General position, minimality, and geometry Martin Bays Joint work - PowerPoint PPT Presentation

General position, minimality, and geometry Martin Bays Joint work with Emmanuel Breuillard 11-04-2018 FPS-Leeds

Geometry of a minimal type A geometry is a pregeometry with cl ( ∅ ) = ∅ and cl ( { x } ) = { x } . If X is a strongly minimal set (or SU-rank 1 type), then { acl ( x ) : x ∈ X } forms a geometry. Definition A geometry ( S , cl ) is modular if for a , b ∈ S and C ⊆ S , if a ∈ cl ( bC ) \ cl ( C ) then there exists c ∈ cl ( C ) such that a ∈ cl ( bc ) . Say a , b ∈ S are non-orthogonal if a ∈ cl ( bc ) for some c ∈ S . Fact (Veblen-Young co-ordinatisation theorem) The modular geometries of dimension ≥ 4 in which every two points are non-orthogonal are precisely the projective geometries P F ( V ) of vector spaces of dimension ≥ 4 over division rings.

Examples Examples of naturally arising projective geometries: ◮ Let n > 1, and let Λ ≤ C n be a generic lattice Λ ∼ = Z 2 n . Then the complex torus T = C n / Λ has no infinite complex analytic subsets. Pillay: T is a modular strongly minimal set in the theory CCM of compact complex manifolds. ◮ Manin kernels in DCF . ◮ In ACFA, { σ ( x ) = η ( x ) } for an appropriate endomorphism η of an abelian variety. ◮ Structure induced from ACF on roots of unity or on the torsion subgroup of a simple abelian variety (Manin-Mumford). Any ω -categorical, and more generally any pseudofinite, strongly minimal set is locally modular.

Pseudofinite sets in fields Hrushovski “On Pseudo-Finite Dimensions” (2013) ◮ U ⊆ P ( ω ) non-principal ultrafilter. ◮ K := C U . ◮ X ⊆ K n is internal if X = � s →U X s for some X s ⊆ C n . s →U | X s | ∈ R U . Then | X | := � ◮ Let ξ ∈ R U with ξ > R . ◮ Coarse pseudofinite dimension: � log ( | X | ) � δ ( X ) = δ ξ ( X ) := st ∈ R ≥ 0 ∪ {−∞ , ∞} . log ( ξ )

L int monster ◮ L int : predicate for each internal X ⊆ K n . ◮ K ≻ K monster model in L int . ◮ δ has unique continuous extension to K , namely δ ( φ ( x , a )) := inf { q ∈ Q : K � ∃ <ξ q x . φ ( x , a ) } . ◮ δ (Φ) := inf { δ ( φ ) : Φ � φ } . ◮ δ ( a / C ) := δ ( tp ( a / C )) . Fact For C ⊆ K small and a , b ∈ K <ω , (i) a ≡ C b = ⇒ δ ( a / C ) = δ ( b / C ) . (ii) δ ( ab / C ) = δ ( a / bC ) + δ ( b / C ) . (iii) A partial type Φ over C has a realisation K � Φ( a ) with δ ( a / C ) = δ (Φ) .

acl 0 Superscript 0 means: reduct to ACF with parameters for C . Definition For B ⊆ K , ◮ acl 0 ( B ) := C ( B ) alg ≤ K ; ◮ dim 0 ( B ) := trd ( B / C ) . ◮ Cb 0 ( a / B ) := Cb ACF ( a / C ( B )) Remark C ⊆ dcl ( ∅ ) , so a ∈ acl 0 ( B ) = ⇒ δ ( a / B ) = 0.

Coarse general position Definition Let W be an irreducible variety over C . A definable set X ⊆ W is in coarse general position (or is cgp ) if 0 < δ ( X ) < ∞ and for any W ′ � W proper subvariety over K , δ ( X ∩ W ′ ) = 0. Example G a complex semiabelian variety, e.g. G = ( C × ) n . Let γ ∈ G ( C ) generic. s →U {− s · γ, . . . , s · γ } is cgp, since | X ∩ W ′ | < ℵ 0 by Then X := � uniform Mordell-Lang.

Coarse general position Definition Let W be an irreducible variety over C . A definable set X ⊆ W is in coarse general position (or is cgp ) if 0 < δ ( X ) < ∞ and for any W ′ � W proper subvariety over K , δ ( X ∩ W ′ ) = 0. Definition a ∈ W ( K ) is cgp if for any B ⊆ K , dim 0 ( a / B ) < dim 0 ( a ) = ⇒ δ ( a / B ) = 0 . If X is cgp, then any a ∈ X is cgp.

Coarse general position Definition a ∈ W ( K ) is cgp if for any B ⊆ K , dim 0 ( a / B ) < dim 0 ( a ) = ⇒ δ ( a / B ) = 0 . Definition P ⊆ K eq is coherent if ◮ every a ∈ P is cgp, and ◮ for any tuple a ∈ P <ω , dim 0 ( a ) = δ ( a ) . Then ( P ; acl 0 ) is a pregeometry.

Szemerédi-Trotter bounds Lemma (Elekes-Szabó, Fox-Pach-Sheffer-Suk-Zahl) Suppose X 1 ⊆ K n 1 and X 2 ⊆ K n 2 are � -definable, and V ⊆ K n 1 + n 2 is K -Zariski closed. Let X := ( X 1 × X 2 ) ∩ V. Suppose for a , b ∈ X 2 with a � = b, we have δ ( X ( a ) ∩ X ( b )) = 0 . 4 n 2 − 1 > 0 and y + := max { 0 , y } , 1 Then with ǫ 0 := δ ( X ) ≤ max ([ 1 2 δ ( X 1 ) + δ ( X 2 )] − ǫ 0 [ δ ( X 2 ) − 1 2 δ ( X 1 )] + , δ ( X 1 ) , δ ( X 2 )) . In particular, if δ ( X 2 ) > 1 2 δ ( X 1 ) > 0 , then δ ( X ) < 1 2 δ ( X 1 ) + δ ( X 2 ) . Hrushovski: Szemerédi-Trotter corresponds to modularity.

Coherent linearity Lemma If P is coherent, a 1 , a 2 , b 1 , . . . , b n ∈ P, dim 0 ( a 1 ) = k = dim 0 ( a 2 ) and 0 a 2 but a 1 � | 0 b a 2 . Let e := Cb 0 ( a / b ) . Then dim 0 ( e ) = k. a 1 | ⌣ ⌣ Proof. X 1 := tp ( a ) , X 2 := tp ( e ) , V := loc 0 ( ae ) . By cgp and canonicity, δ ( X ( e 1 ) ∩ X ( e 2 )) = 0 for e 1 � = e 2 ∈ X 2 . Meanwhile, δ ( X ) − δ ( X 2 ) ≥ δ ( a / e ) ≥ δ ( a / b ) = dim 0 ( a / b ) = 1 2 dim 0 ( a ) = 1 2 δ ( X 1 ) . So by Szemerédi-Trotter bounds, must have δ ( X 2 ) ≤ 1 2 δ ( X 1 ) . Now e ∈ acl 0 ( b ) and b is coherent, and it follows that dim 0 ( e ) ≤ δ ( e ) . So dim 0 ( e ) ≤ δ ( e ) = δ ( X 2 ) ≤ 1 2 δ ( X 1 ) = k .

Coherent modularity Lemma If P is coherent, a 1 , a 2 , b 1 , . . . , b n ∈ P, dim 0 ( a 1 ) = k = dim 0 ( a 2 ) and 0 a 2 but a 1 � | 0 b a 2 . Let e := Cb 0 ( a / b ) . Then dim 0 ( e ) = k. a 1 | ⌣ ⌣ Moreover, { e } is coherent. ccl ( P ) := { x ∈ acl eq ( P ) : { x } is coherent } . If P is coherent, so is ccl ( P ) . Proposition Suppose P = ccl ( P ) is coherent. Then ( P , acl 0 ) is a modular pregeometry.

Projective geometries fully embedded in algebraic geometry Example Suppose G is a complex abelian algebraic group and F ≤ Q ⊗ Z End ( G ) is a division ring. F acts by endomorphisms on G ( K ) / G ( C ) . Let A ⊆ G ( K ) be a set of independent generics. Let V := � A / G ( C ) � F . If b = � x / G ( C ) � F ∈ P F ( V ) , let η ( b ) := acl 0 ( x ) . Then if b ∈ P F ( V ) <ω , we have dim 0 ( η ( b )) = dim ( G ) · dim P F ( V ) ( b ) . Theorem (“Evans-Hrushovski for K eq ”) Suppose P F ( V ) is a projective geometry, and η : P F ( V ) → { L = acl 0 ( L ) } , and k G ∈ N , and dim 0 ( η ( b )) = k G · dim G ( b ) for any b ∈ P F ( V ) <ω . Then η is as in the example. G is unique up to isogeny.

Projective geometries fully embedded in algebraic geometry Proof idea. Abelian group configuration yields G . [ 0 : 1 : 0 ] [ 1 : 1 : 0 ] [ 1 : 0 : 0 ] [ 2 : 1 : 1 ] [ 1 : 1 : 1 ] [ 1 : 0 : 1 ] [ 0 : 0 : 1 ] Version due to Faure of the fundamental theorem of projective geometry embeds F in Q ⊗ Z End ( G ) .

Elekes-Szabó consequences Definition Say a finite subset X of a variety W is τ -cgp if for any proper subvariety W ′ � W of complexity ≤ τ , we have | X ∩ W ′ | < | X | 1 τ . Definition If V ⊆ � i W i are irreducible complex algebraic varieties, with dim ( W i ) = m and dim ( V ) = dm , say V admits a powersaving if for some τ and ǫ > 0 there is a bound � X i ∩ V | ≤ O ( N d − ǫ ) | i for τ -cgp X i ⊆ W i with | X i | ≤ N .

Elekes-Szabó consequences Definition H ≤ G n is a special subgroup if G is a commutative algebraic group and H = ker ( A ) o for some A ∈ Mat ( F ∩ End ( G )) for some division subalgebra F ≤ Q ⊗ Z End ( G ) . Theorem V ⊆ � i W i admits no powersaving iff it is in co-ordinatewise algebraic correspondence with a product of special subgroups.

Elekes-Szabó consequences; detailed statement Definition a ∈ W ( K ) is dcgp if a ∈ X ⊆ W ( K ) for some ∅ -definable cgp X . Theorem Given V ⊆ � i W i , TFAE (a) V admits no powersaving. (b) Exists coherent generic a ∈ V ( K ) with a i dcgp in W i . (c) Exists coherent generic a ∈ V ( K ) . (d) V is in co-ordinatewise algebraic correspondence with a product of special subgroups. Proof. ( a ) ⇔ ( b ) : ultraproducts. ( b ) = ⇒ ( c ) : clear. ( c ) = ⇒ ( d ) : “higher Evans-Hrushovski”. ( d ) = ⇒ ( b ) : see below.

Example ◮ G := ( C × ) 4 . ◮ Q ⊗ Z End ( G ) ∼ = Q ⊗ Z Mat 4 ( Z ) ∼ = Mat 4 ( Q ) . ◮ H Q = ( Q [ i , j , k ] : i 2 = j 2 = k 2 = − 1 ; ij = k ; jk = i ; ki = j ) embeds in Mat 4 ( Q ) via the left multiplication representation. ◮ H Z = Z [ i , j , k ] ⊆ H Q acts on G by endomorphisms: n · ( a , b , c , d ) = ( a n , b n , c n , d n ); i · ( a , b , c , d ) = ( b − 1 , a , d − 1 , c ); j · ( a , b , c , d ) = ( c − 1 , d , a , b − 1 ); k · ( a , b , c , d ) = ( d − 1 , c − 1 , b , a ) . ◮ Then V := { ( x , y , z 1 , z 2 , z 3 ) ∈ G 5 : z 1 = x + y , z 2 = x + i · y , z 3 = x + j · y } is a special subgroup of G 5 .

Example (continued) ◮ V := { ( x , y , z 1 , z 2 , z 3 ) ∈ G 5 : z 1 = x + y , z 2 = x + i · y , z 3 = x + j · y } is a special subgroup of G 5 . ◮ “Approximate H Z -submodules” witness that V admits no powersaving: ◮ H N := { n + mi + pj + qk : n , m , j , k ∈ [ − N , N ] } ⊆ H Z ◮ g ∈ G generic ◮ X N := H N · g = { h · g : h ∈ H N } ⊆ H Z · g ⊆ G . ◮ Then (by uniform Mordell-Lang), for W � G proper closed of complexity ≤ τ , | W ∩ H Z g | ≤ O τ ( 1 ) . ◮ So ∀ τ. ∀ N >> 0 . X N is τ -cgp in G . ◮ But i · X N = X N = j · X N , so | X 5 N ∩ V | ≥ Ω( | X N | 2 ) .