Algebraic structures with unbounded Chern numbers

Abstract

We determine all Chern numbers of smooth complex projective varieties of dimension at least 4 which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of linear combinations of Chern numbers with that property and prove its optimality in dimension 4.

Kähler structures on spin 6-manifolds

“…Moreover, none of the statements generalizes to higher dimensions. Indeed, for any n ≥ 4 there are smooth 2n-manifolds with infinitely many Kähler structures whose Chern numbers are unbounded, see [32]. The method of that paper can be adapted to produce similar examples with spin structures; more precisely, [32, Theorem 1] remains true for spin manifolds.…”

“…Let q ≥ 3 be an odd integer. As in [32], we consider the Dolgachev surface S q , which is the elliptic surface, obtained from a general pencil of plane cubic curves S / / P 1 by a logarithmic transformation of order 2 and q at two smooth fibres. There is an h-cobordism W q between S q and S 3 which induces isomorphisms H 2 (S q , Z) ≃ H 2 (S 3 , Z).…”

“…Since S q and S 3 are h-cobordant, w 2 (S q ) ∈ H 2 (S 3 , Z/ 2Z) does not depend on q and so the mod 2 reduction of c 1 (S q ) + ω vanishes for all q. Since h 2,0 (S q ) = 0, there is a line bundle L q ∈ Pic(S q ) with c 1 (L q ) = ω, see [32,Section 2].…”

Chern Numbers of Uniruled Threefolds

“…Generalising Hirzebruch's question, Kotschick asked [8] (see also [12]) whether the topology of the underlying smooth manifold determines the Chern numbers of smooth complex projective varieties at least up to finite ambiguity. In [19], we have shown that in dimension at least four, this question has in general a negative answer. That is, there are smooth real manifolds that carry infinitely many complex algebraic structures such that the corresponding Chern numbers are unbounded, except for c n , c 1 c n−1 and c 2 2 which are known to be bounded (see [14] for the non-trivial one c 1 c n−1 ).…”

“…The above result should be compared to the fact that all known examples of sequences of homeomorphic varieties with unbounded Chern numbers are Mori fibre spaces, and in fact projective bundles (see [19]). We therefore believe that together with the aforementioned results from [2], the above theorem puts forward strong evidence for the conjecture that the Chern numbers of smooth projective threefolds are determined up to finite ambiguity by the underlying smooth manifold.…”