Degenerations of Calabi-Yau Manifolds and Related Geometries

• Applicant: Professor Dr. Helge Ruddat
• Subject Area: Mathematics
• Term: from 2015 to 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 272561367

• Final Report Year: 2023

Final Report Abstract

Calabi-Yau manifolds form a central geometric class with a plethora of connections and applications to other mathematical areas and mathematical physics. Various structural questions about the particularly interesting three-dimenensional such geometries could not be answered to date, e.g. to which extent mirror symmetry applies and for what fundamental reason it can be observed. As part of the research project, a methodology for studying these questions was developed. The basic approach consists of gaining new information through a maximal degeneration of the geometry. Logarithmic and tropical geometry provided a natural basis for controlling the geometry of degenerations. In mathematical physics, degenerations already came about in connection with the phenomenon of mirror symmetry discovered around 1990 in the context of string theory. This deep-reaching relationship of complex and symplectic geometry of two different Calabi-Yau manifolds was further explored in the course of the project and new insights were revealed. A general theorem about the smoothability of maximally degenerate spaces under mild assumptions was published. For more special degenerating families built from from wall structures, fundamental structural statements such as analyticity and versality were verified. In addition, the project produced a simple formula for period integrals in degeneration families and many new results on tropical curves and their correspondence with Lagrangian submanifolds as well as with algebraic curves and the algebraic structures based on them.

Publications

• Motivic Zeta Functions of the Quartic and its Mirror Dual, Proc. Sympos. Pure Math. 93 String-Math 2014, published 2016, Pages 189–200
  Helge Ruddat, Johannes Nicaise, Peter Overholser
  
• Perverse curves and mirror symmetry. Journal of Algebraic Geometry, 26(1), 17-42.
  Ruddat, Helge
  
• Towards mirror symmetry for varieties of general type. Advances in Mathematics, 308, 208-275.
  Gross, Mark; Katzarkov, Ludmil & Ruddat, Helge
  
• Descendant log Gromov-Witten invariants for toric varieties and tropical curves. Transactions of the American Mathematical Society, 373(2), 1109-1152.
  Mandel, Travis & Ruddat, Helge
  
• Local Gromov-Witten invariants are log invariants. Advances in Mathematics, 350, 860-876.
  van Garrel, Michel; Graber, Tom & Ruddat, Helge
  
• Local uniqueness of approximations and finite determinacy of log morphisms. Preprint, 20 p.
  Helge Ruddat
  
• Logarithmic Enumerative Geometry and Mirror Symmetry. Oberwolfach Reports, 16(2), 1639-1695.
  Abramovich, Dan; van Garrel, Michel & Ruddat, Helge
  
• Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations. Publications mathématiques de l'IHÉS, 132(1), 1-82.
  Ruddat, Helge & Siebert, Bernd
  
• Tropically constructed Lagrangians in mirror quintic threefolds. Forum of Mathematics, Sigma, 8(2020).
  Mak, Cheuk Yu & Ruddat, Helge
  
• A homology theory for tropical cycles on integral affine manifolds and a perfect pairing. Geometry & Topology, 25(6), 3079-3132.
  Ruddat, Helge
  
• Compactifying Torus Fibrations Over Integral Affine Manifolds with Singularities. MATRIX Book Series (2021), 609-622. American Geophysical Union (AGU).
  Ruddat, Helge & Zharkov, Ilia
  
• Smoothing toroidal crossing spaces. Forum of Mathematics, Pi, 9(2021).
  Felten, Simon; Filip, Matej & Ruddat, Helge
  
• Tailoring a pair of pants. Advances in Mathematics, 381, 107622.
  Ruddat, Helge & Zharkov, Ilia
  
• The degeneration formula for stable log maps. manuscripta mathematica, 170(1-2), 63-107.
  Kim, Bumsig; Lho, Hyenho & Ruddat, Helge
  
• Tropical Quantum Field Theory, Mirror Polyvector Fields, and Multiplicities of Tropical Curves. International Mathematics Research Notices, 2023(4), 3249-3304.
  Mandel, Travis & Ruddat, Helge
  
• The proper Landau-Ginzburg potential is the open mirror map. Advances in Mathematics, 447, 109639.
  Gräfnitz, Tim; Ruddat, Helge & Zaslow, Eric