K-homology classes of Dirac operators on Hermitian manifolds

Question

Given a compact Hermitian manifold $M$, we have three canonical pseudo-differential operators on the sections of complexified de Rham complex, namely

1) (d + d$^*,\Omega^{*})$

2) ($\partial$ + $\partial^*,\Omega^{(*,0)})$

3) $(\overline{\partial} + \overline{\partial}^*,\Omega^{(0,*)})$

Question 1: What is the relation between the K-homology classes (see example 3.4) of these three operators? Can we at least say that all three give non-equivalent classes? In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? I.e. does tensoring by a line bundle descent to an operation on K-homology?

Assume in addition $M$ is spin, which according to Atiyah (see this question) means that $\Omega^{(0,n)}$ admits a line bundle. Then we also have an associated Dirac operator, which is the tensor product of the third operator with the square root endowed with a connection.

Question 2: N-lab says that the class of the Dirac is equal to the class of the operator in (1) above.

Question 3: If so, is this operation a permutation?

Answer 1

I am not entirely familiar with $KK$-theory, so please correct me if there are mistakes. I think ultimately you are trying to show the topological $K$-theory class you get from taking the horizontal derivative, vertical derivative, or the total deriative is the same for a double-complex. This can be done by standard comparsion theorems for de Rham cohomology. The usual way to do it is via diagram chasing, or if you strongly prefer using spectral sequences.

Here is a "direct" way to see it analytically. They are all self-adjoint elliptic operators, and their index should all be zero, essentially because $$ Ind(P)=\dim (\ker P)-\dim(\ker P^{*}) $$ There are some analytic subtlies to sort out ($C^{\infty}_{c}(X)$ is dense in the Sobolev space $H^{s}(X)$, which is an extension of the map on $C^{\infty}(X)$, etc). But the result should be the same. My guess is that you meant something more refined like the signature operator or the operator associated with a spin bundle.

Your second question: "In general, given two differential operators which are "representatives" of the same K-homology class, will tensoring them each by a line bundle always land in the same class? " does not make sense to me - how to you tensor a differential operator with a line bundle? Do you mean a "twisted"-differential operator or something? I do not really follow.