Hodge theory—one of the pillars of modern algebraic geometry—is
a deep theory with many applications and open problems, the most
enigmatic of which is the Hodge conjecture, one of the Clay
Institute's seven Millennium Prize Problems. Hodge theory is also
famously difficult to learn, requiring training in many different
branches of mathematics. The present volume begins with an
examination of the precursors of Hodge theory: first, the studies
of elliptic and abelian integrals by Cauchy, Abel, Jacobi, and
Riemann, among many others; and then the studies of two-dimensional
multiple integrals by Poincaré and Picard. Thenceforth, the focus
turns to the Hodge theory of affine hypersurfaces given by tame
polynomials, for which many tools from singularity theory, such as
Brieskorn modules and Milnor fibrations, are used. Another aspect
of this volume is its computational presentation of many well-known
theoretical concepts, such as the Gauss–Manin connection,
homology of varieties in terms of vanishing cycles, Hodge cycles,
Noether–Lefschetz, and Hodge loci. All are explained for the
generalized Fermat variety, which for Hodge theory boils down to a
heavy linear algebra. Most of the algorithms introduced here are
implemented in Singular, a computer algebra system for polynomial
computations. Finally, the author introduces a few problems, mainly
for talented undergraduate students, which can be understood with a
basic knowledge of linear algebra. The origins of these problems
may be seen in the discussions of advanced topics presented
throughout this volume.
General
Imprint: |
International Press of Boston Inc
|
Country of origin: |
United States |
Release date: |
July 2021 |
Authors: |
Hossein Movasati
|
Dimensions: |
254 x 177 x 25mm (L x W x T) |
Format: |
Paperback
|
Pages: |
382 |
ISBN-13: |
978-1-57146-400-2 |
Categories: |
Books
|
LSN: |
1-57146-400-X |
Barcode: |
9781571464002 |
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