Let $K$ be a compact group. For symplectic quotient $M_{\lambda}$ of Hamiltonian K\"ahler $K$-manifold, we show that the induced complex structure on is locally invariant when parameter $\lambda$ varies in $\mathrm{Lie}(K)^*$. To prove such result, take two different approaches: (i) by using geometry properties implosion construction; (ii) investigating variation GIT quotients.

The tadpole conjecture by Bena, Blaback, Grana and Lust effectively states that for string-theory compactifications with a large number of complex-structure moduli, not all these moduli can be stabilized fluxes. In this note we study in the regime using statistical data obtained Demirtas, Long, McAllister Stillman Kreuzer-Skarke list. We estimate lower bound on flux type IIB Calabi-Yau orientif...