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Universität Augsburg
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Universität Augsburg
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Professor Dr. Christian Liedtke
Technische Universität München
spricht am
Montag, 24. November 2014
um
16:00 Uhr
im
Raum 2004 (L1)
über das Thema:
| Abstract: |
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It is generally believed that the geometry of complex manifolds (maybe even symplectic manifolds) is largely determined by the quantitative and qualitative behavior of the rational curves they contain, or, non-trivial maps from genus zero Riemann surfaces to them. The framework of stable maps and Gromov-Witten invariants even gives a precise meaning to counting such maps. In complex dimension two, a conjecture of Bogomolov states the existence of infinitely many such curves on every K3 surface (Calabi-Yau manifold of dimension two) — the GW-count of these maps supports this, but (because of multiple covers) does actually not prove this. Another interesting point is that these rational curves cannot be deformed, and that one does not know of a way of producing new rational curves out of known ones, which says that there must be an infinite number of different reasons if there are infinitely many rational curves. Extending ideas of Bogomolov, Hassett, and Tschinkel, we establish this conjecture in the case the rank of the Picard group is odd (which includes the case of Picard rank one, which is the general case) by using methods from number theory and arithmetic geometry: namely, it suffices to check this conjecture for K3 surfaces that are defined by equations in projective space whose coefficients lie in number fields, and then, one can use reduction to finite fields, and arithmetic methods. This is joint work with Jun Li. |
| Hierzu ergeht herzliche Einladung. |