Cao and Zheng's tropical Griffiths conjecture

From papers

Let qq, nn and dd be positive integers such that q>Mq>M, where

M=(n+dd)1.M=\binom{n+d}{d}-1.

Let f:RTPnf:\mathbb{R}\to\mathbb{TP}^n be a tropical algebraically nondegenerate tropical holomorphic curve. Assume that tropical hypersurfaces VP0,,VPqV_{P_0},\ldots,V_{P_q} are defined by homogeneous tropical polynomials P0,,PqP_0,\ldots,P_q with degrees d0,,dqd_0,\ldots,d_q, respectively, such that the least common multiple of d0,,dqd_0,\ldots,d_q is dd. Let

λ=ddg({PM+1f,,Pqf})\lambda=\operatorname{ddg}(\{P_{M+1}\circ f,\ldots,P_q\circ f\})

and suppose that

lim suprlogTf(r)r=0.\limsup_{r\to\infty}\frac{\log T_f(r)}{r}=0.

Cao and Zheng's tropical Griffiths conjecture. The defect sum satisfies

j=0qδ(VPj,f)n+1+λd.\sum_{j=0}^q\delta(V_{P_j},f)\leq\frac{n+1+\lambda}{d}.

This is proposed as the tropical version of Griffiths' conjecture, following the tropical version of the Shiffman conjecture proposed by Cao and Zheng. The statement concerns defect relations for tropical holomorphic curves intersecting tropical hypersurfaces and is not resolved in the supplied source context.

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Sources & referencesView supporting material

Primary source

Juho Halonen, Risto Korhonen and Galina Filipuk, “Tropical second main theorem and the Nevanlinna inverse problem”, arXiv:2305.13939 (2023).

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