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Calabi-Yau manifolds in SU-bordism theory

Contents

Contents

Idea

Since Calabi-Yau manifolds have, in particular SU-structure, they represent classes in SU-bordism theory. In fact, all non-torsion SUSU-bordism classes are represented by products of linear combinations of classes of Calabi-Yau manifolds. Those in degree 44 are spanned by the K3-surface.

Preliminaries

We write Ω SU\Omega^{SU}_\bullet for the SU-bordism ring.

Proposition

The kernel of the forgetful morphism

Ω SUΩ U \Omega^{SU}_\bullet \longrightarrow \Omega^{\mathrm{U}}_\bullet

from the SU-bordism ring to the complex bordism ring, is pure torsion.

(CLP 19, Thm. 5.8a)

Proposition

The torsion subgroup of the SU-bordism ring is concentrated in degrees 8k+18k+1 and 8k+28k+2, for kk \in \mathbb{N}.

(CLP 19, Thm. 5.11a)

Proposition

Every torsion element in the SU-bordism ring Ω SU\Omega^{SU}_\bullet has order 2.

(CLP 19, Thm. 5.8b)

Proposition

(SU-bordism ring away from 2 is polynomial algebra)

The SU-bordism ring with 2 inverted is the polynomial algebra over [12]\mathbb{Z}\big[\tfrac{1}{2}\big] on one generator in every even degree 4\geq 4:

Ω SU[12][12][{y 2i+4} i]. \Omega^{SU}\big[\tfrac{1}{2}\big] \;\simeq\; \mathbb{Z} \big[ \tfrac{1}{2} \big] \big[ \{ y_{2i+4} \}_{i \in \mathbb{N}} \big] \,.

(due to Novikov 62, review in LLP 17, Thm. 1.2)

Statement

Proposition

(K3-surface spans SU-bordism ring in degree 4)

The degree-4 generator y 4Ω 4 SUy_4 \in \Omega^{SU}_4 in the SU-bordism ring (Prop. ) is represented by minus the class of any (non-torus) K3-surface:

Ω 4 SU[12][K3]. \Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle \,.

(LLP 17, Lemma 1.5, Example 3.1, CLP 19, Theorem 13.5a)

Proposition

(Calabi-Yau manifolds generate the SU-bordism ring away from 2)

The SU-bordism ring away from 2 is multiplicatively generated by Calabi-Yau manifolds.

(LLP 17, Theorem 2.4)

In particular:

Proposition

(Calabi-Yau manifolds in complex dim 4\leq 4 generating the SU-bordism ring in deg8deg \leq 8 away from 2)

There are Calabi-Yau manifolds of complex dimension 33 and 44 whose whose SU-bordism classes equal the generators ±y 6\pm y_6 and ±y 8\pm y_8 in Prop. .

(CLP 19, Theorem 13.5b,c)

References

Survey:

Last revised on November 27, 2020 at 03:48:46. See the history of this page for a list of all contributions to it.