Let X be a complex projective manifold. A manifold is rationally connected if any two general points can be joined by a chain of rational curves. Write Omega^1_X for the cotangent bundle of X. Mumford's conjecture. If H^0 big(X,(Omega^1_X)^otimes mbig)=0 for all m≥ 1, then X is rationally connected. The conjecture characterizes rationally connected manifolds by the vanishing of all tensor powers of their holomorphic one-forms. It is known when dim X≤ 3, while the general higher-dimensional case remains open.

Mumford's conjecture on rational connectedness

About 10 years old · traced to 2016

Let $X$ be a complex projective manifold. A manifold is rationally connected if any two general points can be joined by a chain of rational curves. Write $\Omega^1_X$ for the cotangent bundle of $X$ . Mumford's conjecture. If

H^0\big(X,(\Omega^1_X)^{\otimes m}\big)=0

for all $m\geq 1$ , then $X$ is rationally connected.

The conjecture characterizes rationally connected manifolds by the vanishing of all tensor powers of their holomorphic one-forms. It is known when $\dim X\leq 3$ , while the general higher-dimensional case remains open.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Mumford's conjecture on rational connectedness
Let $X$ be a projective manifold. Rational connectedness means that any two points of $X$ can be connected by a rational curve. Mumford's conjecture. If
$H^0(X,(T^*_X)^{\otimes m})=0, \quad\text{for every}\quad m\geq 1,$
then $X$ is rationally connected. The converse to the vanishing of holomorphic tensor fields on rationally connected projective manifolds would characterize rational connectedness; the source presents this as a well-known conjecture and describes the paper's differential-geometric approach toward it.
source: Xiaokui Yang, “RC-positivity, rational connectedness and Yau's conjecture”, arXiv:1708.06713 (2018).

References

Primary source

Vladimir Lazić and Thomas Peternell, “Rationally connected varieties - on a conjecture of Mumford”, arXiv:1608.04706 (2016).

Additional references

2 papers in this index state this conjecture (2005–2016). The statement above is taken from the most recent of them; the others are arXiv:math/0504266.

Source: https://arxiv.org/abs/1608.04706 Mumford (as cited in Kollár, 1996), Rational Curves on Algebraic Varieties

Posted 1 month ago

Mumford's conjecture on rational connectedness

Mumford's conjecture on rational connectedness

Primary source

Additional references

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