Let $k$ be an algebraically closed field, and $X$ a projective variety over $k$. Let $i : X\subset \mathbf{P}^d\_k$ be a closed immersion into a projective space of high enough dimension.



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> Is there a smooth projective variety $X'$ that is a global complete intersection in some projective space, together with an alteration $X'\to X$?
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In other words, can one alter projective varieties into smooth global complete intersections? Presumably the answer is "no" (for instance when $X$ itself is smooth) and there's some cohomological obstruction that I'm unaware of.



Alterations and smooth complete intersections
As you guessed, there are cohomological obstructions. Indeed, if $f \colon Y \twoheadrightarrow X$ is a surjective morphism of smooth projective varieties and $H$ is a Weil cohomology theory (with coefficients containing $\mathbf Q$), then the pullback $f \colon H^i(X) \to H^i(Y)$ is *injective*; see for instance [Kleiman, Prop. 1.2.4].


So in particular, if $\dim X > 1$ and $H^1(X) \neq 0$, then this is impossible since a smooth complete intersection $Y$ satisfies
$$h^i(Y) = \begin{cases}1, & i \text{ even} \\ 0, & i \text{ odd}\end{cases}$$
for $i < \dim Y$. (Since $H^1(Y)$ is a birational invariant of smooth projective varieties, replacing $Y \twoheadrightarrow X$ by a dominant rational map $Y \dashrightarrow X$ doesn't help.)




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**References.**


[Kleiman] S. L. Kleiman, *Algebraic cycles and the Weil conjectures*. In: Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math. **3**, p. 359-386 (1968). [ZBL0198.25902](https://zbmath.org/?q=an:0198.25902).


