Making tight contact manifolds non-fillable

For any $n\geq 2$, we prove that the $(2n+1)$-dimensional sphere admits a tight non-fillable contact structure that is homotopically standard. By taking connected sums, we deduce that any $(2n+1)$-dimensional manifold that admits a strongly fillable contact structure with torsion first Chern class also admits a tight but not strongly fillable contact structure in the same almost contact class. Lastly, we also obtain infinitely many such structures on Weinstein fillable contact manifolds of dimension at least $11$.Comment: 32 pages, 1 figure. Comments are welcome