Toric Geometry - Our basic research regarding Hodge Conjecture - Dave Ishii

The document addresses the Intersection Cohomology of a Fan and the Hodge Conjecture for Toric Varieties¹.
The problem involves a projective, (potentially singular) toric variety $X_{\Sigma}$ defined by a fan $\Sigma$ in a lattice $N$².
The Intersection Hodge Conjecture for $X_{\Sigma}$ asserts that the "Hodge-theoretic" cycle class map is surjective³.
The map in question is $cl_{IH}:\bigoplus_{k}\mathcal{Z}^{k}(X_{\Sigma})_{\mathbb{Q}}\rightarrow\bigoplus_{k}IH^{2k}(X_{\Sigma},\mathbb{Q})\cap IH^{k,k}(X_{\Sigma})$⁴.
$\mathcal{Z}^{k}(X_{\Sigma})$ represents the group of algebraic cycles of codimension k⁵.
$IH^{*}$ denotes the intersection cohomology⁶.
The fundamental open problem is to find a purely combinatorial description for both Hodge-theoretic and algebraic cycle classes⁷.
This description should be in terms of the fan $\Sigma$ and the lattice $N$⁸.
The problem also requires proving the equivalence of these two descriptions⁹.

Objective 1 (Hodge Side): Develop a combinatorial algorithm using only fan data to compute a basis for "combinatorial Hodge classes"¹⁰.
"Combinatorial Hodge classes" are defined as the rational classes in $IH^{k,k}(X_{\Sigma})$¹¹.
Objective 2 (Algebraic Side): Prove that the space from Objective 1 is spanned precisely by the intersection cohomology classes of torus-invariant subvarieties $V(\tau)$¹².
This must hold for all cones $\tau \in \Sigma$¹³.

This problem is described as a specialized, combinatorial version of one of the deepest unsolved problems in mathematics¹⁴.
The Hodge Conjecture: It states that for a smooth projective variety $X$, any class in $H^{2k}(X,\mathbb{Q})\cap H^{k,k}(X)$ is the class of an algebraic cycle¹⁵.
The conjecture connects the abstract topology of $X$ to its concrete algebraic geometry¹⁶.
Intersection Cohomology: Most toric varieties are singular¹⁷.
For singular varieties, standard cohomology $H^{*}(X)$ lacks good properties like Poincaré Duality¹⁸.
Intersection Cohomology ($IH^{*}(X)$) is the correct replacement for singular varieties¹⁹.
Therefore, the Hodge Conjecture must be reformulated in terms of $IH^{*}(X)$²⁰.
The Fan: For a toric variety $X_{\Sigma}$, every geometric and topological property is completely encoded in the combinatorial data of its fan $\Sigma$²¹.
The "natural" algebraic cycles on $X_{\Sigma}$ are the torus-invariant subvarieties $V(\tau)$²².
These subvarieties correspond one-to-one with the cones $\tau \in \Sigma$²³.
The cohomology of smooth toric varieties is well-understood combinatorially and related to the Stanley-Reisner ring of the fan²⁴.
The combinatorial description of intersection cohomology $IH^{*}(X_{\Sigma})$ is much more complex²⁵.

Challenge 1: Computing $IH^{*}(X_{\Sigma})$ combinatorially is hard, as there is no simple, general formula²⁶. A successful approach might need a new combinatorial invariant that captures the "failure" of $X_{\Sigma}$ to be smooth²⁷.
Challenge 2: A main creative step is to define a "combinatorial Hodge class" by finding a "combinatorial signature" within the fan data that identifies classes of type $(k, k)$²⁸.
This signature would be some new invariant of the cones and their relationships within the lattice $N$²⁹.
Challenge 3: A solution would need to prove surjectivity by showing the new combinatorial description (from Challenge 2) generates a set identical to the set of torus-invariant cycles $V(\tau)$³⁰.
Proving that no other algebraic cycles are needed would be a major breakthrough³¹.

Dave Ishii - Problem Solving Blog