# The structure of algebraic varieties

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1 The structure of algebraic varieties János Kollár Princeton University ICM, August, 2014, Seoul with the assistance of Jennifer M. Johnson and Sándor J. Kovács (Written comments added for clarity that were part of the oral presentation.)

2 Euler, Abel, Jacobi Elliptic integrals (multi-valued): dx x 3 + ax 2 + bx + c. To make it single-valued, look at the algebraic curve C := { (x, y) : y 2 = x 3 + ax 2 + bx + c } C 2. We get the integral dx y Γ for some path Γ on C.

3 General case Let g(x, y) be any polynomial, it determines y := y(x) as a multi-valued function of x. Let h(u, v) be any function. Then h ( x, y(x) ) dx (multi-valued integral) becomes Γ h ( x, y ) dx (single-valued integral) for some path Γ on the algebraic curve C := ( g(x, y) = 0 ) C 2

4 Example: C := ( y 2 = (x + 1) 2 x(1 x) ). Real picture: (Comment: looks like 2 parts, but the real picture can decieve. Complex picture is better.)

5 Example: C := ( y 2 = (x + 1) 2 x(1 x) ). Complex picture: (Comment: the picture is correct in projective space only. The correct picture has 2 missing points at infinity.)

6 Substitution in integrals Question (Equivalence) Given two algebraic curves C and D, when can we transform every integral C h dx into an integral D g dx? Question (Simplest form) Among all algebraic curves C i with equivalent integrals, is there a simplest?

7 Example For C := ( y 2 = (x + 1) 2 x(1 x) ) the substitution x = 1 1 t 2, y = t(2 t2 ) (1 t 2 ) 2 with inverse t = transforms h(x, y)dx into an integral Γ ( 1 h 1 t, t(2 ) t2 ) 2 (1 t 2 ) 2 2t (1 t 2 ) 2 dt. y x(x + 1)

8 Theorem (Riemann, 1851) For every algebraic curve C C 2 we have S: a compact Riemann surface and meromorphic, invertible φ : S C establishing an isomorphism between Merom(C) : meromorphic function theory of C and Merom(S) : meromorphic function theory of S.

9 MINIMAL MODEL PROBLEM Question X any algebraic variety. Is there another algebraic variety X m such that Merom(X ) = Merom ( X m) and the geometry of X m is the simplest possible? Answers: Curves: Riemann, 1851 Surfaces: Enriques, 1914; Kodaira, 1966 Higher dimensions: Mori s program 1981 also called Minimal Model Program many open questions

10 MODULI PROBLEM Question What are the simplest families of algebraic varieties? How to transform any family into a simplest one? Answers: Curves: Deligne Mumford, 1969 Surfaces: Kollár Shepherd-Barron, 1988; Alexeev, 1996 Higher dimensions: the KSBA-method works but needs many technical details

11 Algebraic varieties 1 Affine algebraic set: common zero-set of polynomials X aff = X aff (f 1,..., f r ) C N = { (x 1,..., x N ) : f i (x 1,..., x N ) = 0 i }. Hypersurfaces: 1 equation, X (f ) C N. Complex dimension: dim C N = N (= 1 (topological dim)) 2 Curves, surfaces, 3-folds,...

12 Algebraic varieties 2 Example: x 4 y 4 + z 4 + 2x 2 z 2 x 2 + y 2 z 2 = 0. (Comment: What is going on? Looks like a sphere and a cone together.)

13 Explanation: x 4 y 4 + z 4 + 2x 2 z 2 x 2 + y 2 z 2 = (x 2 + y 2 + z 2 1)(x 2 y 2 + z 2 ) Variety= irreducible algebraic set

14 Algebraic varieties 3 Projective variety: X CP N, closure of an affine variety. Homogeneous coordinates: [x 0 : :x N ] = [λx 0 : :λx N ] p(x 0,..., x N ) makes no sense Except: If p is homogeneous of degree d then p(λx 0,..., λx N ) = λ d p(x 0,..., x N ). Well-defined notions are: Zero set of homogeneous p. Quotient of homogeneous p, q of the same degree f (x 0,..., x N ) = p 1(x 0,..., x N ) p 2 (x 0,..., x N ) Rational functions on CP N, and, by restriction, rational functions on X CP N.

15 Theorem (Chow, 1949; Serre, 1956) M CP N any closed subset that is locally the common zero set of analytic functions. Then M is algebraic: globally given as the common zero set of homogeneous polynomials and every meromorphic function on M is rational: globally the quotient of two homogeneous polynomials. Non-example: M := (y = sin x) C 2 CP 2. (Comment: The closure at infinity is not locally analytic.)

16 Rational maps = meromorphic maps Definition X CP N algebraic variety f 0,..., f M rational functions. Map (or rational map) f : X CP M given by p [ f 0 (p): :f M (p) ] CP M. Where is f defined? away from poles and common zeros, but, as an example, let π : CP 2 CP 1 be given by [x:y:z] [ x z : y z ]. Note that [ x z : y z ] = [ x y : 1] = [ 1 : y x ]. So π is defined everywhere except (0:0:1).

17 Isomorphism Definition X, Y are isomorphic if there are everywhere defined maps f : X Y and g : Y X that are inverses of each other. Denoted by X = Y. Isomorphic varieties are essentially the same.

18 Birational equivalence Unique to algebraic geometry! Definition X, Y are birational if there are rational maps f : X Y and g : Y X such that φ Y φ X := φ Y f and φ X φ Y := φ X g give Merom(X ) = Merom(Y ). Equivalent: There are Z X and W Y such that (X \ Z) = (Y \ W ). Denoted by X bir Y.

19 (Comment: The next 12 slides show that, in topology, one can make a sphere and a torus from a sphere by cutting and pasting. Nothing like this can be done with algebraic varieties. Notice that if we keep the upper cap open and the lower cap closed then the construction is naturally one-to-one on points.)

20 Non-example from topology

21 Non-example from topology

22 Non-example from topology

23 Non-example from topology

24 Non-example from topology

25 Non-example from topology

26 Non-example from topology

27 Non-example from topology

28 Non-example from topology

29 Non-example from topology

30 Non-example from topology

31 Non-example from topology

32 Non-example from topology

33 Example of birational equivalence Affine surface S := (xy = z 3 ) C 3. It is birational to C 2 uv as shown by C 3 C 2 f : (x, y, z) (x/z, y/z) (u 2 v, uv 2, uv) (u, v) : g f not defined if z = 0 g defined but maps the coordinate axes to (0, 0, 0). S = C 2 but S \ (z = 0) = C 2 \ (uv = 0)

34 Basic rule of thumb Assume X bir Y, hence (X \ Z) = (Y \ W ). Many questions about X can be answered by first studying the same question on Y then a similar question involving Z and W. Aim of the Minimal Model Program: Exploit this in two steps: Given a question and X, find Y bir X that is best adapted to the question. This is the Minimal Model Problem. Set up dimension induction to deal with Z and W.

35 When is a variety simple? Surfaces: Castelnuovo, Enriques ( ) Higher dimensions: There was not even a conjecture until Mori, Reid ( ) Kollár Miyaoka Mori (1992) Need: Canonical class or first Chern class We view it as a map: { algebraic curves in X } Z, it is denoted by: C c 1(X ) or (K X C). (Comment: next few slides give the definition.)

36 Volume forms Measure or volume form on R n : s(x 1,..., x n ) dx 1 dx n. Complex volume form: locally written as ω := h(z 1,..., z n ) dz 1 dz n. ( 1 ) n ω gives a real volume form ω ω (Comment: for the signs note that dz d z = (dx + 1dy) (dx 1dy) = 2 1dx dy) 2

37 TENSION Differential geometers want C volume forms: h(z 1,..., z n ) should be C -functions. Algebraic/analytic geometers want meromorphic forms: h(z 1,..., z n ) should be meromorphic functions. Simultaneously possible only for Calabi Yau varieties.

38 Connection: Gauss Bonnet theorem X smooth, projective variety, ω r C volume form, ω m meromorphic volume form, C X algebraic curve. Definition (Chern form or Ricci curvature) c 1 (X, ω r ) := 1 π ij 2 log h r (z) z i z j dz i d z j. Definition (Algebraic degree) deg C ω m := #(zeros of ω m on C) #(poles of ω m on C), zeros/poles counted with multiplicities. (assuming ω m not identically 0 or on C.)

39 Theorem (Gauss Bonnet) X smooth, projective variety, ω r C volume form, ω m meromorphic volume form, C X algebraic curve. Then c 1 (X, ω r ) = deg C ω m is independent of ω r and ω m. C Denoted by C c 1(X ). (Comment on the minus sign: differential geometers prefer the tangent bundle; volume forms use the cotangent bundle.)

40 Building blocks of algebraic varieties Negatively curved: C c 1(X ) < 0 for every curve C X. Largest class of the three. Flat or Calabi Yau: C c 1(X ) = 0 for every curve C X. Important role in string theory and mirror symmetry. Positively curved or Fano: C c 1(X ) > 0 for every curve. Few but occur most frequently in applications. Kähler Einstein metric : pointwise conditions. negative/flat: Yau, Aubin,... positive: still not settled

41 Mixed type I Semi-negatively curved or Kodaira Iitaka type C c 1(X ) 0 for every curve C X. Structural conjecture (Main open problem) There is a unique I X : X I (X ) such that C c 1(X ) = 0 iff C fiber of I X. I (X ) is negatively curved in a suitable sense. Intermediate case: 0 < dim I (X ) < dim X : family of lower dimensional Calabi Yau varieties parametrized by the lower dimensional variety I (X ). (Comment: this is one example why families of varieties are important to study.)

42 Mixed type II Positive fiber type I really would like to tell you that: There is a unique m X : X M(X ) such that C c 1(X ) > 0 if C fiber of m X. M(X ) is semi negatively curved. BUT this is too restrictive. We fix the definition later.

43 Main Conjecture Conjecture (Minimal model conjecture, extended) Every algebraic variety X is birational to a variety X m that is either semi-negatively curved or has positive fiber type. X m is called a minimal model of X (especially in first case) Caveat. X m may have singularities (This was a rather difficult point historically.)

44 Some history

45 Some history Enriques Kodaira

46 Some history Enriques Mori Kodaira Reid

47 Some history Enriques Mori Hacon Kodaira Reid McKernan

48 Rationally connected varieties Theme: plenty of rational curves CP 1 X. Theorem X smooth projective variety. Equivalent: x 1, x 2 X there is CP 1 X through them. x 1,..., x r X there is a CP 1 X through them. x 1,..., x r X + tangent directions v i T xi X there is a CP 1 X through them with given directions. Definition X is rationally connected or RC if the above hold.

49 Properties of rationally connected varieties Positively curved RC (Nadel, Campana, Kollár Miyaoka Mori, Zhang) Birational and smooth deformation invariant (Kollár Miyaoka Mori) Good arithmetic properties: p-adic fields (Kollár), finite fields (Kollár Szabó, Esnault) C(t) (Graber Harris Starr, de Jong Starr). Loop space of RC is RC (Lempert Szabó). Problem Is RC a symplectic property?

50 Positive fiber type Definition X is of positive fiber type if there is a unique m X : X M(X ) such that almost all fibers are rationally connected and M(X ) is semi negatively curved.

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