On the Chern numbers of a smooth threefold.

*(English)*Zbl 1430.14039A question asked by Hirzebruch is: Which linear combinations of Chern numbers on a smooth complex projective variety are topologically invariant? D. Kotschick [J. Topol. 1, No. 2, 518–526 (2008; Zbl 1145.57023); Adv. Math. 229, No. 2, 1300–1312 (2012; Zbl 1244.57048)] answered the question and showed that a rational linear combination of Chern numbers is a homeomorphism invariant of smooth complex projective varieties if and only if it is a multiple of the Euler characteristic. Kotschick also asked the following question for smooth projective threefolds: Does \(c_1^3=-K_X^3\) take only finitely many values on the projective algebraic structures \(X\) with the same underlying 6-manifold?

Motivated by this question, the authors study the Chern numbers of a smooth projective threefold \(X\). By Minimal Model Program, there is a birational map \(\phi: X\dashrightarrow Y\) into a threefold \(Y\) such that either \(X\) is not uniruled (A variety of dimension \(n\) is uniruled if there is a variety \(T\) of dimension \(n-1\) and a dominant map \(T\times\mathbb{P}^1\dashrightarrow X\)) and \(Y\) is minimal (i.e., the canonical divisor \(K_X\) is nef) or \(X\) is uniruled and \(Y\) admits a Mori fiber space structure (i.e., a morphism \(Y\rightarrow Z\) with connected fibers with relative Picard number equal to one and whose general fiber is a nontrivial Fano variety). Their strategy consists in two steps: first find the upper bound of \(K_Y^3\) then estimate the bound of \(K_X^3-K_Y^3\).

By the Bogomolov-Miyaoka-Yau inequality for terminal threefolds, the authors prove their fist result.

Theorem 1.2. Let \(X\) be a smooth complex projective threefold which is not uniruled. Then \[ \mathrm{vol}(X)\leq 6b_2(X)+36b_3(X), \] where \[ \mathrm{vol}(X)=\lim\sup_{m\rightarrow \infty}\frac{n!h^0(X, mK_X)}{m^n}. \] By combining methods in birational geometry, topology, and arithmetic geometry, the authors prove their main result:

Theorem 1.3. Let \(Y\) be a \(\mathbb{Q}\)-factorial terminal threefold with associate cubic form \(F_Y\), and let \(f:Y\rightarrow X\) be a divisorial contraction to a point or a smooth curve contained in the smoothy locus of \(X\) (in the last case assume also that \(\Delta_{F_Y}\neq 0\)).

(1) If \(f\) contracts a divisor to a point, then \[ |K_Y^3-K_X^3|\leq 2S_W+6(b_3(Y)+1), \] where \(S_Y\) is a topology invariant defined by \(F_Y\). Moreover, the same inequality is true after replacing \(b_3(Y)\) by \(Ib_3(Y)=\dim IH^3(Y, \mathbb{Q})\).

(2) The cubic form \(F_X\) is determined up to finite ambiguity by the cubic form \(F_Y\).

An application of these results is: Let \(X\) be a smooth complex projective threefold which is not uniruled and let \(F_X\) be its associate cubic form. Assume that \(\Delta_{F_X}\neq 0\) and that there exists a birational morphism \(f:X\rightarrow Y\) onto a minimal projective threefold \(Y\), which is obtained as a composition of divisorial contractions to points and blow-downs to smooth curves in smooth loci. Then there is a constant \(C\) depending only on the topology of the 6-manifold underlying \(X\) such that \[ |K_X^3|\leq C. \]

Motivated by this question, the authors study the Chern numbers of a smooth projective threefold \(X\). By Minimal Model Program, there is a birational map \(\phi: X\dashrightarrow Y\) into a threefold \(Y\) such that either \(X\) is not uniruled (A variety of dimension \(n\) is uniruled if there is a variety \(T\) of dimension \(n-1\) and a dominant map \(T\times\mathbb{P}^1\dashrightarrow X\)) and \(Y\) is minimal (i.e., the canonical divisor \(K_X\) is nef) or \(X\) is uniruled and \(Y\) admits a Mori fiber space structure (i.e., a morphism \(Y\rightarrow Z\) with connected fibers with relative Picard number equal to one and whose general fiber is a nontrivial Fano variety). Their strategy consists in two steps: first find the upper bound of \(K_Y^3\) then estimate the bound of \(K_X^3-K_Y^3\).

By the Bogomolov-Miyaoka-Yau inequality for terminal threefolds, the authors prove their fist result.

Theorem 1.2. Let \(X\) be a smooth complex projective threefold which is not uniruled. Then \[ \mathrm{vol}(X)\leq 6b_2(X)+36b_3(X), \] where \[ \mathrm{vol}(X)=\lim\sup_{m\rightarrow \infty}\frac{n!h^0(X, mK_X)}{m^n}. \] By combining methods in birational geometry, topology, and arithmetic geometry, the authors prove their main result:

Theorem 1.3. Let \(Y\) be a \(\mathbb{Q}\)-factorial terminal threefold with associate cubic form \(F_Y\), and let \(f:Y\rightarrow X\) be a divisorial contraction to a point or a smooth curve contained in the smoothy locus of \(X\) (in the last case assume also that \(\Delta_{F_Y}\neq 0\)).

(1) If \(f\) contracts a divisor to a point, then \[ |K_Y^3-K_X^3|\leq 2S_W+6(b_3(Y)+1), \] where \(S_Y\) is a topology invariant defined by \(F_Y\). Moreover, the same inequality is true after replacing \(b_3(Y)\) by \(Ib_3(Y)=\dim IH^3(Y, \mathbb{Q})\).

(2) The cubic form \(F_X\) is determined up to finite ambiguity by the cubic form \(F_Y\).

An application of these results is: Let \(X\) be a smooth complex projective threefold which is not uniruled and let \(F_X\) be its associate cubic form. Assume that \(\Delta_{F_X}\neq 0\) and that there exists a birational morphism \(f:X\rightarrow Y\) onto a minimal projective threefold \(Y\), which is obtained as a composition of divisorial contractions to points and blow-downs to smooth curves in smooth loci. Then there is a constant \(C\) depending only on the topology of the 6-manifold underlying \(X\) such that \[ |K_X^3|\leq C. \]

Reviewer: Jing Zhang (University Park)

##### Keywords:

Chern number; divisorial contraction; minimal model program; smooth projective threefold; topological invariant##### References:

[1] | Be\u\i linson, A. A.; Bernstein, J.; Deligne, P., Faisceaux pervers. Analysis and topology on singular spaces, I, Luminy, 1981, Ast\'erisque 100, 5-171, (1982), Soc. Math. France, Paris |

[2] | Bisi, Cinzia; Cascini, Paolo; Tasin, Luca, A remark on the Ueno-Campana’s threefold, Michigan Math. J., 65, 3, 567-572, (2016) · Zbl 1383.14010 |

[3] | Caib\u ar, Mirel, Minimal models of canonical 3-fold singularities and their Betti numbers, Int. Math. Res. Not., 26, 1563-1581, (2005) · Zbl 1082.14017 |

[4] | Cheltsov, Ivan, Nonrational nodal quartic threefolds, Pacific J. Math., 226, 1, 65-81, (2006) · Zbl 1123.14010 |

[5] | Chen, Jungkai A.; Hacon, Christopher D., Factoring 3-fold flips and divisorial contractions to curves, J. Reine Angew. Math., 657, 173-197, (2011) · Zbl 1230.14015 |

[6] | Chen, Jungkai Alfred, Birational maps of 3-folds, Taiwanese J. Math., 19, 6, 1619-1642, (2015) · Zbl 1357.14021 |

[7] | [Che16]Chen16 H.-K. Chen, \emph Betti numbers in three dimensional minimal model program, ArXiv e-prints: 1605.04372 (2016). |

[8] | Cascini, Paolo; Zhang, De-Qi, Effective finite generation for adjoint rings, Ann. Inst. Fourier (Grenoble), 64, 1, 127-144, (2014) · Zbl 1364.14012 |

[9] | Deligne, Pierre, Th\'eorie de Hodge. III, Inst. Hautes \'Etudes Sci. Publ. Math., 44, 5-77, (1974) · Zbl 0237.14003 |

[10] | \bibGKZbook label=GKZ94, author=Gel\cprime fand, I. M., author=Kapranov, M. M., author=Zelevinsky, A. V., title=Discriminants, resultants, and multidimensional determinants, series=Mathematics: Theory & Applications, pages=x+523, publisher=Birkh\"auser Boston, Inc., Boston, MA, date=1994, doi=10.1007/978-0-8176-4771-1, isbn=0-8176-3660-9, review=\MR1264417, |

[11] | Hirzebruch, Friedrich, Some problems on differentiable and complex manifolds, Ann. of Math. (2), 60, 213-236, (1954) · Zbl 0056.16803 |

[12] | Hacon, Christopher D.; McKernan, James, Boundedness of pluricanonical maps of varieties of general type, Invent. Math., 166, 1, 1-25, (2006) · Zbl 1121.14011 |

[13] | Kawakita, Masayuki, Three-fold divisorial contractions to singularities of higher indices, Duke Math. J., 130, 1, 57-126, (2005) · Zbl 1091.14008 |

[14] | Kawamata, Yujiro, On the plurigenera of minimal algebraic \(3\)-folds with \(K≡0\), Math. Ann., 275, 4, 539-546, (1986) · Zbl 0582.14015 |

[15] | Koll\'ar, J\'anos; Mori, Shigefumi, Classification of three-dimensional flips, J. Amer. Math. Soc., 5, 3, 533-703, (1992) · Zbl 0773.14004 |

[16] | Koll\'ar, J\'anos; Mori, Shigefumi, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, viii+254 pp., (1998), Cambridge University Press, Cambridge · Zbl 0926.14003 |

[17] | Koll\'ar, J\'anos, Flops, Nagoya Math. J., 113, 15-36, (1989) · Zbl 0645.14004 |

[18] | Koll\'ar, J\'anos, Effective base point freeness, Math. Ann., 296, 4, 595-605, (1993) · Zbl 0818.14002 |

[19] | Koll\'ar, J\'anos, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math., 113, 1, 177-215, (1993) · Zbl 0819.14006 |

[20] | Kotschick, D., Chern numbers and diffeomorphism types of projective varieties, J. Topol., 1, 2, 518-526, (2008) · Zbl 1145.57023 |

[21] | Kotschick, D., Topologically invariant Chern numbers of projective varieties, Adv. Math., 229, 2, 1300-1312, (2012) · Zbl 1244.57048 |

[22] | Lang, Serge, Fundamentals of Diophantine geometry, xviii+370 pp., (1983), Springer-Verlag, New York · Zbl 0528.14013 |

[23] | Lazarsfeld, Robert, Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 48, xviii+387 pp., (2004), Springer-Verlag, Berlin · Zbl 1093.14501 |

[24] | LeBrun, Claude, Topology versus Chern numbers for complex \(3\)-folds, Pacific J. Math., 191, 1, 123-131, (1999) · Zbl 0951.57010 |

[25] | Libgober, Anatoly S.; Wood, John W., Uniqueness of the complex structure on K\"ahler manifolds of certain homotopy types, J. Differential Geom., 32, 1, 139-154, (1990) · Zbl 0711.53052 |

[26] | Miyaoka, Yoichi, The Chern classes and Kodaira dimension of a minimal variety. Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math. 10, 449-476, (1987), North-Holland, Amsterdam |

[27] | Mori, Shigefumi, On \(3\)-dimensional terminal singularities, Nagoya Math. J., 98, 43-66, (1985) · Zbl 0589.14005 |

[28] | Okonek, Ch.; Van de Ven, A., Cubic forms and complex \(3\)-folds, Enseign. Math. (2), 41, 3-4, 297-333, (1995) · Zbl 0869.14018 |

[29] | Reid, Miles, Young person’s guide to canonical singularities. Algebraic geometry, Bowdoin, 1985, Brunswick, Maine, 1985, Proc. Sympos. Pure Math. 46, 345-414, (1987), Amer. Math. Soc., Providence, RI |

[30] | Saito, Morihiko, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci., 24, 6, 849-995 (1989), (1988) · Zbl 0691.14007 |

[31] | Schreieder, Stefan; Tasin, Luca, Algebraic structures with unbounded Chern numbers, J. Topol., 9, 3, 849-860, (2016) · Zbl 1405.32035 |

[32] | Schreider, Stefan; Tasin, Luca, K\"ahler structures on spin 6-manifolds, Math. Ann. |

[33] | Steenbrink, J. H. M., Mixed Hodge structures associated with isolated singularities. Singularities, Part 2, Arcata, Calif., 1981, Proc. Sympos. Pure Math. 40, 513-536, (1983), Amer. Math. Soc., Providence, RI |

[34] | Sturmfels, Bernd, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, vi+197 pp., (1993), Springer-Verlag, Vienna · Zbl 0802.13002 |

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