What is the total Betti number of a random real hypersurface?
-
Damien Gayet
and
Jean-Yves Welschinger
Abstract.
We bound from above the expected total Betti number of a high degree random real hypersurface in a smooth real projective manifold. This upper bound is deduced from the equidistribution of critical points of a real Lefschetz pencil restricted to the complex domain of such a random hypersurface, equidistribution which we first establish. Our proofs involve Hörmander's theory of peak sections as well as the formula of Poincaré–Martinelli.
The research leading to these results has received funding from the European Community's Seventh Framework Progamme (FP7/2007-2013, FP7/2007-2011) under grant agreement no. 258204, as well as from the French Agence nationale de la recherche, ANR-08-BLAN-0291-02. We are grateful to the referee for fruitful comments on the paper.
© 2014 by Walter de Gruyter Berlin/Boston
Articles in the same Issue
- Frontmatter
- Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa
- On the number of integers in a generalized multiplication table
- Peripheral structures of relatively hyperbolic groups
- What is the total Betti number of a random real hypersurface?
- Parametrization of ideal classes in rings associated to binary forms
- Weakly-exceptional singularities in higher dimensions
- Erratum to A derived approach to geometric McKay correspondence in dimension three (J. reine angew. Math. 636 (2009), 193–236)
Articles in the same Issue
- Frontmatter
- Nonlinear PDE aspects of the tt* equations of Cecotti and Vafa
- On the number of integers in a generalized multiplication table
- Peripheral structures of relatively hyperbolic groups
- What is the total Betti number of a random real hypersurface?
- Parametrization of ideal classes in rings associated to binary forms
- Weakly-exceptional singularities in higher dimensions
- Erratum to A derived approach to geometric McKay correspondence in dimension three (J. reine angew. Math. 636 (2009), 193–236)
