Symplectic geometry of rationally connected threefolds

Abstract

S S S t t t o o o n

Symplectic geometry and rationally connected 4-folds

“…Proof. This essentially follows from the proof of Theorem 1.6 in [Tia12a]. In fact, by the result of Kollár and Ruan (Theorem 4.2.10, [Kol98], Proposition G, Proposition 4.9, [Rua99]), Y is uniruled.…”

“…Thus it suffices to show that there are only finitely many such curves through a general point in E ′ . But every irreducible curve C through a general point in E ′ with −K E ′ · C = 2 lies in the smooth locus of E ′ by Lemma 5.2 in [Tia12a]. Hence such curves have no first order deformation (as a stable map) with one point fixed.…”

Compact Kähler threefolds with non-nef canonical bundle and symplectic geometry

“…In the same direction, Kollár [Kol98] and Ruan [Rua99] showed that uniruledness of Kähler manifolds is a symplectic invariant. The same holds true for rational connectedness of smooth projective varieties of dimension up to three [Voi08,Tia12] and some of dimension four [Tia15].…”

Symplectic invariance of uniruled affine varieties and log Kodaira dimension

“…Work by Kollár and Ruan ([Kol96] and [Rua99]) tells us the property of being uniruled is a symplectic invariant using Gromov Witten theory. Another extremely useful notion in algebraic geometry is rational connectedness and this has been studied from a symplectic viewpoint in [Voi08] and [Tia12].…”