Given a simply connected smooth projective variety (hence a compact Kähler manifold) with singular cohomology generated in even degrees, do we know that there is a Morse function on it such that all its Morse indices are even?
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3$\begingroup$ I do not know anything about Kähler manifolds, but if one only assumes that the manifold is symplectic this is very far from true. Gompf has shown that there exist, for any finitely presented group $G$, a closed symplectic four manifold $M$ with fundamental group $G$. Now take a perfect group $G$, then $H_1(M)=H_3(M)=0$, but it will not admit a Morse function with only even indices. $\endgroup$– Thomas RotCommented Sep 18, 2019 at 9:10
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$\begingroup$ Hmm, I see. I will put an additional condition that the manifold is simply connected. Is it then true? $\endgroup$– FilipCommented Sep 18, 2019 at 10:14
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$\begingroup$ I don't know Filip92. $\endgroup$– Thomas RotCommented Sep 18, 2019 at 11:21
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Assuming that by `cohomology' you mean integer coefficients, there is a general result saying what you want for simply connected manifolds of dimension $5$ or more, without the Kähler condition. According to Smale (Generalized Poincare’s conjecture in dimensions greater than four, Ann. Math. 74, No, 2, 391-406 (1961)) for $n > 5$ and Barden (Simply connected five-manifolds, Ann, Math, 82,365-385 (1965)) if $n = 5$ a simply connected $n$-manifold has a handle decomposition (ie Morse function) with the fewest handles that you need for a chain complex representing the homology.
In your example, having only even dimensional cohomology means the same for homology, and hence there's no torsion. Thus there is a chain complex with only even index generators and so, for complex dimension at least $3$, by Smale a Morse function with the same property.
The case of complex surfaces is not yet established, although there are many concrete examples where there are decompositions with only even index handles. (For example, hypersurfaces in ${\mathbb C}P^3$, elliptic surfaces without multiple fibers...). See the book of Gompf-Stipsicz for some of this. A more recent striking example, due to Akbulut, is the Dolgachev surface, equal to a two-fold log transform of orders $2$ and $3$ on an elliptic surface $E(1)$. This had been previously suggested as a counterexample to your statement.
answered Sep 18, 2019 at 13:56
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$\begingroup$ Oh, that is great, thanks! As for the case of complex surfaces, in my examples I get only Hirzebruch's surfaces so, I guess for them one can do decompositions with even index handles? $\endgroup$– FilipCommented Sep 19, 2019 at 8:14
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1$\begingroup$ Hirzebruch surfaces are very easy! Up to diffeomorphism, they are either $S^2 \times S^2$ or the non-trivial $S^2$ bundle over $S^2$. In either case they are gotten by adding 2-handles to the 4-ball along the Hopf link (and then adding a 4-handle). This is one of the first examples you encounter in 4-manifolds; you can find it in in Gompf-Stipsicz (or do it yourself from the description as a bundle). $\endgroup$ Commented Sep 19, 2019 at 13:48