Consider degenerations of Abelian differentials with prescribed number and multiplicity of zeros and poles. Motivated by the theory of limit linear series, we define twisted canonical divisors on pointed nodal curves to study degenerate differentials, give dimension bounds for their moduli spaces, and establish smoothability criteria. As applications, we show that the spin parity of holomorphic and meromorphic differentials extends to distinguish twisted canonical divisors in the locus of stable pointed curves of pseudocompact type. We also justify whether zeros and poles on general curves in a stratum of differentials can be Weierstrass points. Moreover, we classify twisted canonical divisors on curves with at most two nodes in the minimal stratum in genus three. Our techniques combine algebraic geometry and flat geometry. Their interplay is a main flavor of the paper.

We prove an extension of the nonabelian Hodge theorem [Sim92] in which the underlying objects are twisted torsors over a smooth complex projective variety. In the prototypical case of $GL_n$-torsors, one side of this correspondence consists of vector bundles equipped with an action of a sheaf of twisted differential operators in the sense of Beĭlinson and Bernstein [BB93]; on the other side, we endow them with appropriately defined twisted Higgs data.

The proof we present here is formal, in the sense that we do not delve into the analysis involved in the classical nonabelian Hodge correspondence. Instead, we use homotopy-theoretic methods—chief among them the theory of principal $\infty$-bundles [NSS12a]—to reduce our statement to classical, untwisted Hodge theory [Sim02].

We solve in the affirmative the remaining cases of the Riemann mapping problem of Cherrier–Escobar [35][38] first raised by Cherrier [35] in 1984. Indeed, performing a suitable scheme of the barycenter technique of Bahri–Coron [14] via the Almaraz–Chen’s bubbles [3][34], we completely solve all the cases left open after the work of Chen [34]. Hence, combining our work with the ones of Almaraz [2], Chen [34], Cherrier [35], Escobar [38][40] and Marques [55][56], we have that every compact Riemannian manifold with boundary, of dimension greater or equal than three, and with finite Sobolev quotient, carries a conformal scalar flat metric with constant mean curvature on the boundary.

We prove Hersch’s type isoperimetric inequality for the third positive eigenvalue on $\mathbb{S}^2$. Our method builds on the theory we developed to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.

We show that a compact Kähler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. The key ingredient is the recent solution of this conjecture in the projective case by Wu–Yau.