[Paper Review] Boundedness of elliptic Calabi-Yau threefolds

Stefano Filipazzi, Christopher D. Hacon+1 more|arXiv (Cornell University)|Dec 2, 2021

Geometry and complex manifolds14 references4 citations

TL;DR

This paper establishes that elliptic Calabi–Yau threefolds form a bounded family, meaning they are parametrized by finitely many algebraic families. The authors prove boundedness by fixing the growth rate of pluricanonical forms and the degree of a multisection of the Iitaka fibration, using techniques from birational geometry, log boundedness, and deformation theory of divisors in families.

ABSTRACT

We show that elliptic Calabi--Yau threefolds form a bounded family. We also show that the same result holds for minimal terminal threefolds of Kodaira dimension 2, upon fixing the rate of growth of pluricanonical forms and the degree of a multisection of the Iitaka fibration. Both of these hypotheses are necessary to prove the boundedness of such a family.

Motivation & Objective

• To resolve S.-T. Yau's long-standing question on whether Calabi–Yau threefolds have finitely many topological types.
• To establish boundedness of elliptic Calabi–Yau threefolds, including those with non-rational bases, by fixing key invariants such as the degree of a multisection and the growth rate of pluricanonical forms.
• To extend boundedness results to minimal terminal threefolds of Kodaira dimension 2 under the same invariants, showing that boundedness is conditional on these constraints.
• To demonstrate that the boundedness of such families is optimal, as both hypotheses are necessary for the result to hold.
• To provide a foundation for proving finiteness of topological types in broader classes of Calabi–Yau threefolds, especially those with large Picard rank.

Proposed method

• Use of log boundedness techniques in codimension one for elliptic Calabi–Yau fibrations, building on prior work by Filippini and others.
• Application of deformation theory for divisors in families to control the geometry of the base and total space of the fibration.
• Employment of the Kawamata–Morrison conjecture framework, showing that models of elliptic fibrations are finite up to isomorphism over the base.
• Leveraging the boundedness of log Fano varieties in lower dimensions to deduce boundedness of the base surfaces via the Iitaka fibration.
• Construction of a finite family of models over a base scheme $T$ such that every elliptic Calabi–Yau threefold appears as a fiber after a sequence of flops.
• Use of the theory of minimal models and flops over a base, ensuring that any such Calabi–Yau threefold is isomorphic to a fiber of a finite family of models.

Experimental results

Research questions

• RQ1 Do elliptic Calabi–Yau threefolds form a bounded family when the base is not necessarily rational?
• RQ2 Is the boundedness of elliptic Calabi–Yau threefolds dependent on fixing the degree of a multisection and the growth rate of pluricanonical forms?
• RQ3 Can the boundedness result be extended to minimal terminal threefolds of Kodaira dimension 2 under the same invariants?
• RQ4 Are the hypotheses of fixed multisection degree and fixed growth rate of pluricanonical forms necessary for boundedness?
• RQ5 Can the boundedness of the base variety in the Iitaka fibration be used to deduce boundedness of the total space in the Calabi–Yau case?

Key findings

• Elliptic Calabi–Yau threefolds form a bounded family, meaning they are parametrized by finitely many algebraic families.
• The boundedness result holds even when the base is not rational, including cases where the base is birational to an Enriques surface.
• The hypotheses of fixing the degree of a multisection and the growth rate of pluricanonical forms are necessary for boundedness.
• The set of minimal terminal threefolds of Kodaira dimension 2 is bounded when these invariants are fixed.
• The proof relies on the boundedness of log Fano varieties in dimension two and the log boundedness of fibrations in codimension one.
• Every elliptic Calabi–Yau threefold arises as a fiber of a finite family of models after a finite sequence of flops, confirming boundedness up to flops and hence topological finiteness.