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Mirror symmetric Gamma conjecture for del Pezzo surfaces
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abstract
For a del Pezzo surface of degree $\geq 3$, we compute the oscillatory integral for its mirror Landau-Ginzburg model in the sense of Gross-Hacking-Keel [Mark Gross, Paul Hacking, and Sean Keel, "Mirror symmetry for log Calabi-Yau surfaces I". In: Publ. Math. Inst. Hautes Etudes Sci. 122 (2015), pp. 65-168]. We explicitly construct the mirror cycle of a line bundle and show that the leading order of the integral on this cycle involves the twisted Chern character and the Gamma class. This proves a version of the Gamma conjecture for non-toric Fano surfaces with an arbitrary K-group insertion.
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Relative mirror symmetry, theta functions and the Gamma conjecture
For Fano varieties with an snc anticanonical divisor whose dual intersection complex is pure-dimensional, the mirror symmetric Gamma conjecture holds for the structure sheaf and skyscraper sheaf.
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