If we are given compact complex manifold $X$ and a Kahler class $[\omega]$, can we always find a positive definite representative $\omega \in [\omega]$ that is real analytic?
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1$\begingroup$ For hyperkahler manifold, S.Boucksom showed that a $(1, 1)$ class $ \{\omega\} $ is Kahler if and only if it lies in the positive part of the Beauville-Bogomolov quadratic cone and moreover $\int_C \omega > 0$ for all rational curves $C ⊂ X$. For general case it is a conjecture yet! SEE : Cones positifs des varietes complexes compactes, Thesis, Grenoble 2002. $\endgroup$– user21574Commented Jul 25, 2017 at 3:29
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Yes. Run the Kähler-Ricci heat flow $$ \frac{\mathrm{d}}{\mathrm{d}t}\left(\omega(t)\right) = -\mathrm{Ric}\bigl(\omega(t)\bigr) $$ with initial condition $\omega(0) = \omega$. This will exist for some time interval $[0,T)$, and the $\omega_t$ for $t>0$ will all be be real-analytic with respect to the natural real-analytic structure on $M$ given by the complex structure. Note that we have $$ \left[\omega(t)\right] = \left[\omega(0)\right] - t\,c_1(M). $$
Now, for $0<t_1<t_2<T$, with $t_1$ very small with respect to $t_2$, consider the $2$-form $$ \omega(t_1,t_2) = \frac{\omega(t_1) - (t_1/t_2)\ \omega(t_2)}{1-t_1/t_2} $$ It is easy to see that, if $t_1/t_2>0$ is very small, then $\omega(t_1,t_2)$ is a positive $(1,1)$-form. It is real-analytic and, by construction, it satisfies $$ \left[\omega(t_1,t_2)\right] = \left[\omega(0)\right] = [\omega]. $$
answered Jul 2, 2014 at 13:19
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$\begingroup$ Thank you for the answer. By Kahler Ricci flow you mean $$\partial_{t}\omega_{t}=-Ric(\omega_{t})$$? Sorry but i don't see why $\omega_{0}$ and $\omega_{t}$ are cohomologous. $\endgroup$– ItaloCommented Jul 2, 2014 at 14:23
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$\begingroup$ Oh, sorry, I should have remembered that you have to do a linear combination in general. I'll fix that. Thanks for pointing out that oversight. $\endgroup$ Commented Jul 2, 2014 at 14:55
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$\begingroup$ Many thanks! can you tell me a reference where it is proved that the evolution of a smooth Kahler metric through Kahler Ricci flow is real analytic? $\endgroup$– ItaloCommented Jul 2, 2014 at 18:39
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$\begingroup$ It's in the literature, but I don't remember an explicit reference. It follows immediately, though, from Bando's original 1987 result that any solution of Ricci-flow on a compact Riemannian manifold is, at any positive time, real-analytic in geodesic normal coordinates coupled with the fact that, in the Kähler-Ricci case, the complex structure $J$ is parallel with respect to each metric $g(t)$ associated to $\omega(t)$ for $t>0$. This implies that $J$ is real-analytic in geodesic normal coordinates for each $g(t)$ and hence $J$-holomorphic functions are real-analytic in such coordinates. $\endgroup$ Commented Jul 2, 2014 at 19:06
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$\begingroup$ @RobertBryant No, your answer need to be revised! You may never have such initial metric when $0<kod(X)<\dim X$. See hal.archives-ouvertes.fr/hal-01413754 $\endgroup$– user21574Commented Jul 25, 2017 at 3:18