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• W2963016221 abstract "The Yau-Zaslow conjecture predicts the genus 0 curve counts of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K Baseline 3> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces in terms of the Dedekind <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=eta> <mml:semantics> <mml:mi>η<!-- η --></mml:mi> <mml:annotation encoding=application/x-tex>eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function. The classical intersection theory of curves in the moduli of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K Baseline 3> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces with Noether-Lefschetz divisors is related to 3-fold Gromov-Witten invariants via the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K Baseline 3> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> curve counts. Results by Borcherds and Kudla-Millson determine these classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K Baseline 3> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces), we prove the Yau-Zaslow conjecture for all curve classes on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper K Baseline 3> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>K3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper." @default.
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• W2963016221 date "2010-06-09" @default.
• W2963016221 modified "2023-10-16" @default.
• W2963016221 title "Noether-Lefschetz theory and the Yau-Zaslow conjecture" @default.
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• W2963016221 doi "https://doi.org/10.1090/s0894-0347-2010-00672-8" @default.
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