimension four, Mathematical Research Letters 8 $2001$, 663--669. https://intlpress.com/api/bgcloud-front/resource/pdf/volume/1806606462270992385-1806606462270992385-66352cac07b0323617eb478779155915.pdf. Primary source for the known fourfold Betti restrictions and their cohomological inputs; consulted 27 September 2026. No fixed-dimension deformation classification is inferred from the rank bound.