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### What are the main contributions to the mathematics of general relativity by Sir Roger Penrose, winner of the 2020 Nobel prize?

It seems (as mentioned by Sam Hopkins above) that the Singularity Theorem is the official reason for the Nobel Award. But that is by no means the only (and perhaps not even the most important) ...
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### Algebraic vs. homological equivalence for curves on a smooth complex projective surface

Super vast generalisation: for divisors on a smooth projective variety over an algebraically closed field of any characteristic, the notions of algebraic, homological (for any Weil cohomology theory), ...

I think you have in mind the integrability (a.k.a. "flatness") problem for $G$-structures. Beyond the cases mentioned at that link (symplectic, Kähler, and complex structures, corresponding to $G=Sp(n,... • 28.3k 11 votes ### Are most Kähler manifolds non-projective? I agree that in a sense it's harder to get your hands a non algebraic Kähler manifold, because you can't simply write an equation for one, but I would argue that there are plenty of them. You won't ... • 31.9k 11 votes Accepted ### Is there a Kähler manifold with no anti-holomophic involution? The moduli space$\mathcal{M}_1^\mathbb{R}$of real algebraic curves of genus$1$equals the real part of the moduli space$\mathcal{M}_1=\mathbb{C}$. In particular, the general elliptic curve has no ... • 62.2k 10 votes ### Why can we not always take a Kähler class to be in rational cohomology? Since Artie Prendergast-Smith is not expanding his comment in an answer, let me do it. As I said in the comments, his comment is essentially THE answer to the OP question. But let me give some more ... • 7,672 10 votes ### List of Applications of the$\partial\overline{\partial}$-lemma This is a list of length one :) The$\partial \bar \partial$-Lemma allows a parameterization of the cohomology class$[\omega]$of a compact Kaehler manifold$(M, \omega)$by means of scalar ... • 590 10 votes Accepted ### The logarithm of Kähler metric is not globally defined Question 1: Note,$\omega + \partial\bar{\partial}\phi$cannot be zero. Recall that$\phi$is chosen so that$\omega + \partial\bar{\partial}\phi$is another metric (in particular, a Kähler-Einstein ... • 17.3k 10 votes Accepted ### Atiyah-Singer for Riemannian and Kaehler manifolds I highly recommend the discussion in Shanahan's book, The Atiyah-Singer Index Theorem (An introduction), Lecture Notes in Math 638. In addition to a sketch of the proof, he gives a nice discussion of ... • 17.4k 10 votes Accepted ### The period map and the Kodaira--Spencer map Differential of period map$d P^{p+q,p}$is composition of KS-map$T_{B,0} \to H^1(X_0,T_{X_0})$with natural map$H^1(X_0,T_{X_0}) \to Hom(H^{p,q}(X_0),H^{p-1,q+1}(X_0))\$ (given by the cup product ...

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