On the cohomology of surfaces with $p_g = q = 2$ and maximal Albanese dimension
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- by Johan Commelin and Matteo Penegini PDF
- Trans. Amer. Math. Soc. 373 (2020), 1749-1773
Abstract:
In this paper we study the cohomology of smooth projective complex surfaces $S$ of general type with invariants $p_g = q = 2$ and surjective Albanese morphism. We show that on a Hodge-theoretic level, the cohomology is described by the cohomology of the Albanese variety and a K3 surface $X$ that we call the K3 partner of $S$. Furthermore, we show that in suitable cases we can geometrically construct the K3 partner $X$ and an algebraic correspondence in $S \times X$ that relates the cohomology of $S$ and $X$. Finally, we prove the Tate and Mumford–Tate conjectures for those surfaces $S$ that lie in connected components of the Gieseker moduli space that contain a product-quotient or a mixed surface.References
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Additional Information
- Johan Commelin
- Affiliation: Albert-Ludwigs-Universität Freiburg, Mathematisches Institut, Ernst-ZermeloStraße 1, D-79104 Freiburg, Germany
- MR Author ID: 1198615
- Email: jmc@math.uni-freiburg.de
- Matteo Penegini
- Affiliation: Università degli Studi di Genova, DIMA Dipartimento di Matematica, I-16146 Genova, Italy
- MR Author ID: 948565
- Email: penegini@dima.unige.it
- Received by editor(s): January 18, 2019
- Received by editor(s) in revised form: June 2, 2019, and June 24, 2019
- Published electronically: December 10, 2019
- Additional Notes: The first author was supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 613.001.207 (Arithmetic and motivic aspects of the Kuga–Satake construction) and by the Deutsche Forschungs Gemeinschaft (DFG) under Graduiertenkolleg 1821 (Cohomological Methods in Geometry)
The second author was partially supported by MIUR PRIN 2015 “Geometry of Algebraic Varieties” and also by GNSAGA of INdAM - © Copyright 2019 by the authors under Creative Commons Attribution 4.0 International (CC BY 4.0)
- Journal: Trans. Amer. Math. Soc. 373 (2020), 1749-1773
- MSC (2010): Primary 14C15, 14J10; Secondary 14C30, 11G10, 14J29, 14L30
- DOI: https://doi.org/10.1090/tran/7940
- MathSciNet review: 4068281