In the year 2000, the Clay Mathematics Institute announced the Millennium Prize Problems: seven outstanding questions in pure mathematics that have withstood decades of sustained research. Each problem was selected not only for its intrinsic technical difficulty but also for its potential to reshape large swathes of modern mathematics should it be resolved. These problems span areas ranging from the analysis of partial differential equations (the Navier--Stokes existence and smoothness problem) to number theory (the Riemann Hypothesis, the Birch and Swinnerton-Dyer Conjecture), topology (the Poincar Conjecture, subsequently solved by Perelman), and fundamental structures in geometry and algebra.
Among these, the Hodge Conjecture (HC) occupies a central position in algebraic geometry and complex geometry. Formulated in the mid-20th century by W. V. D. Hodge, the conjecture lies at the intersection of topology, algebra, and geometry, offering a bridge between abstract cohomological invariants and concrete algebraic cycles. Its resolution promises profound consequences for our understanding of algebraic varieties, arithmetic geometry, and even mathematical physics (through mirror symmetry and string theory).
Formally, let XXX be a smooth, projective complex algebraic variety of complex dimension nnn. The cohomology group H2p(X,Q)H^{2p}(X, \mathbb{Q})H2p(X,Q) carries a natural Hodge decomposition:
H2p(X,C)r+s=2pHr,s(X),H^{2p}(X, \mathbb{C}) \cong \bigoplus_{r+s=2p} H^{r,s}(X),H2p(X,C)r+s=2pHr,s(X),
where Hr,s(X)H^{r,s}(X)Hr,s(X) consists of cohomology classes represented by differential forms of type (r,s)(r,s)(r,s). A class H2p(X,Q)\gamma \in H^{2p}(X, \mathbb{Q})H2p(X,Q) is called a Hodge class if its complexification lies in Hp,p(X)H^{p,p}(X)Hp,p(X). That is,
1Hp,p(X)H2p(X,C).\gamma \otimes 1 \in H^{p,p}(X) \subseteq H^{2p}(X, \mathbb{C}).1Hp,p(X)H2p(X,C).
On the other hand, algebraic geometry provides algebraic cycles, formal Z\mathbb{Z}Z-linear combinations of irreducible subvarieties of codimension ppp in XXX. Each algebraic cycle defines a class in cohomology via the cycle class map:
clp:Zp(X)H2p(X,Q),\operatorname{cl}_p : Z^p(X) \longrightarrow H^{2p}(X, \mathbb{Q}),clp:Zp(X)H2p(X,Q),
where Zp(X)Z^p(X)Zp(X) denotes the group of codimension-ppp algebraic cycles on XXX.
The Hodge Conjecture posits that every rational Hodge class is algebraic:
H2p(X,Q)Hp,p(X)Im(clp).H^{2p}(X, \mathbb{Q}) \cap H^{p,p}(X) \; \subseteq \; \operatorname{Im}(\operatorname{cl}_p).H2p(X,Q)Hp,p(X)Im(clp).
