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Michael Albanese
Michael Albanese
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Spin structures on Sasakian manifolds and the KahlerKähler analogy

A Sasakian manifold is often said to be the odd dimensional analogue of a KahlerKähler manifold.

Now for a $2n$-dimensional KahlerKähler manifold we know from Atiyah that it is spin exactly if the line bundle $\Omega^{(0,n)}$ admits a square root ${\cal S}$, and a choice of spin structure is equivalent to a choice of holomorphic structure ${\cal S}$. In this case the associated spinor bundle is the tensor product of ${\cal S}$ with the anti-holomorphic complex. Moreover, the associated Dirac operator is the tensor product of $\overline{\partial} + \overline{\partial}^*$ with the $\overline{\partial}$-operator corresponding to the choice of holomorphic structure.

So does the above analogy present any Sasakian versions of these spin geometry to complex geometry dictionary.?

Spin structures on Sasakian manifolds and the Kahler analogy

A Sasakian manifold is often said to be the odd dimensional analogue of a Kahler manifold.

Now for a $2n$-dimensional Kahler manifold we know from Atiyah that it is spin exactly if the line bundle $\Omega^{(0,n)}$ admits a square root ${\cal S}$, and a choice of spin structure is equivalent to a choice of holomorphic structure ${\cal S}$. In this case the associated spinor bundle is the tensor product of ${\cal S}$ with the anti-holomorphic complex. Moreover, the associated Dirac operator is the tensor product of $\overline{\partial} + \overline{\partial}^*$ with the $\overline{\partial}$-operator corresponding to the choice of holomorphic structure.

So does the above analogy present any Sasakian versions of these spin geometry to complex geometry dictionary.

Spin structures on Sasakian manifolds and the Kähler analogy

A Sasakian manifold is often said to be the odd dimensional analogue of a Kähler manifold.

Now for a $2n$-dimensional Kähler manifold we know from Atiyah that it is spin exactly if the line bundle $\Omega^{(0,n)}$ admits a square root ${\cal S}$, and a choice of spin structure is equivalent to a choice of holomorphic structure ${\cal S}$. In this case the associated spinor bundle is the tensor product of ${\cal S}$ with the anti-holomorphic complex. Moreover, the associated Dirac operator is the tensor product of $\overline{\partial} + \overline{\partial}^*$ with the $\overline{\partial}$-operator corresponding to the choice of holomorphic structure.

So does the above analogy present any Sasakian versions of these spin geometry to complex geometry dictionary?

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Alesandro Levi
Alesandro Levi
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Spin structures on Sasakian manifolds and the Kahler analogy

A Sasakian manifold is often said to be the odd dimensional analogue of a Kahler manifold.

Now for a $2n$-dimensional Kahler manifold we know from Atiyah that it is spin exactly if the line bundle $\Omega^{(0,n)}$ admits a square root ${\cal S}$, and a choice of spin structure is equivalent to a choice of holomorphic structure ${\cal S}$. In this case the associated spinor bundle is the tensor product of ${\cal S}$ with the anti-holomorphic complex. Moreover, the associated Dirac operator is the tensor product of $\overline{\partial} + \overline{\partial}^*$ with the $\overline{\partial}$-operator corresponding to the choice of holomorphic structure.

So does the above analogy present any Sasakian versions of these spin geometry to complex geometry dictionary.