Harnack manifolds

Abstract: In 1876, Axel Harnack proved in a foundational article that

1) every real algebraic curve of degree d in RP^2 has at most (d-1)(d-2)/2 + 1 connected components;

2) for every d there exists a curve of degree d with exactly this number of connected components.

Over the past 150 years, these results have played a central role in the study of the topology of real algebraic varieties. The first part of Harnack’s theorem generalizes to the so-called Smith–Floyd inequality for arbitrary real algebraic varieties: the sum of the Betti numbers of the real part is at most the corresponding sum for the complex part. Despite spectacular advances, the generalization of the second part of Harnack’s theorem remains open in the case of projective hypersurfaces.

For these, however, Ilia Itenberg and Oleg Viro showed that the Smith–Floyd inequality is asymptotically optimal by using the combinatorial patchworking technique. In joint work with Erwan Brugallé and Jean-Yves Welschinger, we show that an elementary generalization of Harnack’s original construction method in dimension 2 yields this asymptotic optimality for any ample line bundle on a real algebraic variety. Beyond Betti numbers, we also describe the diffeomorphism type of an open subset of these topologically rich varieties.

algebraic geometry

Audience: researchers in the discipline

ODTU-Bilkent Algebraic Geometry Seminars