Groups of symplectic involutions on symplectic varieties of Kummer type and their fixed loci
Bibliographic record
Abstract
Abstract We describe the Galois action on the middle $\ell $ -adic cohomology of smooth, projective fourfolds $K_A(v)$ that occur as a fiber of the Albanese morphism on moduli spaces of sheaves on an abelian surface A with Mukai vector v . We show this action is determined by the action on $H^2_{\mathrm {\acute{e}t}}(A_{\bar {k}},{\mathbb Q}_{\ell }(1))$ and on a subgroup $G_A(v) \leqslant (A\times \hat {A})[3]$ , which depends on v . This generalizes the analysis carried out by Hassett and Tschinkel over ${\mathbb C}$ [21]. As a consequence, over number fields, we give a condition under which $K_2(A)$ and $K_2(\hat {A})$ are not derived equivalent. The points of $G_A(v)$ correspond to involutions of $K_A(v)$ . Over ${\mathbb C}$ , they are known to be symplectic and contained in the kernel of the map $\operatorname {\mathrm {Aut}}(K_A(v))\to \mathrm {O}(H^2(K_A(v),{\mathbb Z}))$ . We describe this kernel for all varieties $K_A(v)$ of dimension at least $4$ . When $K_A(v)$ is a fourfold over a field of characteristic 0, the fixed-point loci of the involutions contain K3 surfaces whose cycle classes span a large portion of the middle cohomology. We examine the fixed-point locus on fourfolds $K_A(0,l,s)$ over ${\mathbb C}$ where A is $(1,3)$ -polarized, finding the K3 surface to be elliptically fibered under a Lagrangian fibration of $K_A(0,l,s)$ .
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".