Tropical Griffiths-type defect relation for tropical hypersurfaces

About 8 years old · traced to

Let qq and nn be positive integers with q≥nq\geq n. Let f:R→TPnf:\mathbb{R}\rightarrow\mathbb{TP}^{n} be a tropical algebraically nondegenerated tropical holomorphic curve. Assume that PjP_j (j=1,…,q)(j=1,\ldots,q) are homogeneous tropical polynomials of degrees djd_j, and let dd be the least common multiple of d1,…,dqd_1,\ldots,d_q. Set M=(n+dd)−1M=\binom{n+d}{d}-1. If λ=ddg⁡({PM+2∘f,…,Pq∘f})\lambda=\operatorname{ddg}(\{P_{M+2}\circ f,\ldots,P_q\circ f\}) and

lim sup⁡r→∞log⁡Tf(r)r=0,\limsup_{r\rightarrow\infty}\frac{\log T_f(r)}{r}=0,

Tropical Griffiths-type defect conjecture. One should have

∑j=1qδf(VPj)≤n+1+λd.\sum_{j=1}^{q}\delta_f(V_{P_j})\leq\frac{n+1+\lambda}{d}.

This is proposed as a tropical analogue of the Griffiths conjecture on defect relations for holomorphic curves and hypersurfaces. The source notes that related classical results are known in special cases, but does not state a resolution of this tropical formulation.

References

Primary source

Tingbin Cao and Jianhua Zheng, “Second main theorem with tropical hypersurfaces and defect relation”, arXiv:1806.04294 (2018).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.