Chen, ZhijieZhijieChenCHANG-SHOU LIN2024-01-262024-01-262023-09-010022040Xhttps://scholars.lib.ntu.edu.tw/handle/123456789/638949We study the SU(3) Toda system with singular sources (∆ ∆ u v+ + 2 2 e e vu− − e e uv= = 4 4 π π PPm kmk=0 =0 n n 2 1 ,k,k δ δ p p k kon on E E τ τ, , where Eτ := C/(Z + Zτ) with Im τ > 0 is a flat torus, δpk is the Dirac measure at pk, and ni,k ∈ Z≥0 satisfy Pk n1,k 6≡ Pk n2,k mod 3. This is known as the non-critical case and it follows from a general existence result of [3] that solutions always exist. In this paper we prove that (i) The system has at most m 1 3 × 2m+1Y (n1,k + 1)(n2,k + 1)(n1,k + n2,k + 2) ∈ N k=0 solutions. We have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry. (ii) For m = 0 and p0 = 0, the system has even solutions if and only if at least one of {n1,0, n2,0} is even. Furthermore, if n1,0 is odd, n2,0 is even and n1,0 < n2,0, then except for finitely many τ’s modulo SL(2, Z) action, the system has exactly n1,20+1 even solutions. Differently from [3], our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.journal article10.4310/jdg/16952365922-s2.0-85174735428https://api.elsevier.com/content/abstract/scopus_id/85174735428