Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in number theory. Se mer In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Se mer De Rham cohomology The Hodge theory references the de Rham complex. Let M be a smooth manifold. For a non-negative integer k, let Ω (M) be the real Se mer Let X be a smooth complex projective variety. A complex subvariety Y in X of codimension p defines an element of the cohomology group Se mer • Potential theory • Serre duality • Helmholtz decomposition • Local invariant cycle theorem Se mer The field of algebraic topology was still nascent in the 1920s. It had not yet developed the notion of cohomology, and the interaction between differential forms and topology was poorly understood. In 1928, Élie Cartan published a note entitled Sur les nombres de … Se mer Let X be a smooth complex projective manifold, meaning that X is a closed complex submanifold of some complex projective space CP . By Chow's theorem, complex projective … Se mer Mixed Hodge theory, developed by Pierre Deligne, extends Hodge theory to all complex algebraic varieties, not necessarily smooth or … Se mer Nettet15. mai 2024 · Topology and its Applications. Volume 313, 15 May 2024, 107981. Hodge theory on Alexander invariants – A survey. ... Note that the infinite cyclic cover U ξ is not in general a complex algebraic variety, so …
Applications of Hodge theory to topology and analysis
NettetLimits in Hodge Theory∗ Phillip Gri ths ∗Lecture series at the University of Miami, Spring 2024 1/64. 2/64 Abstract Almost all of the deep results in Hodge theory and its applications to algebraic geometry require understanding the limits in a family of Hodge structures. In the Nettet10. nov. 2024 · I would like to learn p-adic hodge theory, so I have searched for textbooks of p-adic hodge theory, but I wasn't able to find it except for books for specialists who already know a lot about it. I slightly have a background in number theory, abstract algebra and category theory, but no background in differential geometry, hence I would … shelf life of bread in freezer
Cohomological Descent for Faltings
Nettet8. mai 2024 · May 8 - 14, 2024. Organizers: Bhargav Bhatt, University of Michigan. Martin Olsson, University of California, Berkeley. In this final symposium, organizers Bhargav Bhatt and Martin Olsson covered recent advances in all aspects of 𝑝-adic Hodge theory, including both foundational breakthroughs internal to the subject as well as … NettetWe treat generalizations of Koll\'ar's torsion-freeness, vanishing theorem, and so on, for projective morphisms between complex analytic spaces as an application of the theory of variations of mixed Hodge structure. The results will play a crucial role in the theory of minimal models for projective morphisms of complex analytic spaces. In this paper, we … NettetHodge theory originated as an application of harmonic theory to the study of the geometry of compact complex manifolds. The ideas have proved to be quite powerful, … shelf life of box wine