Tropical intersection theory and gravitational descendants

  • This thesis is devoted to two main topics (accordingly, there are two chapters): In the first chapter, we establish a tropical intersection theory with analogue notions and tools as its algebro-geometric counterpart. This includes tropical cycles, rational functions, intersection products of Cartier divisors and cycles, morphisms, their functors and the projection formula, rational equivalence. The most important features of this theory are the following: - It unifies and simplifies many of the existing results of tropical enumerative geometry, which often contained involved ad-hoc computations. - It is indispensable to formulate and solve further tropical enumerative problems. - It shows deep relations to the intersection theory of toric varieties and connected fields. - The relationship between tropical and classical Gromov-Witten invariants found by Mikhalkin is made plausible from inside tropical geometry. - It is interesting on its own as a subfield of convex geometry. In the second chapter, we study tropical gravitational descendants (i.e. Gromov-Witten invariants with incidence and "Psi-class" factors) and show that many concepts of the classical Gromov-Witten theory such as the famous WDVV equations can be carried over to the tropical world. We use this to extend Mikhalkin's results to a certain class of gravitational descendants, i.e. we show that many of the classical gravitational descendants of P^2 and P^1 x P^1 can be computed by counting tropical curves satisfying certain incidence conditions and with prescribed valences of their vertices. Moreover, the presented theory is not restricted to plane curves and therefore provides an important tool to derive similar results in higher dimensions. A more detailed chapter synopsis can be found at the beginning of each individual chapter.
  • Tropische Schnitt-Theorie und gravitational descendants

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Author:Johannes Rau
Advisor:Andreas Gathmann
Document Type:Doctoral Thesis
Language of publication:English
Year of Completion:2009
Year of Publication:2009
Publishing Institute:Technische Universität Kaiserslautern
Granting Institute:Technische Universität Kaiserslautern
Acceptance Date of the Thesis:2009/07/31
Date of the Publication (Server):2009/08/12
Tag:Enumerative Geometrie
GND-Keyword:Tropische Geometrie; Algebraische Geometrie; Schnitttheorie; Gromov-Witten-Invariante
Faculties / Organisational entities:Kaiserslautern - Fachbereich Mathematik
DDC-Cassification:5 Naturwissenschaften und Mathematik / 510 Mathematik
MSC-Classification (mathematics):14-XX ALGEBRAIC GEOMETRY / 14Cxx Cycles and subschemes / 14C17 Intersection theory, characteristic classes, intersection multiplicities [See also 13H15]
14-XX ALGEBRAIC GEOMETRY / 14Nxx Projective and enumerative geometry [See also 51-XX] / 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
52-XX CONVEX AND DISCRETE GEOMETRY / 52Bxx Polytopes and polyhedra / 52B20 Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]
Licence (German):Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011