Hodge level of weighted complete intersections of general type
- Authors: Przyjalkowski V.V.1
-
Affiliations:
- Steklov Mathematical Institute of Russian Academy of Sciences
- Issue: Vol 213, No 12 (2022)
- Pages: 68-85
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133485
- DOI: https://doi.org/10.4213/sm9584
- ID: 133485
Cite item
Abstract
We show that smooth varieties of general type that are well-formed weighted complete intersections of Cartier divisors have the maximal Hodge level, that is, their rightmost middle Hodge numbers do not vanish. We show that this does not hold in the quasi-smooth case.Bibliography: 23 titles.
About the authors
Victor Vladimirovich Przyjalkowski
Steklov Mathematical Institute of Russian Academy of Sciences
Email: victorprz@mi-ras.ru
Doctor of physico-mathematical sciences, no status
Russian FederationReferences
- В. И. Данилов, “Геометрия торических многообразий”, УМН, 33:2(200) (1978), 85–134
- В. В. Пржиялковский, К. А. Шрамов, “Взвешенные полные пересечения Фано большой коразмерности”, Сиб. матем. журн., 61:2 (2020), 377–384
- В. В. Пржиялковский, К. А. Шрамов, “Гладкие полные пересечения Фано основной серии в торических многообразиях”, Матем. заметки, 109:4 (2021), 590–596
- V. V. Batyrev, D. A. Cox, “On the Hodge structure of projective hypersurfaces in toric varieties”, Duke Math. J., 75:2 (1994), 293–338
- J. A. Carlson, “Extensions of mixed Hodge structures”, Journees de geometrie algebrique (Angers, 1979), Sijthoff & Noordhoff, Alphen aan den Rijn–Germantown, MD, 1980, 107–127
- D. I. Cartwright, T. Steger, “Enumeration of the 50 fake projective planes”, C. R. Math. Acad. Sci. Paris, 348:1-2 (2010), 11–13
- Jheng-Jie Chen, Jungkai A. Chen, Meng Chen, “On quasismooth weighted complete intersections”, J. Algebraic Geom., 20:2 (2011), 239–262
- A. Dimca, “Residues and cohomology of complete intersections”, Duke Math. J., 78:1 (1995), 89–100
- I. Dolgachev, “Weighted projective varieties”, Group actions and vector fields (Vancouver, BC, 1981), Lecture Notes in Math., 956, Springer, Berlin, 1982, 34–71
- E. Fatighenti, G. Mongardi, “A note on a Griffiths-type ring for complete intersections in Grassmannians”, Math. Z., 299:3-4 (2021), 1651–1672
- P. Griffiths, “On the periods of certain rational integrals. I, II”, Ann. of Math. (2), 90:3 (1969), 460–495, 496–541
- A. R. Iano-Fletcher, “Working with weighted complete intersections”, Explicit birational geometry of 3-folds, London Math. Soc. Lecture Note Ser., 281, Cambridge Univ. Press, Cambridge, 2000, 101–173
- A. Kasprzyk, “Bounds on fake weighted projective space”, Kodai Math. J., 32:2 (2009), 197–208
- A. R. Mavlyutov, “Cohomology of complete intersections in toric varieties”, Pacific J. Math., 191:1 (1999), 133–144
- D. Mumford, “An algebraic surface with $K$ ample, $(K^2)=9$, $p_g=q=0$”, Amer. J. Math., 101:1 (1979), 233–244
- J. Nagel, “The Abel–Jacobi map for complete intersections”, Indag. Math. (N.S.), 8:1 (1997), 95–113
- M. Pizzato, T. Sano, L. Tasin, “Effective nonvanishing for Fano weighted complete intersections”, Algebra Number Theory, 11:10 (2017), 2369–2395
- V. Przyjalkowski, C. Shramov, “Hodge level for weighted complete intersections”, Collect. Math., 71:3 (2020), 549–574
- V. Przyjalkowski, C. Shramov, Weighted complete intersections, preprint
- M. Rapoport, “Complement à l'article de P. Deligne ‘La conjecture de Weil pour les surfaces $K3$’ ”, Invent. Math., 15 (1972), 227–236
- J. B. Rosser, L. Schoenfeld, “Approximate formulas for some functions of prime numbers”, Illinois J. Math., 6 (1962), 64–94
- M. Rossi, L. Terracini, “Linear algebra and toric data of weighted projective spaces”, Rend. Semin. Mat. Univ. Politec. Torino, 70:4 (2012), 469–495
- J. J. Sylvester, “On subinvariants, i.e. semi-invariants to binary quantics of an unlimited order”, Amer. J. Math., 5:1 (1882), 79–136
Supplementary files

