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Let $X$ be a complex manifold, and let $p:X\to \mathbb{D}$ be a surjective holomorphic map that is a submersion and has compact fibers. That is, $X$ is a family of diffeomorphic compact complex manifolds. In particular, if $X$ is projective, then we have deformation invariance of plurigenera.

What is an example of two diffeomorphic compact Kähler complex manifolds with different plurigenera? That is, why is invariance of plurigenera specific to deformations and not just diffeomorphisms? Of course, one should not expect plurigenera to be a smooth invariant.

Interestingly, Seiberg-Witten theory implies that plurigenera is a diffeomorphism invariant for complex surfaces (not necessarily Kähler), so such an example must come from higher dimensions.

Colin Fan
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Let $X$ be the blowup of $\mathbb{CP}^2$ at $8$ points. Note that $X$ is rational, so $\kappa(X) = -\infty$.

Let $Y$ be a Barlow surface which is a surface of general type which is homeomorphic but not diffeomorphic to $X$. As $Y$ is general type, we have $\kappa(Y) = \dim Y = 2$.

Since $X$ and $Y$ are simply connected four-manifolds with the same intersection form, they are smoothly h-cobordant by a theorem of Wall. Therefore $X\times Y$ and $Y\times Y$ are simply connected, smoothly h-cobordant eight-manifolds, so they are diffeomorphic by the h-cobordism theorem. Since $\kappa(Z_1\times Z_2) = \kappa(Z_1) + \kappa(Z_2)$, we have $\kappa(X\times Y) = -\infty$ while $\kappa(Y\times Y) = 4$. In particular, their plurigenera are vastly different: $P_m(X\times Y) = 0$ for all $m > 0$, while $P_m(Y\times Y)$ grows quartically in $m$.

Using similar ideas, one can obtain many examples of this form, see Răsdeaconu - The Kodaira dimension of diffeomorphic Kähler threefolds.

Michael Albanese
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