Timeline for Do rational maps to abelian varieties extend across rational singularities?
Current License: CC BY-SA 4.0
6 events
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| May 27, 2023 at 7:51 | comment | added | Evgeny Shinder | I believe it is not known whether exceptional divisors for resolutions of rational singularities are always acyclic, as this can depend on singularities of the exceptional divisors themselves. What is known is that for a snc resolution, the fibers are acyclic, see Lemma 2.5 in arxiv.org/pdf/2212.06786.pdf. | |
| May 13, 2023 at 3:23 | comment | added | Ben C | @JasonStarr this is a very interesting strategy,. however I must say I don't understand why the corresponding cohomological vanishing statement for X^1 holds or how you draw your confusion from this equality of hodge structures | |
| May 13, 2023 at 2:27 | comment | added | naf | The pullback map $\mathrm{Pic}^0(X) \to \mathrm{Pic}^0(\tilde{X})$ is an isomorphism since it is always injective for a resolution of a normal variety and then it suffices to know that it is surjective on tangent spaces which follows from the Leray spectral sequence. This is all that is needed to see that the map extends, since for any nonconstant map $f: Z \to A$ where $A$ is an abelian variety---we apply this to a fibre of the resolution---the pullback map on $\mathrm{Pic}^0$ is never zero (reduce to the case $Z$ is a smooth projective curve). | |
| May 12, 2023 at 21:54 | comment | added | Jason Starr | Let $\widetilde{X}^1$ be a desingularization of the closure of the smooth locus in $\widetilde{X}\times_X \widetilde{X}$ with its projections $\text{pr}_i:\widetilde{X}^1 \to \widetilde{X}$, for $i=0,1$. By hypothesis, the natural maps $\mathcal{O}_{\widetilde{X}} \to R\text{pr}_{i,*} \mathcal{O}_{\widetilde{X}^1}$ are quasi-isomorphisms. Thus, via the Leray spectral sequence, the pullback maps $H^q(\widetilde{X},\mathcal{O}_{\widetilde{X}})\to H^q(\widetilde{X}^1,\mathcal{O}_{\widetilde{X}^1})$ maps are also isomorphisms. For $q=1$, this gives equality of the weight-$1$ Hodge structures. | |
| May 12, 2023 at 21:47 | history | edited | Ben C | CC BY-SA 4.0 |
edited title
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| May 12, 2023 at 19:18 | history | asked | Ben C | CC BY-SA 4.0 |