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Algebraic Surfaces with Minimal Betti Numbers

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Algebraic Surfaces with Minimal Betti Numbers Algebraic surfaces with minimal Betti numbers JongHae Keum (Korea Institute for Advanced Study) France-Korea Conference Institute of Mathematics, University of Bordeaux 24-27 November 2019 JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 1 / 28 Outline 1 Q-homology Projective Planes 2 Montgomery-Yang Problem 3 Algebraic Montgomery-Yang Problem 4 Fake Projective Planes JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 2 / 28 In dimension > 1, many invariants: Hodge numbers, Betti numbers i;j j i i h (X) = dim H (X; ΩX ); bi (X) := dim H (X; Q): Given Hodge numbers (and even fixing fundamental group), hard to describe the moduli, in general. Q-homology Projective Planes Classify algebraic varieties up to connected moduli Nonsingular projective algebraic curves =C (compact Riemann surfaces) are classified by the “ mighty" genus 0 1 1 g(C) := (the number of “holes" of C) = dimC H (C; ΩC ) = 2 dimQ H1(C; Q): 1 g(C) = 0 () C =∼ P =∼ (Riemann sphere) = C [ f1g: JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 3 / 28 Q-homology Projective Planes Classify algebraic varieties up to connected moduli Nonsingular projective algebraic curves =C (compact Riemann surfaces) are classified by the “ mighty" genus 0 1 1 g(C) := (the number of “holes" of C) = dimC H (C; ΩC ) = 2 dimQ H1(C; Q): 1 g(C) = 0 () C =∼ P =∼ (Riemann sphere) = C [ f1g: In dimension > 1, many invariants: Hodge numbers, Betti numbers i;j j i i h (X) = dim H (X; ΩX ); bi (X) := dim H (X; Q): Given Hodge numbers (and even fixing fundamental group), hard to describe the moduli, in general. JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 3 / 28 Max Nöther(1844-1921) said [in the book of Federigo Enriques(1871-1946)]: "Algebraic curves are created by god, algebraic surfaces are created by devil." Q-homology Projective Planes Smooth Algebraic Surfaces with pg = q = 0 Long history : Castelnuovo’s rationality criterion, Severi conjecture, ... Here, the geometric genus and the irregularity n 0 n 0;n n;0 pg(X) := dim H (X; OX ) = dim H (X; ΩX ) = h (X) = h (X); 1 0 1 0;1 1;0 q(X) := dim H (X; OX ) = dim H (X; ΩX ) = h (X) = h (X): JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 4 / 28 Q-homology Projective Planes Smooth Algebraic Surfaces with pg = q = 0 Long history : Castelnuovo’s rationality criterion, Severi conjecture, ... Here, the geometric genus and the irregularity n 0 n 0;n n;0 pg(X) := dim H (X; OX ) = dim H (X; ΩX ) = h (X) = h (X); 1 0 1 0;1 1;0 q(X) := dim H (X; OX ) = dim H (X; ΩX ) = h (X) = h (X): Max Nöther(1844-1921) said [in the book of Federigo Enriques(1871-1946)]: "Algebraic curves are created by god, algebraic surfaces are created by devil." JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 4 / 28 Smooth algebraic surfaces with minimal invariants, that is, with b1 = b3 = 0; b0 = b2 = b4 = 1 () pg = q = 0) are P2; 2 fake projective planes (= surfaces of general type with pg = 0, K = 9). Remark. FPP’s are not simply connected. Exotic P2 does NOT exist in complex geometry. Q-homology Projective Planes Smooth Algebraic Surfaces with pg = q = 0 Enriques-Kodaira classification of algebraic surfaces (1940’s): P2, rational ruled surfaces; Enriques surfaces; properly elliptic surfaces with pg = q = 0; 2 surfaces of general type with pg = 0 (these have K = 1; 2;:::; 9); blow-ups of the above surfaces. JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 5 / 28 Remark. FPP’s are not simply connected. Exotic P2 does NOT exist in complex geometry. Q-homology Projective Planes Smooth Algebraic Surfaces with pg = q = 0 Enriques-Kodaira classification of algebraic surfaces (1940’s): P2, rational ruled surfaces; Enriques surfaces; properly elliptic surfaces with pg = q = 0; 2 surfaces of general type with pg = 0 (these have K = 1; 2;:::; 9); blow-ups of the above surfaces. Smooth algebraic surfaces with minimal invariants, that is, with b1 = b3 = 0; b0 = b2 = b4 = 1 () pg = q = 0) are P2; 2 fake projective planes (= surfaces of general type with pg = 0, K = 9). JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 5 / 28 Q-homology Projective Planes Smooth Algebraic Surfaces with pg = q = 0 Enriques-Kodaira classification of algebraic surfaces (1940’s): P2, rational ruled surfaces; Enriques surfaces; properly elliptic surfaces with pg = q = 0; 2 surfaces of general type with pg = 0 (these have K = 1; 2;:::; 9); blow-ups of the above surfaces. Smooth algebraic surfaces with minimal invariants, that is, with b1 = b3 = 0; b0 = b2 = b4 = 1 () pg = q = 0) are P2; 2 fake projective planes (= surfaces of general type with pg = 0, K = 9). Remark. FPP’s are not simply connected. Exotic P2 does NOT exist in complex geometry. JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 5 / 28 3 3 Any cubic surface in P with 3A2 is isom. to (w = xyz). 2 If S has A1 or A2-singularities only, S = P (1; 2; 3) or one of the above. In this talk, we assume S has at worst quotient singularities. 2 Then S is a Q-homology P if b2(S) = 1. For a minimal resolution S0 ! S, 0 0 pg(S ) = q(S ) = 0: Q-homology Projective Planes Q-homology P2 Definition 2 2 A normal projective surface S is called a Q-homology P if bi (S) = bi (P ) for all i, i.e. b1 = b3 = 0; b0 = b2 = b4 = 1. If S is smooth, then S = P2 or a fake projective plane. ∼ 2 3 If S has A1-singularities only, then S = (w = xy) ⊂ P . If S has A2-singularities only, then S has 3A2 or 4A2 and S =∼ P2=G or FPP=G, where G =∼ Z=3 or (Z=3)2. JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 6 / 28 2 If S has A1 or A2-singularities only, S = P (1; 2; 3) or one of the above. In this talk, we assume S has at worst quotient singularities. 2 Then S is a Q-homology P if b2(S) = 1. For a minimal resolution S0 ! S, 0 0 pg(S ) = q(S ) = 0: Q-homology Projective Planes Q-homology P2 Definition 2 2 A normal projective surface S is called a Q-homology P if bi (S) = bi (P ) for all i, i.e. b1 = b3 = 0; b0 = b2 = b4 = 1. If S is smooth, then S = P2 or a fake projective plane. ∼ 2 3 If S has A1-singularities only, then S = (w = xy) ⊂ P . If S has A2-singularities only, then S has 3A2 or 4A2 and S =∼ P2=G or FPP=G, where G =∼ Z=3 or (Z=3)2. 3 3 Any cubic surface in P with 3A2 is isom. to (w = xyz). JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 6 / 28 In this talk, we assume S has at worst quotient singularities. 2 Then S is a Q-homology P if b2(S) = 1. For a minimal resolution S0 ! S, 0 0 pg(S ) = q(S ) = 0: Q-homology Projective Planes Q-homology P2 Definition 2 2 A normal projective surface S is called a Q-homology P if bi (S) = bi (P ) for all i, i.e. b1 = b3 = 0; b0 = b2 = b4 = 1. If S is smooth, then S = P2 or a fake projective plane. ∼ 2 3 If S has A1-singularities only, then S = (w = xy) ⊂ P . If S has A2-singularities only, then S has 3A2 or 4A2 and S =∼ P2=G or FPP=G, where G =∼ Z=3 or (Z=3)2. 3 3 Any cubic surface in P with 3A2 is isom. to (w = xyz). 2 If S has A1 or A2-singularities only, S = P (1; 2; 3) or one of the above. JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 6 / 28 For a minimal resolution S0 ! S, 0 0 pg(S ) = q(S ) = 0: Q-homology Projective Planes Q-homology P2 Definition 2 2 A normal projective surface S is called a Q-homology P if bi (S) = bi (P ) for all i, i.e. b1 = b3 = 0; b0 = b2 = b4 = 1. If S is smooth, then S = P2 or a fake projective plane. ∼ 2 3 If S has A1-singularities only, then S = (w = xy) ⊂ P . If S has A2-singularities only, then S has 3A2 or 4A2 and S =∼ P2=G or FPP=G, where G =∼ Z=3 or (Z=3)2. 3 3 Any cubic surface in P with 3A2 is isom. to (w = xyz). 2 If S has A1 or A2-singularities only, S = P (1; 2; 3) or one of the above. In this talk, we assume S has at worst quotient singularities. 2 Then S is a Q-homology P if b2(S) = 1. JongHae Keum (KIAS) Algebraic surfaces with minimal Betti numbers 6 / 28 Q-homology Projective Planes Q-homology P2 Definition 2 2 A normal projective surface S is called a Q-homology P if bi (S) = bi (P ) for all i, i.e. b1 = b3 = 0; b0 = b2 = b4 = 1. If S is smooth, then S = P2 or a fake projective plane. ∼ 2 3 If S has A1-singularities only, then S = (w = xy) ⊂ P . If S has A2-singularities only, then S has 3A2 or 4A2 and S =∼ P2=G or FPP=G, where G =∼ Z=3 or (Z=3)2. 3 3 Any cubic surface in P with 3A2 is isom.
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