TY - BOOK

T1 - Hodge theory and algebraic cycles

AU - Paulsen, Matthias Christoph Bernhard

PY - 2023

Y1 - 2023

N2 - This thesis tackles different problems related to the connection between geometric and Hodge theoretic aspects of algebraic varieties. One of the main results, joint with Stefan Schreieder and Remy van Dobben de Bruyn, concerns the construction problem for Hodge numbers. We realize all Hodge diamonds in ℤ/m for arbitrary m ≥ 2 by smooth complex projective varieties. This results in a full answer to a question by Kollár about universal polynomial relations between Hodge numbers. Then we investigate the case of positive characteristic, where Hodge symmetry may fail. In this setting, we are able to realize even all asymmetric Hodge diamonds in ℤ/m. Therefore, we completely understand polynomial relations between Hodge numbers in arbitrary characteristic. Another main result of this thesis solves the first instances of a conjecture by Griffiths and Harris from 1985 about the degree of curves on very general hypersurfaces. Specifically, a very general complex hypersurface in ℙ⁴ of degree d ≥ 6 is conjectured to contain only curves of degree divisible by d. Based on a degeneration technique developed by Kollár in 1991, we prove this conjecture and its higher-dimensional generalizations for infinitely many values of d. The conjecture by Griffiths and Harris was not known for any d previously. Using the link between this problem and the failure of the integral Hodge conjecture, our result shows that the cokernel of the cycle class map is precisely ℤ/d for these hypersurfaces. In the last part of the thesis, we consider another counterexample to the integral Hodge conjecture, namely the first unirational fourfold with a non-algebraic Hodge class, recently found by Stefan Schreieder. We construct a smooth resolution of Schreieder's conic bundle and study a certain unramified cohomology class on it through a geometric description of the norm residue map in Borel–Moore homology. Our explicit approach allows to get a better understanding of this example and might help to decide in the future whether the constructed non-algebraic class is torsion.

AB - This thesis tackles different problems related to the connection between geometric and Hodge theoretic aspects of algebraic varieties. One of the main results, joint with Stefan Schreieder and Remy van Dobben de Bruyn, concerns the construction problem for Hodge numbers. We realize all Hodge diamonds in ℤ/m for arbitrary m ≥ 2 by smooth complex projective varieties. This results in a full answer to a question by Kollár about universal polynomial relations between Hodge numbers. Then we investigate the case of positive characteristic, where Hodge symmetry may fail. In this setting, we are able to realize even all asymmetric Hodge diamonds in ℤ/m. Therefore, we completely understand polynomial relations between Hodge numbers in arbitrary characteristic. Another main result of this thesis solves the first instances of a conjecture by Griffiths and Harris from 1985 about the degree of curves on very general hypersurfaces. Specifically, a very general complex hypersurface in ℙ⁴ of degree d ≥ 6 is conjectured to contain only curves of degree divisible by d. Based on a degeneration technique developed by Kollár in 1991, we prove this conjecture and its higher-dimensional generalizations for infinitely many values of d. The conjecture by Griffiths and Harris was not known for any d previously. Using the link between this problem and the failure of the integral Hodge conjecture, our result shows that the cokernel of the cycle class map is precisely ℤ/d for these hypersurfaces. In the last part of the thesis, we consider another counterexample to the integral Hodge conjecture, namely the first unirational fourfold with a non-algebraic Hodge class, recently found by Stefan Schreieder. We construct a smooth resolution of Schreieder's conic bundle and study a certain unramified cohomology class on it through a geometric description of the norm residue map in Borel–Moore homology. Our explicit approach allows to get a better understanding of this example and might help to decide in the future whether the constructed non-algebraic class is torsion.

U2 - 10.15488/15542

DO - 10.15488/15542

M3 - Doctoral thesis

CY - Hannover

ER -