Mirror symmetry for weighted projective planes and their noncommutative deformations

TL;DR

This paper proves the homological mirror symmetry for weighted projective planes and extends it to their noncommutative deformations, linking derived categories of coherent sheaves with those of Lagrangian vanishing cycles.

Contribution

It establishes the homological mirror symmetry for weighted projective planes and extends the correspondence to noncommutative deformations, broadening the scope of mirror symmetry results.

Findings

Homological mirror symmetry holds for weighted projective planes.

Derived categories of coherent sheaves are equivalent to those of vanishing cycles.

Mirror symmetry extends to noncommutative deformations of these spaces.

Abstract

We study the derived categories of coherent sheaves of weighted projective spaces and their noncommutative deformations, and the derived categories of Lagrangian vanishing cycles of their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves (B-branes) on the weighted projective plane $\CP^2(a,b,c)$ is equivalent to the derived category of vanishing cycles (A-branes) on the affine hypersurface $X=\{x^ay^bz^c=1\}\subset (\C^*)^3$ equipped with an exact symplectic form and the superpotential $W=x+y+z$. Hence, the homological mirror symmetry conjecture holds for weighted projective planes. Moreover, we also show that this mirror correspondence between derived categories can be extended to toric noncommutative deformations of $\CP^2(a,b,c)$ where B-branes are concerned, and their mirror counterparts, non-exact deformations of the symplectic structure of $X$ where A-branes are concerned. We also obtain similar results for other examples such as weighted projective lines or Hirzebruch surfaces.