Abstract
Describimos deformaciones de 3-variedades Calabi-Yau no compactasWk := Tot(OP1 (-k) ⊕ OP1 (k - 2)),para k = 1, 2, 3. Concretamente, calculamos las deformaciones a través del primer grupo de cohomología H1(Wk, TWk) vía cohomología de Čech. Mostramos que para cada k = 1, 2, 3 las estructuras asociadas son cualitativamente distintas y, además, comentamos sobre sus diferencias con las estructuras análogas de las 2-variedades no compactas Tot(OP1 (-k)).
Highlights
Our motivation to study deformations of Calabi–Yau threefolds comes from mathematical physics
We focus on the case of Calabi–Yau threefolds
When looking for deformations of noncompact manifolds one needs to keep in mind the caveat that cohomology calculations are generally not enough to decide questions of existence of infinitesimal deformations, as the following example illustrates
Summary
Our motivation to study deformations of Calabi–Yau threefolds comes from mathematical physics. Deformations of complex structures of Calabi– Yau threefolds enter as terms of the integrals defining the action of the theories of Kodaira–Spencer gravity [3]. In general our threefolds will have infinite-dimensional deformation spaces, allowing for rich applications. We consider smooth Calabi–Yau threefolds Wk containing a line ∼= P1. W1 is the space appearing in the basic flop. The basic flop is described by the diagram: W p1. The basic flop is the rational map from W − to W +. It is famous in algebraic geometry for being the first case of a rational map that is not a blow-up. Zk := Tot OP1 (−k) for comparison in Sections 4 and 5
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