L-Equivalence of Derived-Equivalent Simply Connected Varieties

Definitions and mathematical statement

Let $k$ be a field of characteristic zero. Write $K_0(\operatorname{Var}/k)$ for the Grothendieck ring of varieties over $k$: its additive generators are isomorphism classes $[V]$ of separated reduced schemes of finite type over $k$, with relations $[V]=[Z]+[V\setminus Z]$ for closed reduced subschemes $Z\subseteq V$, and multiplication $[V][W]=[V\times_k W]$. Set $\mathbb L=[\mathbb A^1_k]$. Two varieties $X,Y$ are $L$-equivalent if there exists an integer $r\geq 0$ such that \[ ([X]-[Y])\mathbb L^r=0\qquad\text{in }K_0(\operatorname{Var}/k). \] In the usual $k$-linear interpretation, $D$-equivalence means the existence of an exact $k$-linear equivalence \[ D^b(\operatorname{Coh}(X))\simeq D^b(\operatorname{Coh}(Y)), \] where these are the bounded derived categories of coherent sheaves.

The supplied question asks: for every field $k$ of characteristic zero and every pair of smooth projective simply connected $D$-equivalent varieties $X,Y$ over $k$, must $X$ and $Y$ be $L$-equivalent? The exponent $r$ may depend on $k,X,Y$, and $r=0$ is allowed.

The source leaves the meaning of "simply connected" over an arbitrary characteristic-zero field unspecified. One possible interpretation requires geometric connectedness and trivial geometric $\acute{e}$tale fundamental group $\pi_1^{\mathrm{\acute{e}t}}(X_{\bar k})$, and similarly for $Y$. Over ${\mathbb{C}}$, topological simple connectedness instead concerns the fundamental group of the associated complex analytic space. These interpretations must not be silently identified. The source also does not explicitly specify $k$-linearity of the derived equivalence. Thus the question above preserves the supplied scope, but its precise hypothesis requires clarification of these conventions.

Short English statement

Must every pair of smooth projective simply connected derived-equivalent varieties over a characteristic-zero field have equal Grothendieck classes after multiplication by some nonnegative power of the affine-line class? The meanings of simple connectedness and linearity of the derived equivalence need clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003609$, source problem code catalogue entry, titled "L-Equivalence of Derived-Equivalent Simply Connected Varieties," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2017.
• Supplied report citation: Algebraic Geometry: Birational Classification, Derived Categories, and Moduli Spaces (2018), Oberwolfach Reports, pp. 2703--2767, DOI https://doi.org/10.4171/owr/2017/45. The supplied literature list labels this reference "Oberwolfach Report 45 2017."
• Supplied literature reference labeled "Kuznetsov Shinder": https://arxiv.org/abs/1612.07193. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes the general implication for smooth projective simply connected varieties as a conjecture supported by examples. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.