Rational 1-Connectedness Under Higher Fano Conditions

Definitions and mathematical statement

Let $X$ be a smooth projective Fano variety, so $-K_X$ is ample. In the usual complex, characteristic-zero interpretation, define the codimension-two class \[ \eta_X=c_2(\Omega_X^1)-\tfrac12c_1(K_X)^2. \] The verified question asks whether negativity of this class forces $X$ to be rationally $1$-connected. The source gives as a concrete negativity criterion \[ \eta_X\cdot[S]<0\quad\text{for every integral surface }S\subset X. \] Here the product is a numerical degree on the surface. Since $c_2(\Omega_X^1)=c_2(T_X)$, this criterion is equivalently strict positivity of \[ \operatorname{ch}_2(T_X)=\tfrac12c_1(T_X)^2-c_2(T_X)=-\eta_X \] on every such surface. No uniform lower bound or separate ampleness condition on a codimension-two class is added.

A conventional rational-$1$-connectedness formulation uses an effective curve class $\beta$ and an irreducible family $M\subset\overline{\mathcal M}_{0,2}(X,\beta)$ of genus-zero stable maps with two marked points. Its evaluation map \[ \operatorname{ev}:M\longrightarrow X\times X,\qquad [f:C\to X,p_1,p_2]\longmapsto(f(p_1),f(p_2)) \] should be dominant and have rationally connected geometric generic fiber. Rational connectedness means that two general points of that fiber can be joined by a rational curve. The question asks whether the Fano and stated Chern-negativity conditions guarantee the appropriate rational-$1$-connectedness property.

The extract does not specify the base field, the full meaning of "negative" beyond its illustrative surface-intersection test, or whether rational $1$-connectedness means the existence of one such component or a stronger assertion in all sufficiently positive degrees. The displayed component formulation is an explicit standard interpretation; those quantifiers must match the report before a stronger version is asserted.

The literature assessment checked on 21 August 2026 reports important low-degree and general-hypersurface results but no theorem resolving the general Chern-class criterion. Those special cases are not a proof for every smooth Fano variety satisfying the displayed inequality. The open label is dated triage requiring independent verification.

Short English statement

Does a smooth Fano variety with negative $c_2(\Omega_X^1)-\frac12K_X^2$, in particular negative intersection with every surface, have rationally connected families of rational curves through two general points? The exact rational-$1$-connectedness degree quantifiers need source confirmation.

Sources

• Arithmetic Algebraic Geometry (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/37, pp. 1971--2014, https://doi.org/10.4171/owr/2004/37; proposed year supplied as 2004.
• Literature assessment checked 21 August 2026: original report, https://doi.org/10.4171/owr/2004/37; Starr, "rational curves on low-degree hypersurfaces," https://arxiv.org/abs/math/0305432.
• Local UnsolvedMath registry, permanent ID $30000103$, exact source code catalogue entry, created 25 August 2026 and updated 16 September 2026. Its clean statement is labelled corrected and verified, with the precise Chern-class hypothesis and illustrative surface-intersection criterion recovered from the report.