Geodesic
Invariants of the smooth structure of an algebraic surface arising from the Dirac operator
Izvestiya. Mathematics , Tome 40 (1993) no. 2, pp. 267-351

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We construct invariants of the smooth structure of an algebraic surface in terms of coupled Dirac operators. The invariants allow us to distinguish between del Pezzo surfaces and fake del Pezzo surfaces by their smooth structure. 
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     author = {V. Ya. Pidstrigach and A. N. Tyurin},
     title = {Invariants of the smooth structure  of an algebraic surface arising from the {Dirac} operator},
     journal = {Izvestiya. Mathematics },
     pages = {267--351},
     publisher = {mathdoc},
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     number = {2},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1993_40_2_a1/}
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V. Ya. Pidstrigach; A. N. Tyurin. Invariants of the smooth structure  of an algebraic surface arising from the Dirac operator. Izvestiya. Mathematics , Tome 40 (1993) no. 2, pp. 267-351. http://geodesic.mathdoc.fr/item/IM2_1993_40_2_a1/