Residue mirror symmetry for Grassmannians

Motivated by recent works on localizations in A-twisted gauged linear sigma models, we discuss a generalization of toric residue mirror symmetry to complete intersections in Grassmannians.Comment: 41 pages. v3: revised following the suggestions of the referee

Complex Kuranishi structures and counting sheaves on Calabi-Yau 4-folds, II

We develop a theory of complex Kuranishi structures on projective schemes. These are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the theory to projective moduli spaces M of stable sheaves on Calabi-Yau 4-folds. Using real derived differential geometry, Borisov-Joyce produced a virtual homology cycle on M. In the prequel to this paper we constructed an algebraic virtual cycle on M. We prove the cycles coincide in homology after inverting 2 in the coefficients. And when Borisov-Joyce's real virtual dimension is odd, their virtual cycle is 2-torsion.Comment: Incorrect proof (that Borisov-Joyce class is 2-torsion when virtual dimension is odd) replaced by a correct one. 63 page

Counting sheaves on Calabi-Yau 4-folds, I

Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localisation of Edidin-Graham's square root Euler class for $SO(r,\mathbb C)$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a $K$-theoretic refinement by defining $K$-theoretic square root Euler classes and their localised versions. In a sequel we prove our invariants reproduce those of Borisov-Joyce.Comment: 63 pages. Fixed sign problem pointed out by J{\o}rgen Rennemo and convergence problem pointed out by Andrei Okounko

Simultaneous reconstruction of the forearm extensor compartment tendon, soft tissue, and skin

Malignant peripheral nerve sheath tumor (MPNST) is a very rare type of sarcoma, with an incidence of 0.001%. MPNST has a 5-year survival rate near 80%, so successful reconstruction techniques are important to ensure the patient’s quality of life. Sarcoma of the forearm is known for its poor prognosis, which leads to wider excision, making reconstruction even more challenging due to the unique anatomical structure and delicate function of the forearm. A 44-year-old male presented with a large mass that had two aspects, measuring 9×6 cm and 7×5 cm, on the dorsal aspect of the right forearm. The extensor compartment muscles (EDM, EDC, EIP, EPB, EPL, ECRB, ECRL, APL) and invaded radius were resected with the mass. Tendon transfer of the entire extensor compartment with skin defect coverage using a 24×8 cm anterolateral thigh (ALT) perforator free flap was performed. The patient was discharged after 18 days without wound complications, and has not complained of discomfort during supination, pronation, or wrist extension/flexion through 3 years of follow-up. To our knowledge, this is the first report of successful reconstruction of the entire forearm extensor compartment with ALT free flap coverage after resection of MPNST