Likely intersections

Sebastian Eterović

doi:10.1017/fms.2025.10114

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Likely intersections

2025-12-15

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Sebastian Eterović

Thomas Scanlon *

DOI：10.1017/fms.2025.10114

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Abstract

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We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient spaces definable in some fixed o-minimal expansion of the ordered field of real numbers. For an instance of our general result, consider the case of subvarieties of Shimura varieties. Let S be a Shimura variety. Let pi : D -> Gamma\D = S realize S as a quotient of D, a homogeneous space for the action of a real algebraic group G, by the action of Gamma < G, an arithmetic subgroup. Let S' subset of S be a special subvariety of S realized as pi(D') for D' subset of D a homogeneous space for an algebraic subgroup of G. Let X subset of S be an irreducible subvariety of S not contained in any proper weakly special subvariety of S. Assume that the intersection of X with pi(gD') is persistently likely as g ranges through G with pi(gD') a special subvariety of S, meaning that whenever zeta : S-1 -> S and xi : S-1 -> S-2 are maps of Shimura varieties (regular maps of varieties induced by maps of the corresponding Shimura data) with finite, dim xi zeta X-1 + dim xi zeta(-1 )pi(gD') >= dim xi S-1. Then X boolean AND U-g is an element of G,U- pi(gD') is special pi(gD') is dense in X for the Euclidean topology.

Keywords:

AX-SCHANUEL

VARIETIES