We will work through the 2001 Inventiones paper with the same name of Borisov and Gunnels and learn how to construct modular forms from the combinatorial data associated to toric varieties. Although the main results speak about the modularity of toric modular forms and some of their properties, the proofs use techniques from homological algebra and the theory of toric varieties.

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Schedule and list of speakers:

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08.10.18: Markus Schwagenscheidt, „Overview: Toric Varieties and Modular Forms“

15.10.18: Isabelle Charton, „Toric Varieties I: Basic definitions“

22.10.18: Isabelle Charton, „Toric Varieties II: Vector bundles and cohomology"

29.10.18: TBA, „Toric Varieties III: Hirzebruch-Riemann-Roch Theorem"

05.11.18: TBA, „Toric Forms: Definitions and Examples"

12.11.18: TBA, „A homological interpretation of toric forms"

19.11.18: TBA, „Convergence of toric forms"

26.11.18: Jonas Kaszian, „Modular forms and theta functions"

03.12.18: Alexander Caviedes Castro, „Representation of toric forms by theta functions"

10.12.18: TBA, „Transformation behaviour of toric forms"

17.12.18: TBA, „Hecke operators“

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Bibliography:

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[1] Lev A. Borisov and Paul E. Gunnels. Toric manifolds and modular forms. Invent. Math., 144, 2001.

[2]  David A. Cox, John B. Little, and Henry K. Schenck. Toric varieties. volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2011.

[3]  William Fulton. Introduction to toric varieties. volume 131 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1993.

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