I am actually a brainlet who likes to put bounties on random PDE questions that I see.

Location might not be correct.

Hodge conjecture: Let $X$ be a non-singular complex projective manifold. Then every Hodge class on $X$ is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of $X$.

Proof: Follows from Corollary 8.2.

$\downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow$

Corollary 8.2

Proof: Using Theorem 8.1, we can conclude that (...).

$\downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow$

Theorem 8.1

Proof: From lemma 7.6 we can see (...).

$\downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow$

Lemma 7.6

Proof: (...) and by Proposition 4.3 (...).

$\downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow$

Proposition 4.3

Proof: (...) and the last inequality is a direct consequence of Theorem 6 from (...) by Evans. See references.

$\downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow$

(...), Evans: Theorem 6

Proof: (...) "Without loss of generality" we may (...). "Obviously", (...). Assertion (ii) follows "similarly". (...) and "temporarily suppose" that (...). Now we will "reduce to the special case" as above (...). See Stein [SE, Chapter IV] for "the theory" of (...) and by Theorem 4 we have that (...).

$\downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow\ \ \ \ \ \downarrow$

(...), Evans: Theorem 4

Proof: The proof is trivial and left as an exercise to the reader (Problem 11).