nLab Calabi-Yau manifolds in SU-bordism theory

Idea

Since Calabi-Yau manifolds have, in particular SU-structure, they represent classes in SU-bordism theory. In fact, all non-torsion $SU$-bordism classes are represented by products of linear combinations of classes of Calabi-Yau manifolds. Those in degree $4$ are spanned by the K3-surface.

Preliminaries

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

Proposition

The kernel of the forgetful morphism

$$\Omega^{SU}_\bullet \longrightarrow \Omega^{\mathrm{U}}_\bullet$$

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 $8k+1$ and $8k+2$, for $k \in \mathbb{N}$.

(CLP 19, Thm. 5.11a)

Proposition

Every torsion element in the SU-bordism ring $\Omega^{SU}_\bullet$ 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 $\mathbb{Z}\big[\tfrac{1}{2}\big]$ on one generator in every even degree $\geq 4$:

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

(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 $y_4 \in \Omega^{SU}_4$ in the SU-bordism ring (Prop. ) is represented by minus the class of any (non-torus) K3-surface:

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

(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 $\leq 4$ generating the SU-bordism ring in $deg \leq 8$ away from 2)

There are Calabi-Yau manifolds of complex dimension $3$ and $4$ whose whose SU-bordism classes equal the generators $\pm y_6$ and $\pm y_8$ in Prop. .

(CLP 19, Theorem 13.5b,c)

References

• Ivan Limonchenko, Zhi Lu, Taras Panov, Calabi-Yau hypersurfaces and SU-bordism, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 (arXiv:1712.07350)

Survey:

• Georgy Chernykh, Ivan Limonchenko, Taras Panov, $SU$-bordism: structure results and geometric representatives, Russian Math. Surveys 74 (2019), no. 3, 461-524 (arXiv:1903.07178)

• Taras Panov, A geometric view on $SU$-bordism, talk at Moscow State University 2020 (webpage, pdf, pdf)