Shibata's refined multiplicity conjecture for log canonical complete intersection singularities

Let XX be an nn-dimensional variety that is locally a complete intersection with log canonical singularities. For a closed point xx of XX, let emb(X,x)\mathrm{emb}(X,x) be the embedding dimension at xx and let lct(mx)\mathrm{lct}(\mathfrak m_x) be the log canonical threshold of the maximal ideal mx\mathfrak m_x. Write e(OX,x)e(\mathcal O_{X,x}) for the Hilbert–Samuel multiplicity. Shibata's refined multiplicity conjecture.

e(OX,x)2nlct(mx),e(\mathcal O_{X,x})\le 2^{n-\lceil\mathrm{lct}(\mathfrak m_x)\rceil},

and equality holds if and only if

emb(X,x)=2nlct(mx).\mathrm{emb}(X,x)=2n-\lceil\mathrm{lct}(\mathfrak m_x)\rceil.

The source presents this as a refinement of Watanabe's conjecture, allowing log canonical rather than canonical singularities and incorporating the log canonical threshold and embedding dimension. No resolution is supplied in the given text.

Sources & referencesView supporting material

Primary source

Kohsuke Shibata, “Bounds of the multiplicity of abelian quotient complete intersection singularities”, arXiv:1908.01218 (2019).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.