Signature operator

Remarks

The operator $d + \delta$ is elliptic with square the Hodge Laplacian, so $D^+$ is an elliptic operator of order $1$. Its index equals the signature $\sigma(M)$ of the intersection form on $H^{2k}(M; \R)$. Indeed, the harmonic forms of degrees $p \ne 2k$ contribute equally to the kernel and the cokernel and cancel. On harmonic $2k$-forms $\tau = \star$, whose $\pm 1$ eigenspaces are the self-dual and anti-self-dual forms. The Atiyah–Singer index theorem applied to $D^+$ gives the Hirzebruch signature theorem.

Example: on $2k$-forms, $\tau = i^{2k(2k-1) + 2k} \star = i^{4k^2} \star = \star$. For $M = \C P^2$ ($k = 1$), the harmonic $2$-forms are the multiples of the Kähler form $\omega_{FS}$, which is self-dual, so $\sigma(\C P^2) = 1$ and the index of $D^+$ is $1$.

Non-example: the operator $d + \delta \colon \Omega^{\mathrm{even}}(M) \to \Omega^{\mathrm{odd}}(M)$ uses the splitting by degree instead of the splitting by $\tau$. It is the de Rham or Euler operator, and its index is the Euler characteristic, not the signature; for $\C P^2$ it is $3$.