Cfrgtkky

I believe the following is true and well known.

Theorem (?). Let $X$ and $Y$ be smooth, irreducible, projective varieties over $mathbb{C}$. Let
$$
fcolon Xrightarrow Y
$$
be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,wedge^kOmega_X)cong H^0(Y,wedge^kOmega_Y)$ for all $kge 0$.

I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).

By spectral sequence, we just need to prove that $R^if_*mathcal{O}_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I].

Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I].

Welcome new contributor. The answer above by @ChenJiang using Hodge theory and vanishing theorems is perfectly correct. I am posting a different answer that uses no Hodge theory nor vanishing theorems since this question is related to an open conjecture of David Mumford. Before starting, let me mention that in positive characteristic, the conclusion of @ChenJiang's argument above is open, so definitely the "Hodge dual" result involving pullbacks of $ell$-forms is better understood.

Open problem. For a smooth, projective morphism $f^o:X to Y$ of smooth, quasi-projective $k$-schemes, if the geometric fibers are separably rationally connected, do the higher direct image sheaves $R^ell f^o_*mathcal{O}_X$ vanish for $ell>0$?

Kollár proves this in characteristic $0$. The main result in positive characteristic is a beautiful theorem of Frank Gounelas that proves this for $ell=1$. Gounelas reviews other work on this problem in his paper.

MR3268754

Gounelas, Frank

The first cohomology of separably rationally connected varieties.

C. R. Math. Acad. Sci. Paris 352 (2014), no. 11, 871–873.
https://arxiv.org/pdf/1303.2222.pdf

Mumford's Conjecture relates the common zeroes of all contravariant tensors to the fibers of the maximally rationally connected fibration, proved to exist by Campana and by Kollár-Miyaoka-Mori.

Definition. For every smooth, projective $mathbb{C}$-variety $overline{X}$, a maximally rationally connected fibration (also known as a rational quotient) of $overline{X}$ is a surjective, projective morphism defined on a dense Zariski open subset of $overline{X}$, $$f:Xto Y,$$ whose geometric generic fiber is rationally connected and such that $f$ contracts all free rational curve in $overline{X}$ that intersect $X$.

Mumford's Conjecture. The relative tangent sheaf of $f$, $$T_{f}:= text{Ker}(df:T_{X/mathbb{C}} to f^*T_{Y/mathbb{C}}) cong Omega_f^vee,$$ equals the restriction to $X$ of the simultaneous kernel of $mathcal{O}_{overline{X}}$-module homomorphisms, $$phi:T_{overline{X}/mathbb{C}} to Omega_{overline{X}/mathbb{C}}^{otimes (ell-1)},$$ as $ell$ varies over all positive integers, and as $phi$ varies over all $mathcal{O}_{overline{X}}$-module homomorphisms.

This conjecture says, roughly, that we can recover the maximally rationally connected fibration from the global contravariant tensors on $overline{X}$. Mumford's Conjecture is open in dimensions $>3$, and in fact we do not even know whether the simultaneous kernel is involutive, i.e., stable under Lie bracket. Mumford's Conjecture follows from the Uniruledness Conjecture (negative Kodaira dimension implies uniruledness -- this conjecture is also sometimes attributed to Mumford), but there are examples of varieties where Mumford's Conjecture is known yet the Uniruledness Conjecture is open. However, one direction of Mumford's Conjecture is known in all dimensions, and this is related to the question in the post: the simultaneous kernel contains $T_f$, and the pullback maps are isomorphisms. In fact, that holds in arbitrary characteristic.

Let $k$ be an algebraically closed field of arbitrary characteristic.

Theorem. Let $X$ and $Y$ be smooth, connected, quasi-projective $k$-schemes. Let $$f:Xto Y$$ be a surjective, projective $k$-morphism whose geometric generic fiber is smooth and separably rationally connected. Then for every integer $ell>0$, the pullback map $$a_f^{otimes ell}:H^0(Y,Omega^{otimes ell}_{Y/k}) to H^0(X,Omega^{otimes ell}_{X/k})$$ is an isomorphism.

Proof. Denote by $X_{f,text{sm}}$, resp. $X_{f,text{sing}}$, the maximal open subscheme of $X$ on which $f$ is smooth, resp. the closed complement. Stated differently, the closed subset $X_{f,text{sing}}$ is the degeneracy locus of the pullback map of locally free $mathcal{O}_X$-modules,
$$alpha:f^*Omega_{Y/k} to Omega_{X/k}.$$ Thus, the open complement $X_{f,text{sm}}$ is the maximal open on which the cokernel of $alpha$ is locally free of rank $n:=text{dim}(X)-text{dim}(Y)$.

Injectivity of $a_f^{otimes ell}$. By hypothesis, the open set $X_{f,text{sm}}$ contains the generic fiber of $f$. It follows that $alpha$ is injective. Thus, also the $ell$-fold self-tensor product of this map of locally free $mathcal{O}_X$-modules is also injective, $$alpha^{otimes ell}: f^*Omega_{Y/k}^{otimes ell} to Omega_{X/k}^{otimes ell}.$$ Therefore the pullback map $a^{otimes ell}$ is injective.

Reduction of surjectivity of $a_f^{otimes ell}$ to $X_{f,text{sm}}$. Since also the following restriction map is injective, $$H^0(X,Omega_{X/k}^{otimes ell}) to H^0(X^o,Omega_{X^o/k}^{otimes ell}),$$ surjectivity of $a_f^{otimes ell}$ is equivalent to surjectivity of the following pullback map, $$a^{otimes ell}_{f,text{sm}}:H^0(Y,Omega_{Y/k}^{otimes ell}) to H^0(X_{f,text{sm}},Omega_{X/k}^{otimes ell}).$$

Loci of good reduction and bad reduction. Since the restriction of $f$ to $X_{f,text{sm}}$ is smooth, the image of this set in $Y$ is an open subset, $Y_{f,text{sm}}$. Since $f$ is proper and since $X_{f,text{sing}}$ is a closed subset of $X$, the image of $X_{f,text{sing}}$ in $Y$ is a closed subset $Y_{f,text{b}}$. Thus, the complement $Y_{f,{text{g}}}$ is an open subset, contained in $Y_{f,text{sm}}$ (both of these opens are relevant in the argument below). The closed subset $Y_{f,text{b}}$ is usually called the "discriminant locus" or the locus of "bad reduction". The open complement $Y_{f,text{g}}$ is usually called the locus of "good reduction". This is the maximal open subset of $Y$ such that for the inverse image $X_{f,text{g}}$, the restriction of $f$ is smooth and projective, $$f^o:X_{f,text{g}} to Y_{f,text{g}}.$$ (There is also a locus of "potentially good reduction", but this is not needed in the argument below.)

Filration of tensors on $X$ associated to $f$. The inverse image under $f$ of $Y_{f,text{g}}$ is an open subset $X_{f,text{g}}$ that is contained in $X_{f,text{sm}}$ and that contains the generic fiber of $f$. Since the restriction $f^o$ is smooth, the pullback map gives a locally split short exact sequence of locally free sheaves, $$0to f^*Omega_{Y_{f,text{g}}} to Omega_{X_{f,text{g}}} to Omega_{f^o} to 0.$$ Since this is locally split, all of the induced filtrations of the $ell$-fold self-tensor-product $Omega_{X_{f,text{g}}}^{otimes ell}$ are also locally split. The deepest subsheaf in this filtration is $f^*Omega_{Y_{f,text{g}}}^{otimes ell}$. By the projection formula, every sections on $X_{f,text{g}}$ of $f^*Omega_{Y_{f,text{g}}}^{otimes ell}$ is the pullback of a unique section on $Y_{f,text{g}}$ of $Omega_{Y_{f,text{g}}}^{otimes ell}$. The goal is to prove that every section on $X_{f,text{g}}$ of $Omega_{X_{f,text{g}}}^{otimes ell}$ is a section of the subsheaf $f^*Omega_{Y_{f,text{g}}}^{otimes ell}$.

Every other associated graded of this filtration is canonically isomorphic to a direct sum of copies of sheaves $$Omega^{ell,m}:= Omega_{f^o}^{otimes m} otimes_{mathcal{O}_X} f^*Omega_{X_{f,text{g}}}^{otimes(ell -m)},$$ for some integer $0<mleq ell$. Thus, equivalently, we want to prove that $Omega^{ell,m}$ has only the zero section on $X_{f,text{g}}$ for every integer $m$ with $0<mleq ell$.

Surjectivity of $a_f^{otimes ell}$ on the good locus $X_{f,text{g}}$. Note that the restriction of $f^*Omega_{X_{f,text{g}}}$ to any fiber of $f^o$ is isomorphic to a direct sum of copies of the structure sheaf of that fiber. Thus, the restriction of $Omega^{ell,m}$ to any fiber is isomorphic to a direct sum of copies of $Omega_{f^o}^{otimes m}$.

The restriction of $Omega_{f^o}$ to any very free curve in the fiber is anti-ample (by the definition of "very free curve"). Thus, the restriction of $Omega_{f^o}^{otimes m}$ is also anti-ample. So every global section of $Omega_{f^o}^{otimes m}$ on the fiber restricts to the zero section on each very free curve. Yet very free curves sweep out a dense open subset of the generic fiber, by the hypothesis that this fiber is separably rationally connected.

Thus, for every $0<mleq ell$, every global section of $Omega^{ell,m}$ on $X_{f,text{g}}$ restricts as the zero section on the generic fiber. Since $Omega^{ell,m}$ is a locally free sheaf (hence torsion free), the global section of $Omega^{ell,m}$ equals $0$. Therefore, every section $s_{text{g}}$ of $Omega_{X/k}^{otimes ell}$ on $X_{f,text{g}}$ is the pullback of a unique section $t_{text{g}}$ of $Omega_{Y/k}^{otimes ell}$ on $Y_{f,text{g}}$.

Extension of sections of $Omega_{Y/k}^{otimes ell}$ from the good locus to all of $Y$. Every global section $s$ of $Omega_{X/k}^{otimes ell}$ restricts on $X_{f,text{g}}$ to the pullback of a unique section $t_{text{g}}$ of $Omega_{Y/k}^{otimes ell}$ on $Y_{f,text{g}}$. Does $t_{text{g}}$ extend to a global section $t$ of $Omega_{Y/k}^{otimes ell}$?

If so, then both $s$ and the pullback of $t$ have the same restriction on the dense Zariski open subset $X_{f,text{g}}$. Since $Omega_{X/k}^{otimes ell}$ is locally free (hence torsion free), it follows that $s$ and the pullback of $t$ are equal on all of $X$. Thus, to finish the proof of surjectivity, it suffices to prove that $t_{text{g}}$ extends to all of $t$. Finally, by $S2$ extension, it suffices to prove that $t_{text{g}}$ extends to all codimension $1$ points of $Y$.

Rationally Connected Fibration Theorem. By the main theorem of the following (first paper in characteristic $0$, second paper in positive characteristic), the open subset $Y_{f,text{sm}}$ contains every codimension $1$ point of $Y$.

MR1937199 (2003m:14081)

Graber, Tom; Harris, Joe; Starr, Jason

Families of rationally connected varieties.

J. Amer. Math. Soc. 16 (2003), no. 1, 57–67.
https://www.ams.org/journals/jams/2003-16-01/S0894-0347-02-00402-2/S0894-0347-02-00402-2.pdf

MR1981034 (2004h:14018)

de Jong, A. J.; Starr, J.

Every rationally connected variety over the function field of a curve has a rational point.

Amer. J. Math. 125 (2003), no. 3, 567–580.

Thus, every codimension $1$ point $y$ of $Y$ is the image of a codimension $1$ point $x$ of $X_{f,text{sm}}$. Since $f$ is smooth at $x$, the image of $f^*Omega_{Y/k}$ in $Omega_{X/k}$ is saturated on a dense open subset that contains $x$. Thus, since the rational section $t_{text{g}}$ of $Omega_{Y/k}$ pulls back to a regular section of $Omega_{X/k}$ at $x$, it follows that $t_{text{g}}$ is also regular at $y$. QED

Remark. When $k$ has characteristic $0$, the natural action of the symmetric group $mathfrak{S}_{ell}$ on the tensor product $Omega^{otimes ell}$ decomposes as a direct sum of Schur functors. Pullback preserves these summands. Thus, the theorem above for tensor products implies the result for exterior products.