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The rational connectivity of (complex) Fano manifolds ($c_1(T_X) > 0$) is one of the major, and surely most memorable achievements of Mori's bend-and-break method. To this day, despite intensive work on the side of complex geometers, no complex analytic proof has been discovered; instead, one produces rational curves by deformation theory amplified with the use of Frobenius twists along the closed fibres of a spreading of the variety over $\mathrm{Spec} \, \mathbb{Z}$.

Then there is Graber, Harris and Starr's famous fibration theorem, again obtained by coherent sheaf cohomology and deformation theory: the total space of a rationally connected fibration over a rationally connected base is rationally connected. This shows that the class of rationally connected varieties is extremely well behaved geometrically, affirming its central role in the structure theory of algebraic varieties. But, viewed more arithmetically, the theorem also turns out to be equivalent to the following diophantine statement: Over $K := \mathbb{C}(x)$, every regular and projective rationally connected variety has a $K$-rational point.

The combination of these two (both deep and rather different!) results yields a geometric extension of the old theorem of Tsen that a system of homogeneous polynomial equations in $n+1$ variables over $K$ has a non-trivial solution if the sum of the degrees does not exceed $n$ -- in the "generic" (geometric) case that the latter system cuts out a regular complete intersection in $\mathbb{P}_{K}^n$. It should of course be mentioned that both results have perfect analogs over a finite field $K$, though proved completely differently by $l$-adic or $p$-adic cohomology methods. On the geometric side, this is a theorem of Esnault, and on the algebraic side, it is the classical Chevalley-Warning theorem. This similarity portrays the one-dimensional feel of the fields $\mathbb{C}(x)$ and $\mathbb{F}_p$.

Now, the Graber-Harris-Starr theorem has been viewed as an algebro-geometric enrichment of the basic topological observation that a fibration over $S^1$ having connected and path connected fibre has a section. Over a two-dimensional CW complex as base, the same conclusion holds if the fibre is also simply connected, and this has lead algebraic geometers to look for the right notion of rational simple connectivity that would, similarly, allow a geometric extension over a two-dimensional base $K := \mathbb{C}(x,y)$ of Tsen's (and Lang's) algebraic theorem -- which now states that a system of degrees $d_i$ has a non-trivial solution if $\sum d_i^2 \leq n$. Much progress here has been made by Starr, de Jong, and others, although (if my understanding is correct) the definition of rational simple connectivity is only tentative unless $\mathrm{Pic}(X) = \mathbb{Z}$.

But how about a possibly different geometric extension of the Tsen-Lang theorem, which does not pass through higher rational connectivity? If $X$ is a smooth complete intersection of degree $(d_1,\ldots,d_m)$ in $\mathbb{P}_{\mathbb{C}}^n$ then the $r$-th graded piece of the Chern character of $X$ is just $$ \mathrm{ch}_r(X) = \frac{1}{r!} \Big( n+1 - \sum d_i^r \big)c_1(H)^r. $$ This way, Tsen and Lang's $C_r$-condition $\sum d_i^r \leq n$ may be read geometrically: it means that the class $\mathrm{ch}_r(X)$ is strictly positive on every $r$-dimensional integral subscheme of $X$.

Thus one (or at least I) can't help wondering if the following could be a general geometric extension of the Tsen-Lang theorem:

Let $X$ be a regular and projective variety over $K := \mathbb{C}(x_1,\ldots,x_r)$ whose general fibres are complex manifolds having $\mathrm{ch}_i(X) > 0$ for all $i = 0,1,\ldots,r$. If the Colliot-Thelene and Sansuc elementary obstruction vanishes, could it follow that $X(K) \neq \emptyset$?

At least for $r = 2$, I think such manifolds go in the literature by the name of higher Fano varieties. My Question, then, is whether this naive proposition has any known counterexamples; whether it cannot possibly hold in this general form, or whether on the contrary it has been considered as a plausible geometric extension of Tsen's and Lang's theorems.

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    $\begingroup$ Aren't Severi-Brauer varieties counterexamples? $\endgroup$
    – dhy
    May 1 '15 at 23:15
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    $\begingroup$ @dhy: Good point. Added to the statement there must be some vanishing of Brauer type obstruction. This was not seen by one-dimensional fields such as $\mathbb{C}(x)$ and $\mathbb{F}_p$. $\endgroup$ May 1 '15 at 23:18
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    $\begingroup$ You are not supposed to write "Harris, Graber and Starr". You are supposed to write "Graber, Harris and Starr". Silly, I know, but those are the conventions we live by. $\endgroup$ May 2 '15 at 0:18
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    $\begingroup$ The Brauer type obstruction is the "elementary obstruction" of Colliot-Th'el`ene and Sansuc. $\endgroup$ May 2 '15 at 0:20
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    $\begingroup$ One issue is, what precisely do you mean by "positivity" of the higher graded pieces of the Chern character? For complete intersections in projective space, this is simple enough since the cohomology groups are principally generated: every class is either a positive, zero or negative multiple of an "ample" generator. But when the ranks of these groups are bigger than $1$, there are multiple, inequivalent notions of "positive", cf. "Ample Subvarieties of Algebraic Varieties", the work of Debarre-Ein-Lazarsfeld-Voisin, Brian Lehmann, Mihai Fulger, etc. $\endgroup$ May 2 '15 at 0:26
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There are some cases (slight generalizations of Tsen-Lang) where this has been proved. If you look in Kollár's "Rational Curves on Algebraic Varieties", he proves the result if the geometric generic fiber $X_{\overline{K}}$ is a complete intersection in a weighted projective space (where "complete intersection" means that the inverse image in the universal torsor / Cox space is a complete intersection). I believe that precisely the same proof as there works for $X_{\overline{K}}$ a complete intersection in a toric variety with (at worst) finite quotient singularities.

Beyond these examples, there are only a few other known examples of Fano manifolds where $\text{ch}_2(X)$ is positive on all surfaces, cf. work of Araujo-Castravet. Even Grassmannians (usually) fail to have $\text{ch}_2(X)$ positive. This makes it difficult to test conjectures. One approach is to weaken the hypothesis (replace "positivity" of $\text{ch}_2(X)$ by a weaker positivity hypothesis) which certainly gives many more examples. But already, finding which complete intersections in Grassmannians are rationally simply connected is surprisingly subtle (Rob Findley's thesis), and this remains open for complete intersections in other projective homogeneous spaces (e.g., in orthogonal and symplectic Grassmannians).

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