The Picard-Lefschetz formula and a conjecture of Kato, the case of Lefschetz fibrations

A conjecture of Kato says that the monodromy operator on the cohomology of a semi-stable degeneration of projective varieties is represented by an algebraic cycle on the special fiber of a normal crossing model of the fiber product degeneration. We prove this conjecture in the simple case of a semi-stable degeneration arising from a Lefschetz fibration

A model of toxic neuropathy in Drosophila reveals a role for MORN4 in promoting axonal degeneration

Axonal degeneration is a molecular self-destruction cascade initiated following traumatic, toxic, and metabolic insults. Its mechanism underlies a number of disorders including hereditary and diabetic neuropathies and the neurotoxic side effects of chemotherapy drugs. Molecules that promote axonal degeneration could represent potential targets for therapy. To identify such molecules, we designed a screening platform based on intoxication of Drosophila larvae with paclitaxel (taxol), a chemotherapeutic agent that causes neuropathy in cancer patients. In Drosophila, taxol treatment causes swelling, fragmentation, and loss of axons in larval peripheral nerves. This axonal loss is not due to apoptosis of neurons. Taxol-induced axonal degeneration in Drosophila shares molecular execution mechanisms with vertebrates, including inhibition by both NMNAT (nicotinamide mononucleotide adenylyltransferase) expression and loss of wallenda/DLK (dual leucine zipper kinase). In a pilot RNAi-based screen we found that knockdown of retinophilin (rtp), which encodes a MORN (membrane occupation and recognition nexus) repeat-containing protein, protects axons from degeneration in the presence of taxol. Loss-of-function mutants of rtp replicate this axonal protection. Knockdown of rtp also delays axonal degeneration in severed olfactory axons. We demonstrate that the mouse ortholog of rtp, MORN4, promotes axonal degeneration in mouse sensory axons following axotomy, illustrating conservation of function. Hence, this new model can identify evolutionarily conserved genes that promote axonal degeneration, and so could identify candidate therapeutic targets for a wide-range of axonopathies

Degenerations and limit Frobenius structures in rigid cohomology

We introduce a "limiting Frobenius structure" attached to any degeneration of projective varieties over a finite field of characteristic p which satisfies a p-adic lifting assumption. Our limiting Frobenius structure is shown to be effectively computable in an appropriate sense for a degeneration of projective hypersurfaces. We conjecture that the limiting Frobenius structure relates to the rigid cohomology of a semistable limit of the degeneration through an analogue of the Clemens-Schmidt exact sequence. Our construction is illustrated, and conjecture supported, by a selection of explicit examples.Comment: 41 page