nLab Dolbeault-Dirac operator

Idea

On a Kähler manifold $(X,\omega)$ with a choice of spin structure given by a Theta characteristic $\sqrt{\Omega^{0,n}}$, the sum of the Dolbeault operator $\overline{\partial}$ with its adjoint $\overline{\partial}^\ast$ are identified with the corresponding Dirac operator

$$\overline{\partial} + \overline{\partial}^\ast \;\colon\; S_X\simeq \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{0,n}} \to \Omega^{0,\bullet}_X \otimes \sqrt{\Omega^{0,n}} \simeq S_X \,.$$

Here the action of $\overline{\partial} + \overline{\partial}^\ast$ on $\Omega^{0,\bullet}$ is the canonical one. For the action on $\sqrt{\Omega^{n,0}}$ choose any connection $\nabla$ on this line bundle. Then on a local coordinate patch the action is given by differentiating along a coordinate vector, multiplying with the corresponding Clifford element and projecting on the antiholomorphic part, then summing this over all coordinate vectors (e.g. Friedrich 97, p. 79).

Properties

Index and genus

The index is the Todd genus (see there).

Related concepts

• Kähler-Dirac operator

• spin^c Dirac operator

References

Textbook accounts include

section 3.4 (specifically around p. 79) of

• Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate studies in mathematics 25, AMS (1997)