gpt 6 astra pro attempt on Math-11

Generalized Hodge Conjecture

Definitions and mathematical statement

A rational Hodge substructure $V\subseteq H^k(X,{\mathbb{Q}})$ has Hodge coniveau at least $c$ if $V^{a,b}=0$ whenever $a<c$ or $b<c$. Let $M^cH^k$ be the sum of all such substructures. Define geometric coniveau by \[ N^cH^k(X,{\mathbb{Q}})=\sum_{\substack{Z\subseteq X\text{ closed algebraic}\\ \operatorname{codim}Z\ge c}} \ker\left(H^k(X,{\mathbb{Q}})\to H^k(X\setminus Z,{\mathbb{Q}})\right). \] The conjecture is $M^cH^k(X,{\mathbb{Q}})=N^cH^k(X,{\mathbb{Q}})$ for every smooth projective complex $X$ and all relevant $k,c$. Finite sums can be supported on the union of their supporting algebraic sets.

Formulation and concept references: [S1].

Short English statement

Does every part of cohomology whose Hodge types allow high-codimension support actually have such algebraic support?

Research attempt

Current frontier.

Research check: 27 September 2026. The general problem is about rational sub-Hodge structures, not merely individual Hodge classes. Voisin's 2025 survey is the recorded formulation; her earlier author-hosted treatment gives the support/Gysin description and the inclusion $N^c\subseteq M^c$ in Theorem 1.9 and Corollary 1.10. [S1], [E1]. The 2026 moduli-space manuscript proves specified degree/genus ranges, with separate hypotheses for its arithmetic statements; its Theorems A--C are not a theorem for arbitrary smooth projective varieties. [S3]. The calculation below is an independent investigation of the realization step. No historical novelty is asserted for its constituent observations.

Attack route.

Three routes suggest themselves: produce a supporting subvariety directly; realize the effective twist of a Hodge substructure on a smaller-weight cohomology group and algebraize the resulting correspondence; or obtain support from a degeneration. The first needs geometric existence, the second needs both realization and algebraicity, and the third needs a specialization statement that preserves support codimension. We pursue the second route because it cleanly separates two often conflated obstructions.

Attempt: a correspondence realization criterion.

Fix $X$ of dimension $n$, $0\leq c\leq k/2$, and $V\subseteq H^k(X,{\mathbb{Q}})$ of Hodge coniveau at least $c$. Put $m=k-2c$. Suppose there are a smooth projective $Y$ of dimension $d$, and a morphism of rational Hodge structures \[ u:H^m(Y,{\mathbb{Q}})(-c)\longrightarrow H^k(X,{\mathbb{Q}}),\qquad \operatorname{im}u=V. \tag{24.1} \] Assume additionally that the specific Hodge class representing $u$ in $H^{2d+2c}(Y\times X,{\mathbb{Q}})$ is algebraic. Then $V\subseteq N^cH^k(X,{\mathbb{Q}})$.

Here are all the dimension and support details. Poincare duality identifies $H^m(Y)^*$ with $H^{2d-m}(Y)(d)$. Consequently $u$, viewed without the source twist as a map of bidegree $(c,c)$, determines a rational Kunneth class \[ \gamma_u\in H^{2d-m}(Y,{\mathbb{Q}})\otimes H^k(X,{\mathbb{Q}}) \subset H^{2d+2c}(Y\times X,{\mathbb{Q}}) \tag{24.2} \] of type $(d+c,d+c)$. Its action is $\alpha\mapsto (p_X)_*(p_Y^*\alpha\smile\gamma_u)$. The duality identification can be chosen with its usual graded signs, so that this action is exactly $u$, not just its image up to an unspecified scalar. Algebraicity supplies a rational cycle $\Gamma=\sum_i a_i\Gamma_i$ of codimension $d+c$ with class $\gamma_u$. Every component has dimension $n-c$. Properness of $p_X$ shows that \[ Z=\bigcup_i p_X(\Gamma_i)\quad\hbox{is closed},\qquad \dim Z\leq n-c. \tag{24.3} \] On $Y\times(X\setminus Z)$ the cycle is empty. Compatibility of cycle classes, restriction, and proper pushforward therefore makes the correspondence action zero after restriction to $X\setminus Z$. This proves the criterion. It uses one finite supporting union for the whole image, not a potentially infinite union for individual classes.

A precise sufficient package for the full conjecture is now visible: every $V(c)$ must be a quotient of some $H^{k-2c}(Y,{\mathbb{Q}})$, and the classes $24.2$ must be algebraic. The ordinary Hodge conjecture on all products would supply the second statement, but the first is a further geometric realization requirement. Calling the twist "effective" supplies only nonnegative Hodge indices, not such a $Y$.

Attempt: the first two remaining weights.

For $m=0$, $V(c)$ is a sum of copies of ${\mathbb{Q}}(0)$, so a finite disjoint union of points supplies $24.1$. Algebraicity of $24.2$ is then precisely the ordinary codimension-$c$ Hodge-class problem on $X$. Thus the criterion contains, rather than bypasses, the ordinary Hodge conjecture.

For $m=1$, $V(c)$ is polarizable of type $(1,0)+(0,1)$. The classical weight-one Hodge/abelian-variety correspondence realizes it as $H^1(A,{\mathbb{Q}})$ up to isogeny. Equivalently, this realization follows by dualizing a rational lattice, using the polarization for the Riemann form, and forming the resulting projective complex torus. A rational complement in $H^1(A)$, if needed, is obtained from a polarization. [E2], Theorem 6.20. Hence in degree $k=2c+1$ the outstanding step in this route is algebraicity of a particular codimension $\dim A+c$ class on $A\times X$. It is not enough that divisors on $A$ are algebraic: that class has a mixed Kunneth component involving $H^k(X)$.

There is also an unconditional boundary range, worth using as a normalization check. If $n\leq k\leq2n$, put $r=k-n$ and $m'=2n-k$. A smooth intersection $i:Y\hookrightarrow X$ of $r$ sufficiently ample divisors has $\dim Y=m'$. With $h$ their common hyperplane class, \[ i_*i^*\alpha=h^r\alpha,\qquad h^r:H^{m'}(X,{\mathbb{Q}})\xrightarrow{\sim}H^k(X,{\mathbb{Q}}). \tag{24.4} \] The second equality is hard Lefschetz. Thus every class in $H^k(X)$ has support of codimension $r$, and $N^rH^k=M^rH^k=H^k$. This familiar special case is proved here to test the support bookkeeping; it says nothing about the larger values $c>r$.

Stress test.

A rational Hodge projector is not automatically an algebraic projector: replacing the class $24.2$ by a "cycle representing it" without the additional assumption is exactly the missing step. Nor does the semisimplicity of polarizable Hodge structures manufacture a smooth projective variety realizing an arbitrary effective twist. The argument has not deduced that realization from a dimension count. For $k<2c$, both sides vanish; for $c\leq0$, both are all of $H^k$ under the usual coniveau conventions. Negative or out-of-range cohomological degrees are zero. These harmless ranges do not conceal a missing positive coniveau case. Finally, mod-$2$ or integral coniveau obstructions would not be counterexamples to this rational statement.

Outcome.

Outcome: Conditional reduction. The attempt proves a support-producing criterion with explicit cycle codimension, isolates realization separately from algebraicity, and checks the low-weight and high-degree boundary cases. A universal resolution still requires the two stated inputs. In particular, even the weight-zero case would resolve the ordinary Hodge conjecture; that dependency is a warning against treating the correspondence class as already algebraic.

Sources

• S1 Claire Voisin, Hodge and generalized Hodge conjectures, coniveau and algebraic cycles, Journal of Open Mathematical Problems 1$1$ $2025$, 16-51. https://jomprob.org/index.php/jomp/article/view/Vol-1Issue-1Paper-2. Catalogue formulation source.
• S2 Incidence equivalence, a survey. https://arxiv.org/abs/2607.22233. Catalogue research/status reference; not a proof endorsement.
• S3 On the Hodge and Tate conjectures for moduli spaces of curves. https://arxiv.org/html/2605.20453v1. Catalogue research/status reference; not a proof endorsement.
• E1 Claire Voisin, Hodge Structures, Coniveau and Algebraic Cycles, author-hosted manuscript, Theorem 1.9 and Corollary 1.10. https://webusers.imj-prg.fr/~claire.voisin/Articlesweb/23-.pdf. Added research source. Support, Gysin maps and the coniveau inclusion. The new realization calculation is given in the text. Consulted 27 September 2026.
• E2 Pierre Deligne and James S. Milne, Tannakian Categories, in Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900 $1982$, 101--228; corrected author version dated 15 August 2012, Theorem 6.20. https://www.jmilne.org/math/xnotes/tc.pdf. Added research source. Weight-one polarizable Hodge structures of abelian-variety type. Consulted 27 September 2026.