en.wikipedia.org/wiki/Hodge_conjecture
3 corrections found
using the pullback function i ∗ : X → Z
This misstates what the pullback map is. The inclusion is the map \(i:Z\to X\); the induced pullback \(i^*\) goes the other way on differential forms or cohomology, from objects on \(X\) to objects on \(Z\).
Full reasoning
The article correctly defines the inclusion map as (i: Z \to X), but then incorrectly writes the pullback as though it were another map (X \to Z).
For differential forms, pullback is not a map between the underlying spaces at all. Given a smooth map (F:M\to N), the induced pullback on forms is
- (F^* : \Omega^r(N) \to \Omega^r(M)),
and on cohomology similarly goes from classes on the target back to classes on the source. So for an inclusion (i:Z\to X), the pullback is (i^: \Omega^r(X)\to\Omega^r(Z)), not (i^:X\to Z).
That matters here because the integral (\int_Z i^*\alpha) is formed by first pulling the form (\alpha) back from (X) to (Z), then integrating on (Z).
1 source
- Differential Geometry notes (University of Kentucky)
Definition 99. Any smooth map F : M → N ... defines a pullback map F* : Ω^r(N) → Ω^r(M).
If for some p < n 2 the Hodge conjecture holds for Hodge classes of degree p , then the Hodge conjecture holds for Hodge classes of degree 2 n − p
The degree indices are off by a factor of 2. The standard Hard Lefschetz consequence is: if the Hodge conjecture holds in degree \(2p\), then it holds in degree \(2n-2p\) (equivalently, in codimension \(p\) and codimension \(n-p\)).
Full reasoning
This theorem statement drops a factor of 2 from both degrees.
For a smooth complex projective variety of dimension (n), Hodge classes in codimension (p) live in cohomological degree (2p). The standard Hard Lefschetz implication is therefore:
- if the Hodge conjecture holds in degree (2p),
- then it holds in degree (2n-2p).
A standard reference states this explicitly: "If the Hodge conjecture holds for (X) in degree (2p) ... then it holds in degree (2n-2p)." So the article's version with degrees (p) and (2n-p) is not the usual statement and is mathematically incorrect as written.
2 sources
- Arapura, Algebraic Geometry Part II notes
Proposition 13.5.4. If the Hodge conjecture holds for X in degree 2p with p < n = dim X ... then it holds in degree 2n − 2p.
- Skauli, Curve classes on Calabi–Yau complete intersections in toric varieties
By the Hard Lefschetz theorem, this also implies that Hodge conjecture holds for degree 2n−2 classes.
this integral is zero if ( p , q ) ≠ ( n , n )
The bidegree here is wrong. When integrating over a complex submanifold \(Z\) of complex dimension \(k\), the only type that can contribute is \((k,k)\), not \((n,n)\).
Full reasoning
The surrounding discussion is about integrating a ((p,q))-form over a complex submanifold (Z\subset X) of complex dimension (k). In that situation, the relevant real dimension is (2k), so only forms of total degree (2k) can contribute. Moreover, the restriction to (Z) vanishes if either (p>k) or (q>k). Combining those facts forces the only potentially nonzero case to be
- (p+q=2k), and
- (p=q=k).
So the conclusion should be that the integral is zero unless the form has type ((k,k)). Writing ((n,n)) is incorrect: (n) is the complex dimension of the ambient manifold (X), not of the submanifold being integrated over.
This is exactly the standard argument used to show that the fundamental class of a codimension-(c) complex subvariety has Hodge type ((c,c)): if (\dim X=n) and (\dim Z=n-c), then only type ((n-c,n-c)=(k,k)) can contribute on (Z).
2 sources
- Peters & Steenbrink, Mixed Hodge Structures
The restriction to Y of a (p, q)-form with p+q = 2n−2c necessarily vanishes ... if either p > n−c or q > n−c ... We conclude Proposition 1.11: the fundamental cohomology class cl(Y) of a codimension c subvariety ... has pure type (c, c).
- Claire Voisin, Some aspects of the Hodge conjecture
The current annihilates all forms of type (p, q), p ≠ q, p + q = 2n−2k, n = dim X, and it follows dually that its class is of type (k, k).