Numeri di Chern fra topologia e geometria birazionale

Giornate di Geometria Algebrica ed Argomenti Correlati XII (Torino, 2014) Numeri di Chern fra topologia e geometria birazionale (Work in progress with P. Cascini) Luca Tasin (Max Planck Institute for Mathematics) Luca Tasin (MPIM) Chern numbers of 3-folds 1 / 19

Set-up Introduction X smooth projective variety of dimension n over C. c i H 2i (X, Z) the Chern classes of the tangent bundle T X. Luca Tasin (MPIM) Chern numbers of 3-folds 2 / 19

Set-up Introduction X smooth projective variety of dimension n over C. c i H 2i (X, Z) the Chern classes of the tangent bundle T X. c 1 = K X H 2 (X, Z), c n Z is the Euler characteristic of X. The product of Chern classes of total degree 2n are called Chern numbers. Luca Tasin (MPIM) Chern numbers of 3-folds 2 / 19

The main problem Theorem (Kotshick, 2012) A rational linear combination of Chern numbers is a topological invariant iff it is a multiple of the Euler characteristic c n. Luca Tasin (MPIM) Chern numbers of 3-folds 3 / 19

The main problem Theorem (Kotshick, 2012) A rational linear combination of Chern numbers is a topological invariant iff it is a multiple of the Euler characteristic c n. dim X = 2. c1 2 has two possible values. dim X = 3. By Hirzebruch-Riemann-Roch we have 1 24 c 1c 2 = χ(o X ) = 1 h 1,0 +h 2,0 h 3,0 1+b 1 +b 2 +b 3. Luca Tasin (MPIM) Chern numbers of 3-folds 3 / 19

The volume If X is a variety of dimension n, then the volume of X is defined as n!h 0 (X, mk X ) vol(x) := lim sup m + m n, and X is called of general type if vol(x) > 0. Luca Tasin (MPIM) Chern numbers of 3-folds 4 / 19

The volume If X is a variety of dimension n, then the volume of X is defined as n!h 0 (X, mk X ) vol(x) := lim sup m + m n, and X is called of general type if vol(x) > 0. The volume is a birational invariant, but not a topological invariant if dim X 2. Theorem 1 Let X be a smooth projective 3-fold of general type. Then vol(x) 64(b 1 (X) + b 3 (X) + b 2 (X)). The volume takes finitely many values on projective algebraic structures of general type with the same underlying 6-manifold. Luca Tasin (MPIM) Chern numbers of 3-folds 4 / 19

MMP Introduction Let X be a smooth 3-fold of general type. Then there exists a sequence of birational maps X = X 0 X 1 X m = Y such that each X i has terminal Q-factorial singularities, Luca Tasin (MPIM) Chern numbers of 3-folds 5 / 19

MMP Introduction Let X be a smooth 3-fold of general type. Then there exists a sequence of birational maps X = X 0 X 1 X m = Y such that each X i has terminal Q-factorial singularities, each map φ i : X i X i+1 is either a divisorial contraction to a point or a divisorial contraction to a curve or a flip (which is an isomorphism in codimension 2) Luca Tasin (MPIM) Chern numbers of 3-folds 5 / 19

Proof of Theorem 1 I Let X = X 0 X 1 X m = Y be an MMP for X. Luca Tasin (MPIM) Chern numbers of 3-folds 6 / 19

Proof of Theorem 1 I Let X = X 0 X 1 X m = Y be an MMP for X. B.M.Y. inequality for minimal model of general type (Tian-Wang 2011): ( K n 2 Y. KY 2 2n + 1 ) n c 2(Y ) 0 KY 3 8 3 K Y.c 2 (Y ) Luca Tasin (MPIM) Chern numbers of 3-folds 6 / 19

Proof of Theorem 1 I Let X = X 0 X 1 X m = Y be an MMP for X. B.M.Y. inequality for minimal model of general type (Tian-Wang 2011): ( K n 2 Y. KY 2 2n + 1 ) n c 2(Y ) 0 KY 3 8 3 K Y.c 2 (Y ) Riemann-Roch for terminal 3-folds (Kawamata, Reid): χ(y, O Y ) = 1 24 K Y.c 2 (Y ) + p B(Y ) r(p) 2 1 24r(p) Luca Tasin (MPIM) Chern numbers of 3-folds 6 / 19

Proof of Theorem 1 II Hence vol(x) = vol(y ) = KY 3 8 3 K Y.c 2 (Y ) = 8 χ(y, O Y ) + r(p) 2 1. 3 24r(p) p B(Y ) Luca Tasin (MPIM) Chern numbers of 3-folds 7 / 19

Proof of Theorem 1 II Hence vol(x) = vol(y ) = KY 3 8 3 K Y.c 2 (Y ) = 8 χ(y, O Y ) + r(p) 2 1. 3 24r(p) p B(Y ) χ(y, O Y ) = χ(x, O X ) b 1 (X) + b 3 (X). Luca Tasin (MPIM) Chern numbers of 3-folds 7 / 19

Proof of Theorem 1 II Hence vol(x) = vol(y ) = KY 3 8 3 K Y.c 2 (Y ) = 8 χ(y, O Y ) + r(p) 2 1. 3 24r(p) p B(Y ) χ(y, O Y ) = χ(x, O X ) b 1 (X) + b 3 (X). Topological bound on the singularities of Y (Cascini-Zhang, 2012): p B(Y ) r(p) 2 1 r(p) 2b 2 (X). Luca Tasin (MPIM) Chern numbers of 3-folds 7 / 19

The main theorem To any threefold X we can associate an integral cubic form F X Z[x 1,..., x b ], which comes from the trilinear intersection form H 2 (X, Z) H 2 (X, Z) H 2 (X, Z) Z. Denote by FX the discriminant of F X. Theorem 2 Let X be a smooth 3-fold of general type. Assume that FX 0 and that there is an MMP for X composed only by divisorial contractions to points and blow-downs to smooth curves. Then there exists a topological invariant D X such that K 3 X D X. Luca Tasin (MPIM) Chern numbers of 3-folds 8 / 19

Strategy Introduction Let X = X 0 X 1 X m = Y be an MMP as in Theorem 2. Luca Tasin (MPIM) Chern numbers of 3-folds 9 / 19

Strategy Introduction Let X = X 0 X 1 X m = Y be an MMP as in Theorem 2. K 3 Y 64(b 1(X) + b 3 (X) + b 2 (X)). At each step we want to bound K 3 X i K 3 X i+1 with a topological invariant of X. Luca Tasin (MPIM) Chern numbers of 3-folds 9 / 19

Strategy Introduction Let X = X 0 X 1 X m = Y be an MMP as in Theorem 2. K 3 Y 64(b 1(X) + b 3 (X) + b 2 (X)). At each step we want to bound K 3 X i K 3 X i+1 with a topological invariant of X. The number of steps in bounded by b 2. (With flips it is bounded by 2b 2.) Luca Tasin (MPIM) Chern numbers of 3-folds 9 / 19

A quick remark Z = P 3 and C a smooth rational curve of degree d. π : W Z be blow-up along C. b 1 (W ) = b 3 (W ) = b 5 (W ) = 0, b 2 (W ) = b 4 (W ) = 2. Luca Tasin (MPIM) Chern numbers of 3-folds 10 / 19

A quick remark Z = P 3 and C a smooth rational curve of degree d. π : W Z be blow-up along C. b 1 (W ) = b 3 (W ) = b 5 (W ) = 0, b 2 (W ) = b 4 (W ) = 2. KW 3 = K Z 3 2K Z.C + 2 2g(C) = 62 + 8d The Betti numbers are in general not enough to bound KX 3. Luca Tasin (MPIM) Chern numbers of 3-folds 10 / 19

Divisorial contractions to curves I Let f : W Z be a blow-down to a smooth curve with exceptional divisor E. K W = f K Z + E and H 2 (W, Z) = Z[E] H 2 (Z, Z). Luca Tasin (MPIM) Chern numbers of 3-folds 11 / 19

Divisorial contractions to curves I Let f : W Z be a blow-down to a smooth curve with exceptional divisor E. K W = f K Z + E and H 2 (W, Z) = Z[E] H 2 (Z, Z). Let E 1,..., E n be the pull-back of a basis of H 2 (Z, Z). E.E i.e j = E i E.E j E = 0 and K 3 W K 3 Z = 2E 3 + 6 6g(C). Let x 0,..., x n be coordinates on H 2 (W, Z) with respect to E, E 1,..., E n. Then n F W (x 0,..., x n ) = ax0 3 + 3x 0 2 ( b i x i ) + F Z (x 1,..., x n ), where a = E 3 and b i Z. i=1 Luca Tasin (MPIM) Chern numbers of 3-folds 11 / 19

An arithmetic result Theorem 3 Let F Z[x 0,..., x n ] be a cubic form such that F 0. Then, modulo the action of GL(Z, n) on (x 1,..., x n ), there are only finitely many triples (a, (b 1,..., b n ), G) such that a, b i Z, G(x 1,..., x n ) is a cubic form and F can be written as Moreover G 0. F = ax 3 0 + ( b i x i )x 2 0 + G(x 1,..., x n ). Luca Tasin (MPIM) Chern numbers of 3-folds 12 / 19

Divisorial contractions to curves II Let f : W Z be a blow-down to a smooth curve C of genus g. χ(w \ E) = χ(z \ C) b 3 (W ) = b 3 (Z ) + 2g. Luca Tasin (MPIM) Chern numbers of 3-folds 13 / 19

Divisorial contractions to curves II Let f : W Z be a blow-down to a smooth curve C of genus g. χ(w \ E) = χ(z \ C) b 3 (W ) = b 3 (Z ) + 2g. Then KW 3 K Z 3 = 2E 3 + 6 6g 2S W + 6(b 3 (W ) + 1). Luca Tasin (MPIM) Chern numbers of 3-folds 13 / 19

Divisorial contractions to curves II Let f : W Z be a blow-down to a smooth curve C of genus g. χ(w \ E) = χ(z \ C) b 3 (W ) = b 3 (Z ) + 2g. Then K 3 W K 3 Z = 2E 3 + 6 6g 2S W + 6(b 3 (W ) + 1). The cubic form on Z is determined (up to finitely many possibilities) by the cubic form on W. Luca Tasin (MPIM) Chern numbers of 3-folds 13 / 19

Divisorial contractions to points I Let f : W Z be a divisorial contraction to a point in a minimal model program for X. Luca Tasin (MPIM) Chern numbers of 3-folds 14 / 19

Divisorial contractions to points I Let f : W Z be a divisorial contraction to a point in a minimal model program for X. K W = f K Z + de, where E is the exceptional divisor and d is a positive rational number. K 3 W K 3 Z = d 3 E 3 Luca Tasin (MPIM) Chern numbers of 3-folds 14 / 19

Divisorial contractions to points II f : W Z divisorial contraction to a point. H 2 (W, Q) = Q[E] H 2 (Z, Q), but H 2 (W, Z)/H 2 (Z, Z) may have torsion. Luca Tasin (MPIM) Chern numbers of 3-folds 15 / 19

Divisorial contractions to points II f : W Z divisorial contraction to a point. H 2 (W, Q) = Q[E] H 2 (Z, Q), but H 2 (W, Z)/H 2 (Z, Z) may have torsion. This torsion depends on the singularities of W and Z. Admitting rational coefficients with bounded denominators we have something like F W (x 0,..., x n ) = ax 3 0 + F Z (x 1,..., x n ). Luca Tasin (MPIM) Chern numbers of 3-folds 15 / 19

Reduced forms Let F be a cubic form such that F 0. We want to prove that there are only finitely many possible reduced forms for F. Luca Tasin (MPIM) Chern numbers of 3-folds 16 / 19

Reduced forms Let F be a cubic form such that F 0. We want to prove that there are only finitely many possible reduced forms for F. 1 By proving that V F = {p P n : rkh F (p) 2} {F 0} is a finite union of points, line, plane conics and plane cubics, we reduce the problem to binary and ternary cubics. 2 To prove our result for binary and ternary cubics we need Siegel and Faltings theorems on the finiteness of integral and rational points on algebraic curves. Luca Tasin (MPIM) Chern numbers of 3-folds 16 / 19

A basic case I Let F Z[x, y, z] be a ternary cubic such that 0. The algebra of the invariants of F (under the action of SL(3, Z)) is generated by two polynomials S and T in the coefficients of F and = T 2 64S 3. Assume that S 0 and write F = Ax 3 + (By + Cz)x 2 + G(y, z). 1 We can assume that G (modulo SL(2, Z)) is fixed. 2 Passing to a number field K and acting with SL(2, K ) on (y, z) we may assume that G = dy 3 + z 3 and F = Ax 3 + (B 1 y + C 1 z)x 2 + G(y, z). Luca Tasin (MPIM) Chern numbers of 3-folds 17 / 19

A basic case II 4 We get S = bcd and T = 27a 2 d 2 + 4b 3 d + 4c 3 d 2. Luca Tasin (MPIM) Chern numbers of 3-folds 18 / 19

A basic case II 4 We get S = bcd and T = 27a 2 d 2 + 4b 3 d + 4c 3 d 2. 5 Consider the curve C P 3 given by the ideal I = (Sx 2 3 dx 1x 2, Tx 3 3 27d 2 x 2 0 x 3 4dx 3 1 4d 2 x 3 2 ). Luca Tasin (MPIM) Chern numbers of 3-folds 18 / 19

A basic case II 4 We get S = bcd and T = 27a 2 d 2 + 4b 3 d + 4c 3 d 2. 5 Consider the curve C P 3 given by the ideal I = (Sx 2 3 dx 1x 2, Tx 3 3 27d 2 x 2 0 x 3 4dx 3 1 4d 2 x 3 2 ). 6 Since p g (C) = 3, by Faltings theorem we have only a finite number of K -rational points [a, b, c, 1] on C. Luca Tasin (MPIM) Chern numbers of 3-folds 18 / 19

Flops Introduction Theorem (Chen-Hacon, 2012) Any MMP of X can be factored into a sequence of divisorial contractions to points (or their inverses), blow-downs to smooth curves and flops. Let f : W Z be a flop. W and Z have the same singularities and the same Betti numbers (Kollár). K 3 W = K 3 Z. Luca Tasin (MPIM) Chern numbers of 3-folds 19 / 19