[Paper Review] Tautological Hilbert scheme invariants of Calabi-Yau 4-folds via virtual pullback

Yalong Cao, F. Qu|arXiv (Cornell University)|Dec 8, 2020

Algebraic Geometry and Number Theory37 references4 citations

TL;DR

This paper proves a conjecture by Cao-Kool on tautological invariants of Hilbert schemes on Calabi-Yau 4-folds using virtual pullback techniques. It establishes a precise correspondence between invariants of the 4-fold and those of a smooth divisor, confirming that the generating series of these invariants matches the MacMahon function weighted by the first Chern class and third Chern class of the 4-fold.

ABSTRACT

Let $X$ be a Calabi-Yau 4-fold and $D$ a smooth connected divisor on it. We consider tautological bundles of $L=\mathcal{O}_X(D)$ on Hilbert schemes of points on $X$ and their counting invariants defined by integrating the Euler classes against the virtual classes. We relate these invariants to Maulik-Nekrasov-Okounkov-Pandharipande's invariants of Hilbert schemes of points on $D$ by virtual pullback technique and confirm a conjecture of Cao-Kool. The same strategy is also applied to obtain a virtual pullback formula for tautological invariants of one dimensional stable sheaves. This in turn gives a nontrivial identity on primary and descendent invariants as studied in the works of Cao, Maulik and Toda.

Motivation & Objective

• To prove a conjecture by Cao-Kool relating tautological invariants of Hilbert schemes on Calabi-Yau 4-folds to the MacMahon function.
• To establish a virtual pullback formula connecting invariants of the 4-fold X to those of a smooth divisor D ⊂ X.
• To extend the virtual pullback technique to one-dimensional stable sheaves and derive new identities between primary and descendent invariants.
• To confirm a nontrivial relation between genus zero Gopakumar-Vafa invariants of X and its divisor D.

Proposed method

• Applies the virtual pullback technique to a compatible diagram of obstruction theories between Hilbert schemes on X and D.
• Uses the derived moduli space structure of Hilbert schemes and their shifted symplectic structures to define virtual classes.
• Relies on the Borisov-Joyce construction of DT4 virtual classes and Oh-Thomas’s algebraic cycle lift over Z[1/2].
• Employs Grothendieck-Riemann-Roch and determinant line bundle isomorphisms to relate tautological bundles on X to those on D.
• Applies the normalized universal sheaf and Chern character decomposition to express invariants in terms of descendent insertions.
• Derives a virtual pullback formula for one-dimensional stable sheaves and uses it to obtain identities among invariants.

Experimental results

Research questions

• RQ1 Does the generating series of tautological invariants on Hilbert schemes of a Calabi-Yau 4-fold match the MacMahon function as conjectured by Cao-Kool?
• RQ2 Can the virtual pullback technique be used to relate invariants on a Calabi-Yau 4-fold to those on a smooth divisor?
• RQ3 What is the precise relationship between primary and descendent invariants of one-dimensional stable sheaves on a Calabi-Yau 4-fold and its divisor?
• RQ4 How do genus zero Gopakumar-Vafa invariants of X and D relate under the virtual pullback framework?
• RQ5 Can the conjectural formula for descendent invariants be verified using the virtual pullback method?

Key findings

• The conjecture by Cao-Kool is confirmed: for any Calabi-Yau 4-fold X and line bundle L with c₁(L) = c₁(𝒪_X(D)) for a smooth connected divisor D, the generating series of invariants Iₙ(X,L) equals M(−q)^∫_X c₁(L)·c₃(X), where M(q) is the MacMahon function.
• The virtual pullback formula establishes a direct correspondence: ∫_{[Hilbⁿ(X)]^vir} e(L^{[n]}) = ∫_{[Hilbⁿ(D)]^vir} 1, up to orientation sign, under the given conditions.
• The method yields a nontrivial identity: ½⟨τ₀(c₁²(L))⟩_β + ¹⁄₁₂⟨τ₀(c₂(X))⟩_β + ⟨τ₁(c₁(L))⟩_β + ⟨τ₂(1)⟩_β = n₀,β(D), linking invariants on X and D.
• The result extends to one-dimensional stable sheaves, showing that the virtual pullback formula holds for such moduli spaces.
• The paper confirms a conjectural relation between genus zero Gopakumar-Vafa invariants of X and D, with the invariants on D arising as combinations of those on X.
• The proof is valid for all effective divisors on smooth sextic 4-folds in ℙ⁵, as such divisors exist by Bertini’s theorem and satisfy the required conditions.