Is there any references concerning the computation of the fundamental groups and Hodge numbers of Fano schemes of lines in a smooth hypersurface in $\mathbb{P}^n$?



The topology of Fano schemes of lines
As Daniel Loughran points out, my claim regarding vanishing of the fundamental group is wrong. In all of the following, $F$ is the Fano scheme parameterizing lines on a degree $d$ hypersurface $X$ in $\mathbb{P}^n$. I am assuming that $F$ is smooth of the expected dimension $2n-d-3$. Here is what I can prove.


(1) When $d$ equals $2n-4$, except for $n=3$ and $d=2$, the Fano scheme of lines is a smooth, projective, geometrically connected curve, cf. Theorem V.4.3 of Kollár's "Rational Curves on Algebraic Varieties". In this case, there is an enumerative computation for the genus $g$ of this curve: $2g-2 = a\_{n-1,n-1}-a\_{n-2,n}$ where
$$
(n+1-d(d+1)/2)(x+y)\cdot \prod\_{e=0}^d ((d-e)x+ey) = \sum\_{l=0}^{2n-2} a\_{l,2n-2-l}x^ly^{2n-2-l}.
$$


(2) **When $d\leq n-4$, then $F$ is simply connected**. Since the fundamental group of smooth, projective schemes is stable under specialization, we may assume that $X$ is sufficiently general. Denote by $$\rho:\mathcal{C}\to F, \ \ \text{ev}:\mathcal{C} \to X,$$
the universal curve over $F$. Since $\mathcal{C}$ is a $\mathbb{P}^1$-bundle over $\mathcal{C}$, the induced map of fundamental groups,$$\pi\_1(\mathcal{C}) \to \pi\_1(F),$$ is an isomorphism. Thus, it suffices to prove that $\mathcal{C}$ is simply connected.


Since $X$ is sufficiently general, $\text{ev}$ is flat of relative dimension $n-d-1$, the generic fiber of $\text{ev}$ is smooth, and, over codimension $1$ points of $X$, the singular fibers of $\text{ev}$ have only ordinary double point singularities (this follows by considering the incidence correspondence of all hypersurfaces $X$ and lines contained in those hypersurfaces). Denote by $B\subset X$ the subvariety of codimension $\geq 2$ where the fibers of $\text{ev}$ have worse than ordinary double points, and denote by $U\subset X$ the open complement. Since codimension $2$ subvarieties do not affect the fundamental group, it suffices to prove that $\text{ev}^{-1}(U)$ is simply connected.


The fibers of $\text{ev}$ are naturally complete intersections of ample divisors in projective space. Since the dimension is $\geq 3$, by the usual Lefschetz hyperplane theorem (for singular complete intersections), the fibers of $\text{ev}$ over $U$ are simply connected. Probably, by analyzing more carefully the singularities, one can get the same result for $d=n-3$. Because the fibers of $\text{ev}$ over $U$ are simply connected, and because $U$ is simply connected, it follows that $\text{ev}^{-1}(U)$ is simply connected. 


I suspect that $F$ is simply connected in a larger degree range.


**Edit**. By the way, $\mathcal{C}$ is an iterated intersection of $d+1$ Cartier divisors in the projectivized tangent bundle, $\mathbb{P}T\_{\mathbb{P}^n}$. This gives an approach to compute Hodge numbers via residues. Since Hodge numbers are invariant under smooth deformations (in characteristic $0$), we can perform the computation for a general complete intersection (not just the special iterated intersections arising as universal curves over Fano schemes). 


**Second Edit**. To prove (algebraic) simple connectedness of $\mathcal{C}$, it suffices to prove that the higher cohomology of $\mathcal{O}\_{\mathcal{C}}$ is zero. Since the coherent sheaf $\mathcal{O}\_{\mathcal{C}}$ has a Koszul resolution in the projective homogeneous space $\mathbb{P}T\_{\mathbb{P}^n}$ where the locally free sheaves in the complex have filtrations with invertible subquotients, Bott vanishing will imply the vanishing of the higher cohomology of $\mathcal{O}\_{\mathcal{C}}$ in an appropriate degree range. Probably this is a broader degree range than $d\leq n-4$. 


I just checked, and the approach via Bott vanishing **does not work**. The problem is that there is nonvanishing higher cohomology of the structure sheaf whenever $d(d+1)/2$ is at least $n+1$; just apply the Hodge symmetries and observe that $h^0(F,\omega\_F)$ is nonzero. This gives a worse degree range than the computation above when $d\leq n-4$. 


**Third Edit**. I realized that the Lefschetz hyperplane theorem from (2) above applies whenever $d\leq n-3$, not just when $d\leq n-4$. The ambient scheme must have depth $\geq 3$, but the ample divisor may have depth $2$. Based on examples, I strongly suspect that also $F$ is simply connected when $d$ equals $n-2$.


When $d\geq n-1$, then there is an enumerative approach to proving non-simple connectedness. For $S$ a smooth complete intersection in $F$, or in $\mathcal{C}$, of ample divisors, if $S$ is a surface, then $S$ is simply connected if and only if $F$ is simply connected. Now, if $\chi(S,\mathcal{O}\_S) = (c\_1(T\_S)^2+c\_2(T\_S))/12$ is negative, then $S$ is not simply connected. Via "adjunction", we can reduce the computation of these Chern numbers to an intersection theory computation on $F$ or $\mathcal{C}$.


