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I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. Can we deduce out of here some meaningful information about dimension of the deformation space of $V$?

For example, in the case of curves on a toric surface we can find the dimension of the deformation space of $V$ as its self-intersection. Similar arguments work for any hypersurface: its deformation space is encoded by its $n+1$-th power, where $n$ is its dimension.

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  • $\begingroup$ Output of deformation theory (tangent space, obstruction space) is encoded in various cohomology groups. Vanishing theorems, when they hold, may imply that the dimensions of those groups are Euler characteristics. In principle, these Euler haracteristics can be computed via Riemann-Roch, within the Chow group (or cohomology) only. This program is likely to work for toric varieties but details have to be written. $\endgroup$
    – ACL
    Commented Mar 8, 2014 at 8:37

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The Chow group $A_k(X)$ of an arbitrary toric variety $X$ with defining fan $\Delta$ is generated by the classes of the orbits closures $V(\sigma)$ of the cones $\sigma$ of dimension $n-k$ of $\Delta$. Then i will assume that your variety $V$ is a toric subvariety and you have an equivariant embedding $V\rightarrow X$.

You can find results about the deformations of affine toric varieties on the paper "Deformation of toric varieties via Minkowski sum decompositions of polyhedral complexes - Anvar. R. Mavlyutov".

available on arxiv: http://arxiv.org/pdf/0902.0967v3.pdf

In a more general setting, results about equivariant deformations and toric intersection theory of affine normal varieties $X$ with an effective torus action can be find on the paper "Geometry of $T$-varieties - Altmann et al".

available on arxiv: http://arxiv.org/pdf/1102.5760.pdf

I hope you find this useful.

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    $\begingroup$ A tip: it is generally best to link to the abstract pages for papers on arXiv rather directly to the PDF. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Mar 8, 2014 at 7:48

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