Mathematical Physics and Algebraic Geometry Seminar

Speaker: Chuck Doran, Harvard CMSA

Title: Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds

Abstract: Smooth $M_m$-polarized K3-fibered Calabi-Yau (CY) 3-folds have been classified in terms of the choice of a generalized functional invariant and, in the case $m=1$, a generalized homological invariant. The resulting geometries generally exhibit a small number of complex structure moduli greater or equal to two. A concrete choice of these invariants realizes (almost all of) the known Calabi-Yau geometries with exactly two moduli and allows us to describe completely the structure of the corresponding moduli spaces. The corresponding variations of Hodge structure are entirely determined by the regular periods, for which we obtain a generic expression in terms of $m$ and three integers $i,j,s$. Using the form of this period and Batyrev-Borisov mirror symmetry we explicitly construct the corresponding mirror CY 3-folds with two Kaehler moduli and show consistency with the DHT conjecture. In the cases with $s=0$, the mirror CY 3-folds are again K3-fibered but with the mirror $<2m>$-polarization. The generic form of the periods allows us to derive generic modular expressions for the A-model topological string free energies and we argue that those are a consequence of a Tyurin degeneration of the generalized functional invariant with the central fiber being an $M_m$-polarized K3. This is joint work with Boris Pioline and Thorsten Schimannek.