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3. Properties of higher singularities and minimal exponents


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      Problem 3.05.

      [Mihnea Popa] Let X be a variety, and H a Cartier divisor on X. Suppose H is m-Du Bois. Is X m-Du Bois around H?
        1. Remark. [Mircea Mustaţă] There is a counter-example for the pre-m-Du Bois version; see [MOPW23, Example 1.7]. The idea is that if the minimal exponent is large (i.e m-Du Bois for some $m$), then there is non-vanishing for cohomology sheaf of some Du Bois complex. To construct a counter-example, take a pencil of hypersurfaces generated by f and g, where f has large minimal exponent while g has quotient singularity. Then g is pre-m-Du Bois for all m, while a general fiber is not.

          References: [MOPW23] M. Mustaţă, S. Olano, M. Popa, and J. Witaszek, The Du Bois complex of a hypersurface and the minimal exponent, Duke Math. J. 172 (2023), no. 7, 1411–1436.


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              Problem 3.1.

              [Zhengze Xin] What can we say about higher singularities for pure morphisms (e.g. direct summands, linearly reductive group quotients)?


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                  Problem 3.15.

                  [Wanchun Shen] What is the relation between non-smoothability and 1-Du Bois singularities?


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                      Problem 3.2.

                      [Sridhar Venkatesh] Let X be a normal variety of dimension n. X is pre-$\lceil \frac{n}{2} \rceil$-Du Bois. Then is X pre-n-Du Bois? X is pre-$\lfloor \frac{n}{2} \rfloor$-rational. Then is X pre-n-rational?


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                          Problem 3.25.

                          [Sung Gi Park] Are there universal bounds on the multiplicity of a p-DB and p-rational singularities?

                          For p=0, dim X = n, embedding dimension of x=e, then $mult_x \le \binom{e}{n}$ when X is Du Bois, and $\binom{e-1}{n-1}$ when X is rational?


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                              Problem 3.3.

                              [Mircea Mustaţă] Let H = Z(f), $p\in H$. Is the minimal exponent $\tilde{\alpha}_p(H) \le lct(\mathfrak{m}_p \cdot Jac(f))$?

                              This is related to Tessier’s conjecture and also to the work of Dano Kim who proved that this is true if $\tilde{\alpha}_p(H) = lct(H)$. Mircea also commented that it is easy to reduce to the case of isolated singularities.


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                                  Problem 3.35.

                                  [Alexandra Viktorova] Consider a family of hypersurfaces with non-isolated singularities. Is there a condition for the minimal exponent to be constant in the family? (When the hypersurfaces have isolated singularities, there is a condition in terms of Milnor numbers.)


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                                      Problem 3.4.

                                      [Radu Laza] How do we decide if a hypersurface H is m-Du Bois or m-rational in the non-isolated case?


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                                          Problem 3.45.

                                          [Mihnea Popa] What are the $(n+1)\times (n+1)$ tables with entries either 0 or 1, depending on whether $\mathcal(H)^q \underline{\Omega}_X^p$ is zero or not?

                                          Most simple case: Is there a surface X with $\mathcal(H)^1 \underline{\Omega}_X^1 = 0$, but $\mathcal(H)^1 \underline{\Omega}_X^0 \neq 0$?


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                                              Problem 3.5.

                                              [Wanchun Shen] Maxim-Yang prove a result characterizing $m$-Du Bois and $m$-rational hypersurface singularities in terms of characteristic classes. Can we extend this to the lci case?


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                                                  Problem 3.55.

                                                  [Jakub Witaszek] Can we prove that a variety with ‘enough’ endomorphisms is pre-$m$-Du Bois for all $m$? This would support a conjecture which predicts that such varieties have to be toric.


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                                                      Problem 3.6.

                                                      Study the global behavior of period map on $m$-Du Bois loci.


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                                                          Problem 3.65.

                                                          Understand the higher Du Bois singularities in the case of cubics and see what are some examples of Hodge structures in the case of cubics.

                                                              Cite this as: AimPL: Higher Du Bois and higher rational singularities, available at http://aimpl.org/higherdubois.