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Topological Abel-Jacobi Map for Hypersurfaces in Complex Projective Four-Space

Zhang, Yilong
http://rave.ohiolink.edu/etdc/view?acc_num=osu1650527960799903

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
The beginning of the study of the Abel-Jacobi map originates from an attempt to solve the indefinite integral $$\int\frac{dx}{\sqrt{x^3+ax^2+bx+c}}.$$ For a long time, mathematicians were unable to solve. In the point of view of algebraic geometry, the integral can be expressed as a line integral $$ \int_q^p\frac{dx}{y} $$ on the cubic curve $C$ defined by equation $y^2=x^3+ax^2+bx+c$. Since $C$ is topologically a torus and has nontrivial homology group, the integral \eqref{Abs_eqn_2} depends on the homotopy classes of the paths on $C$ joining $q$ to $p$. In fact, since the first homology group of $C$ is generated by loops $\gamma$ and $\delta$, the line integral is only well-defined modulo the periods $n\int_{\gamma}\frac{dx}{y}+m\int_{\delta}\frac{dx}{y}$, where $m,n\in\Z$. This gives definition of Abel-Jacobi map for genus one curve. In general, the Abel-Jacobi map for a genus $g$ curve $C_g$ is to integrate holomorphic 1-forms along unions of paths on $C_g$ modulo periods. In formal language, it is a group homomorphism sending a divisor of degree zero to the Jacobian variety of $C_g$. The Jacobi inversion theorem says that the Abel-Jacobi map is surjective. Based on this fact in families, Lefschetz showed that any integral $(1,1)$ class on a smooth projective surface is algebraic. In higher dimensions, Griffiths (1969) defined an Abel-Jacobi map sending algebraic cycles that are homologously trivial to intermediate Jacobians. However, the Griffiths' Abel-Jacobi map is rarely surjective. This is one of the difficulties of the Hodge conjecture in higher dimensions. Generalizing Griffiths' Abel-Jacobi map, Zhao (2015) defined the \textit{topological Abel-Jacobi map}, for a smooth complex projective variety $X$ of odd dimension and it has certain surjectivity property. To define such a map, choose a projective embedding of $X$. The domain of the topological Abel-Jacobi map is extended to vanishing cycles of smooth hyperplane sections of $X$, and the target is a "smaller" intermediate Jacobian associated with the middle-dimensional primitive cohomology of $X$. In this thesis, we first compare Zhao's definition to an alternative definition of the topological Abel-Jacobi map using extensions of the mixed Hodge structures suggested by Schnell. Second, we study the domain of the topological Abel-Jacobi map and its compactifications. More precisely, by deforming a hyperplane section in the universal family of smooth hyperplane sections of $X$, the integral vanishing cohomology forms a local system $\mathcal{H}_{van}$, whose \'etale space has a distinguished component $T_v$ containing a (primitive) vanishing cycle. $T_v$ is our preferred domain for the topological Abel-Jacobi map. When $X$ is a cubic threefold, a vanishing cycle on a hyperplane section is the difference of two skew lines. We characterize the compactifications of $T_v$ by understanding an irreducible component of the Hilbert scheme of $X$ containing a pair of skew lines. We determined when the topological Abel-Jacobi map extends to the compactifications. Besides, we relate to the ADE singularities on hyperplane sections of $X$ and some Bridgeland moduli spaces. When $X$ is a hypersurface of $\mathbb P^4$ of degree at least four, the $H^2$ on the hyperplane section has a mixed type, and the vanishing cycles are not algebraic in general. We majorly focus on the topological properties. We showed that the topology of $T_v$ is "complicated enough" to generate the $H^3(X,\mathbb Q)$ via the topological Abel-Jacobi map. This is related to tube mapping defined by Schnell.
Herb Clemens (Advisor)
Eric Katz (Committee Member)
David Anderson (Committee Member)
198 p.
Mathematics
Abel-Jacobi Map, Cubic Threefold, Vanishing Cycle, Hilbert Scheme, ADE singularities

Recommended Citations

Citations

  • Zhang, Y. (2022). Topological Abel-Jacobi Map for Hypersurfaces in Complex Projective Four-Space [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1650527960799903

    APA Style (7th edition)

  • Zhang, Yilong. Topological Abel-Jacobi Map for Hypersurfaces in Complex Projective Four-Space. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1650527960799903.

    MLA Style (8th edition)

  • Zhang, Yilong. "Topological Abel-Jacobi Map for Hypersurfaces in Complex Projective Four-Space." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1650527960799903

    Chicago Manual of Style (17th edition)

osu1650527960799903
126
© 2022, some rights reserved.Creative Commons License Topological Abel-Jacobi Map for Hypersurfaces in Complex Projective Four-Space by Yilong Zhang is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. Based on a work at etd.ohiolink.edu.
This open access ETD is published by The Ohio State University and OhioLINK.