Displaying the cohomology of toric line bundles
- Authors: Altmann K.1, Ploog D.2
-
Affiliations:
- Freie Universität Berlin, Institut für Mathematik
- Universität Hannover, Institut für Mathematik
- Issue: Vol 84, No 4 (2020)
- Pages: 66-78
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/142293
- DOI: https://doi.org/10.4213/im8948
- ID: 142293
Cite item
Open Access
Access granted
Subscription Access
Abstract
There is a standard approach to calculate the cohomology of torus-invariant sheaves$\mathcal{L}$ on a toric variety via the simplicial cohomology of the associated subsets$V(\mathcal{L})$ of the space $N_\mathbb{R}$ of 1-parameter subgroups of the torus.For a line bundle $\mathcal{L}$ represented by a formal difference $\Delta^+-\Delta^-$ of polyhedrain the character space $M_\mathbb{R}$, [1] contains a simpler formula for the cohomology of $\mathcal{L}$, replacing $V(\mathcal{L})$ by the set-theoretic difference $\Delta^- \setminus \Delta^+$.Here, we provide a short and direct proof of this formula.
Keywords
About the authors
Klaus Altmann
Freie Universität Berlin, Institut für Mathematik
Email: izv@mi-ras.ru
David Ploog
Universität Hannover, Institut für Mathematik
Author for correspondence.
Email: izv@mi-ras.ru
Doctor of physico-mathematical sciences
References
- K. Altmann, J. Buczynski, L. Kastner, A.-L. Winz, “Immaculate line bundles on toric varieties”, Pure Appl. Math. Q. (to appear)
- D. A. Cox, J. B. Little, H. K. Schenck, Toric varieties, Grad. Stud. Math., 124, Amer. Math. Soc., Providence, RI, 2011, xxiv+841 pp.
- M. Demazure, “Sous-groupes algebriques de rang maximum du groupe de Cremona”, Ann. Sci. Ecole Norm. Sup. (4), 3:4 (1970), 507–588
- G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Math., 339, Springer-Verlag, Berlin–New York, 1973, viii+209 pp.
- В. И. Данилов, “Геометрия торических многообразий”, УМН, 33:2(200) (1978), 85–134
- T. Oda, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties, Ergeb. Math. Grenzgeb. (3), 15, Springer-Verlag, Berlin, 1988, viii+212 pp.
- W. Fulton, Introduction to toric varieties, The 1989 W. H. Roever lectures in geometry, Ann. of Math. Stud., 131, Princeton Univ. Press, Princeton, NJ, 1993, xii+157 pp.
- Р. Хартсхорн, Алгебраическая геометрия, Мир, М., 1981, 600 с.
- Р. Ботт, Л. В. Ту, Дифференциальные формы в алгебраической топологии, Наука, М., 1989, 336 с.
- D. Huybrechts, Fourier–Mukai transforms in algebraic geometry, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press, Oxford, 2006, viii+307 pp.
- D. Orlov, “Remarks on generators and dimensions of triangulated categories”, Mosc. Math. J., 9:1 (2009), 143–149
