Asymptotically log Fano varieties
The authors introduce asymptotically log Fano pairs, a generalisation of varieties of Fano type. Let \(X\) be a projective variety such that \(-K_X\) is \(\mathbb Q\)-Cartier, and let \(D\) be a divisor of the form \(D= \sum_{i=1}^r D_i\), where the \(D_i\)s are distinct prime \(\mathbb Q\)-Cartier divisors. The pair \((X, D)\) is (strongly) asymptotically log Fano if for (all) \((\beta_1,\dots, \beta_r)\in (0, 1]^r\) sufficiently small, \((X, \sum (1-\beta_i)D_i)\) has mild (i.e.~klt) singularities and \(-(K_X+\sum (1-\beta_i)D_i)\) is ample. The main result of the article is a complete classification of strongly asymptotically log del Pezzo surfaces (i.e.~\(2\)-dimensional strongly asymptotically log Fano pairs), which is obtained by classical methods in the birational geometry of surfaces. The authors then reorganise their classification by dividing strongly asymptotically log del Pezzo surfaces \((S, C)\) into four families corresponding to the possible positivity properties of \(-(K_S+C)\) (that is: \(-(K_S+C)\sim 0\), \(-(K_S+ C)\not \sim 0\) nef but not big, \(-(K_S+ C)\) nef and big and \(-(K_S+ C)\) ample). The authors formulate an asymptotic logarithmic version of Calabi's conjecture based on this classification. They conjecture that if \((S, C)\) is an asymptotically log del Pezzo surface with \(C\) smooth and irreducible, then \(S\) admits Kähler-Einstein edge (KEE) metrics with angle \(\beta\) along \(C\) for all sufficiently small \(\beta\in (0;1]\) if and only if \(-(K_S+C)\) is not big. After stating some motivation for the conjecture, the authors prove a few cases by direct methods. Some non-existence results are obtained by generalising a criterion of \textit{Y. Matsushima} [Nagoya Math. J. 11, 145--150 (1957; Zbl 0091.34803)] that implies that the automorphism groups of asymptotically log del Pezzo surfaces with KEE metrics are reductive while some existence results are consequences of results of \textit{T. Jeffres} et al. [Ann. Math. (2) 183, No. 1, 95--176 (2016; Zbl 1337.32037)]. Finally, the authors make a conjecture on the existence of KEE metrics on strongly asymptotically log Fano pairs in arbitrary dimension.
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