Riv. Mat. Univ. Parma, Vol. 6, No. 2, 2015

Yoshiaki Fukuma[1]

Sectional class of ample line bundles on smooth projective varieties

Pages: 215-240
Accepted: 2 February 2015
Mathematics Subject Classification (2010): Primary 14C20; Secondary 14C17, 14J30, 14J35, 14J40, 14J60, 14M99, 14N15.
Keywords: Ample vector bundle, (multi-)polarized manifold, class, sectional Euler number, sectional Betti number.
[1] : Department of Mathematics, Faculty of Science, Kochi University, Akebono-cho, Kochi 780-8520, Japan

Abstract: Let $$X$$ be an $$n$$-dimensional smooth projective variety defined over the field of complex numbers, let $$L_{1}, \dots , L_{n-i}, A_{1}$$ and $$A_{2}$$ be ample line bundles on $$X$$. In this paper, we will define the sectional class $$\mbox{cl}_{i}(X,L_{1}, \dots , L_{n-i}; A_{1},A_{2})$$ for every integer $$i$$ with $$0\leq i\leq n$$, and we will investigate this invariant. In particular, for every integer $$i$$ with $$0\leq i\leq n$$, by setting $$L_{1}=\cdots=L_{n-i}=L$$ and $$A_{1}=A_{2}=L$$, we give a classification of polarized manifolds $$(X,L)$$ by the value of $$\mbox{cl}_{i}(X,L):=\mbox{cl}_{i}(X,\underbrace{L, \cdots , L}_{n-i};L,L)$$.

References

[1] E. Ballico, M. Bertolini and C. Turrini, On the class of some projective varieties, Collect. Math. 48 (1997), 281-287. [MR1475803]
[2] M. C. Beltrametti and A. J. Sommese, The adjunction theory of complex projective varieties, de Gruyter Expositions in Math. 16, Walter de Gruyter & Co., Berlin 1995. [MR1318687]
[3] T. Fujita, Classification of polarized manifolds of sectional genus two, in "Algebraic geometry and commutative algebra" Vol. I (in Honor of Masayoshi Nagata), Kinokuniya, Tokyo 1988, 73-98. [MR0977755]
[4] T. Fujita, Ample vector bundles of small $$c_1$$-sectional genera, J. Math. Kyoto Univ. 29 (1989), 1-16. [MR0988059]
[5] T. Fujita, Classification theories of polarized varieties, London Math. Soc. Lecture Note Ser. 155, Cambridge University Press, Cambridge, 1990. [MR1162108]
[6] Y. Fukuma, On polarized $$3$$-folds $$(X,L)$$ with $$g(L) = q(X) + 1$$ and $$h^0(L) \geq 4$$, Ark. Mat. 35 (1997), 299-311. [MR1478782]
[7] Y. Fukuma, On sectional genus of quasi-polarized $$3$$-folds, Trans. Amer. Math. Soc. 351 (1999), 363-377. [MR1487615]
[8] Y. Fukuma, On complex manifolds polarized by an ample line bundle of sectional genus $$q(X) + 2$$, Math. Z. 234 (2000) 573-604. [MR1774098]
[9] Y. Fukuma, On the sectional geometric genus of quasi-polarized varieties, I, Comm. Algebra 32 (2004), 1069-1100. [MR2063799]
[10] Y. Fukuma, On the second sectional $$H$$-arithmetic genus of polarized manifolds, Math. Z. 250 (2005), 573-597. [MR2179612]
[11] Y. Fukuma, On the sectional invariants of polarized manifolds, J. Pure Appl. Algebra 209 (2007), 99-117. [MR2292120]
[12] Y. Fukuma, A classification of polarized manifolds by the sectional Betti number and the sectional Hodge number, Adv. Geom. 8 (2008), 591-614. [MR2456639]
[13] Y. Fukuma, Invariants of ample line bundles on projective varieties and their applications, I, Kodai Math. J. 31 (2008), 219-256. [MR2435893]
[14] Y. Fukuma, Sectional invariants of scroll over a smooth projective variety, Rend. Sem. Mat. Univ. Padova 121 (2009), 93-119. [MR2542136]
[15] Y. Fukuma, Invariants of ample line bundles on projective varieties and their applications, II, Kodai Math. J. 33 (2010), 416-445. [MR2754330]
[16] Y. Fukuma, Invariants of ample vector bundles on smooth projective varieties, Riv. Mat. Univ. Parma 2 (2011), 273-297. [MR2906120]
[17] Y. Fukuma, Classiffcation of polarized manifolds by the second sectional Betti numbers, Hokkaido Math. J. 42 (2013), 463-472. [MR3137396]
[18] Y. Fukuma, Calculations of sectional classes of special polarized manifolds, preprint, http://www.math.kochi-u.ac.jp/fukuma/Cal-SC.html
[19] Y. Fukuma and H. Ishihara, Complex manifolds polarized by an ample and spanned line bundle of sectional genus three, Arch. Math. (Basel) 71 (1998), 159-168. [MR1631496]
[20] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 2, Springer-Verlag, Berlin, 1984. [MR0732620]
[21] S. L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. 84 (1966), 293-344. [MR0206009]
[22] S. L. Kleiman, Tangency and duality, in "Vancouver conference in algebraic geometry"', CMS Conf. Proc., 6, Amer. Math. Soc., Providence, RI, 1986,163-225. [MR0846021]
[23] A. Lanteri, On the class of a projective algebraic surface, Arch. Math. (Basel) 45 (1985), 79-85. [MR0799452]
[24] A. Lanteri, On the class of an elliptic projective surface, Arch. Math. (Basel) 64 (1995), 359-368. [MR1319008]
[25] A. Lanteri and F. Russo, A footnote to a paper by A. Noma, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Natur. Rend. Lincei, IX Ser., Mat. Appl. 4 (1993), no. 2, 131-132. [MR1277004]
[26] A. Lanteri and F. Tonoli, Ruled surfaces with small class, Comm. Algebra 24 (1996), 3501-3512. [MR1405268]
[27] A. Lanteri and C. Turrini, Projective threefolds of small class, Abh. Math. Sem. Univ. Hamburg 57 (1987), 103-117. [MR0927167]
[28] A. Lanteri and C. Turrini, Projective surfaces with class less than or equal to twice the degree, Math. Nachr. 175 (1995), 199-207. [MR1355018]
[29] R. Lazarsfeld, Positivity in algebraic geometry I, II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Springer-Verlag, Berlin, 2004. [MR2095471/2]
[30] A. Noma, Classification of rank-2 ample and spanned vector bundles on surfaces whose zero loci consist of general points, Trans. Amer. Math. Soc. 342 (1994), 867-894. [MR1181186]
[31] M. Palleschi and C. Turrini, On polarized surfaces with a small generalized class, Extracta Math. 13 (1998), 371-381. [MR1695560]
[32] C. Turrini and E. Verderio, Projective surfaces of small class, Geom. Dedicata 47 (1993), 1-14. [MR1230102]

Home Riv.Mat.Univ.Parma