[Paper Review] Updates on Hirzebruch's 1954 Problem List

D. Kotschick|arXiv (Cornell University)|May 20, 2013

Logic, programming, and type systems6 references3 citations

TL;DR

This paper updates Hirzebruch’s 1954 problem list with recent progress, particularly on the topological invariance of Chern and Hodge numbers in complex algebraic geometry. Using the unitary bordism ring and a newly defined Chern–Hodge ring, the authors fully characterize which rational linear combinations of Hodge and Chern numbers are invariant under oriented or unoriented homeomorphism and diffeomorphism, resolving Problem 31 completely by showing such invariants are generated by Betti and Pontryagin numbers, modulo Hirzebruch–Riemann–Roch relations.

ABSTRACT

We present updates to the problems on Hirzebruch's 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about contact structures, and about (complementary pairs of) foliations.

Motivation & Objective

• To provide updated solutions and progress on open problems from Hirzebruch’s 1954 problem list, especially those concerning topological invariance of characteristic numbers in complex algebraic geometry.
• To resolve Problem 31 on the diffeomorphism and homeomorphism invariance of Chern and Hodge numbers in smooth complex projective varieties.
• To extend previous results on Chern number invariance by incorporating Hodge numbers systematically through the construction of a mixed Chern–Hodge ring.
• To clarify which linear combinations of Hodge and Chern numbers are topological invariants under oriented or unoriented diffeomorphism and homeomorphism.
• To correct omissions in earlier updates and highlight open-ended problems that remain active areas of research.

Proposed method

• The authors use the unitary bordism ring tensored with ℚ to classify diffeomorphic algebraic varieties with distinct Chern numbers, enabling systematic bookkeeping of characteristic classes.
• They construct special basis sequences involving formal differences of orientation-reversing homeomorphic algebraic surfaces and projective bundles to generate examples with differing Thom–Milnor numbers.
• A new mathematical object, the Chern–Hodge ring of Kähler manifolds, is introduced to track both Hodge and Chern numbers simultaneously, generalizing the role of the unitary bordism ring.
• The method relies on refining the Hirzebruch–Riemann–Roch relations to identify all linear dependencies between Hodge and Chern numbers.
• By combining results from [Ko09, Ko12] with new joint work [KS13], the authors achieve completeness in characterizing invariants via the Chern–Hodge ring.
• The solution involves proving that any rational linear combination of Hodge and Chern numbers invariant under oriented diffeomorphism or homeomorphism must reduce to a combination of Betti and Pontryagin numbers, up to correction terms from the χ(Ω^p) − Td_p relations.

Experimental results

Research questions

• RQ1 Which rational linear combinations of Hodge and Chern numbers are invariant under oriented diffeomorphism or homeomorphism of smooth complex projective varieties?
• RQ2 Are the Hodge numbers of Kähler manifolds diffeomorphism invariants, and if not, what linear combinations are?
• RQ3 What is the complete set of relations that determine when a linear combination of Hodge and Chern numbers is a topological invariant?
• RQ4 How can one systematically classify invariants of complex algebraic varieties that are preserved under diffeomorphism or homeomorphism?
• RQ5 What is the role of the Hirzebruch–Riemann–Roch relations in determining the structure of the ring of invariants for Hodge and Chern numbers?

Key findings

• A rational linear combination of Hodge and Chern numbers is an oriented homeomorphism or diffeomorphism invariant if and only if it reduces to a linear combination of Betti and Pontryagin numbers after adding suitable multiples of the differences χ(Ω^p) − Td_p.
• In complex dimension n ≠ 2, a rational linear combination of Hodge and Chern numbers is an unoriented homeomorphism or diffeomorphism invariant if and only if it reduces to a linear combination of Betti numbers after adding suitable multiples of χ(Ω^p) − Td_p.
• The solution to Problem 31 is complete: the only topological invariants among rational linear combinations of Hodge and Chern numbers are those generated by Betti and Pontryagin numbers, modulo Hirzebruch–Riemann–Roch relations.
• The Chern–Hodge ring provides a complete algebraic framework for tracking both Hodge and Chern numbers, resolving the lack of bordism invariance for Hodge numbers.
• The results show that Hodge numbers are not diffeomorphism invariants in general, as demonstrated by constructed examples of diffeomorphic varieties with different Hodge numbers.
• The framework confirms that in complex dimension 2, both c₂ and c₁² are diffeomorphism invariants, but this fails in higher dimensions.