{ "id": "eppmdhr3m7", "slug": "fano-variety", "title": "Fano variety", "updated": "2026-10-15", "topic_path": [ { "id": "physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical" }, { "id": "physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math" }, { "id": "physical.math.number", "label": "Numbers and algebra", "api_url": "https://www.edgepedia.org/api/v1/topics/physical.math.number" } ], "geo": [ { "id": "geo.nongeo.nodate.physical.math.number", "label": "Non-geographic · Not dated: Numbers and algebra", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical.math.number", "path": [ { "id": "geo.nongeo", "label": "Non-geographic", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo" }, { "id": "geo.nongeo.nodate", "label": "Non-geographic · Not dated", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate" }, { "id": "geo.nongeo.nodate.physical", "label": "Physical world and mathematics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical" }, { "id": "geo.nongeo.nodate.physical.math", "label": "Mathematics and statistics", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical.math" }, { "id": "geo.nongeo.nodate.physical.math.number", "label": "Numbers and algebra", "api_url": "https://www.edgepedia.org/api/v1/geo/geo.nongeo.nodate.physical.math.number" } ] } ], "excerpt": "A Fano variety is a smooth projective algebraic variety whose anticanonical divisor is ample, placing it at the positive-curvature end of the classification of algebraic varieties.", "snippet": "A Fano variety is a smooth projective algebraic variety whose anticanonical divisor is ample, placing it at the positive-curvature end of the classification of algebraic varieties.", "node": "physical.math.number", "markdown": "# Fano variety\n\nA **Fano variety** is a smooth projective algebraic variety whose anticanonical divisor \\( -K_X \\) is ample, a positivity condition that makes it a higher-dimensional relative of the sphere and places it at the positive-curvature end of the classification of algebraic varieties[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf). The class is named after the Italian geometer [Gino Fano](https://www.edgepedia.org/gino-fano) (1871–1952), a student of [Corrado Segre](https://www.edgepedia.org/corrado-segre) who first studied varieties with this property[2](https://mathshistory.st-andrews.ac.uk//Biographies/Fano/). Fano varieties appear naturally in the birational classification of varieties of negative [Kodaira dimension](https://www.edgepedia.org/kodaira-dimension) and are very close to rational ones[3](https://link.springer.com/book/9783540614685).\n\n| Key fact | Detail |\n|---|---|\n| Definition | A smooth projective variety \\( X \\) whose anticanonical divisor \\( -K_X \\) is ample; equivalently, the first Chern class is represented by a positive definite form[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf) |\n| Curvature | Compact Kähler manifolds with positive definite Ricci curvature[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf) |\n| Dimension 1 | The only Fano curve is \\( \\mathbb{P}^1 \\)[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf) |\n| Dimension 2 | Fano surfaces are the del Pezzo surfaces, of degree \\( d = (-K_X)^2 \\) with \\( 1 \\le d \\le 9 \\)[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf) |\n| Dimension 3 | 105 deformation families of smooth Fano threefolds, 17 of them prime (Picard rank 1)[4](https://link.springer.com/article/10.1007/s13163-021-00401-2) |\n| Rational connectedness | Fano manifolds are rationally connected; \\( -K_X^n \\le (n(n+1))^n \\), and there are finitely many deformation families in each dimension[5](https://link.springer.com/article/10.1007/s10711-023-00882-z) |\n| Kähler–Einstein metrics | A Fano manifold admits a Kähler–Einstein metric if and only if it is K-polystable[6](https://epubs.siam.org/doi/10.1137/25M1804017) |\n\n## Definition and first examples\n\nThe definition is stated over an algebraically closed field: a smooth complete irreducible algebraic variety \\( X \\) whose anticanonical sheaf \\( K_X^{-1} \\) is ample[7](https://encyclopediaofmath.org/index.php?title=Fano_variety). In the complex setting, the anticanonical bundle is the determinant of the tangent bundle, and ampleness means the first [Chern class](https://www.edgepedia.org/chern-class) can be represented by a positive definite differential form[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf).\n\nBasic examples include all projective spaces \\( \\mathbb{P}^n \\), all smooth complete intersections in \\( \\mathbb{P}^n \\) of hypersurfaces of degrees \\( d_1, \\ldots, d_c \\) provided \\( d_1 + \\cdots + d_c \\le n \\), and all homogeneous projective varieties under a connected linear algebraic group, together with their linear sections of small enough degree[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf). A finite product of Fano varieties is again a Fano variety[8](https://citeseerx.ist.psu.edu/document?doi=55d80e22054936de15edab83c35833f268d6bf46&repid=rep1&type=pdf).\n\nTwo invariants organize the theory. The index is the greatest integer \\( r \\) such that \\( K_X \\equiv_{\\mathrm{num}} -rH \\) for some ample divisor \\( H \\), called a fundamental divisor; Kodaira vanishing then gives \\( H^i(X, sH) = 0 \\) for \\( s > -r \\), so dimensions of sections of multiples of \\( H \\) can be computed by Riemann–Roch[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf). The Picard number \\( \\rho(X) \\) equals the second [Betti number](https://www.edgepedia.org/betti-number) \\( b_2(X) \\) by Kodaira vanishing, and Fano varieties with \\( \\rho = 1 \\) are called prime[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf).\n\n## Low dimensions\n\n**Curves.** For a smooth projective curve of genus \\( g \\), the degree of \\( -K_X \\) is \\( 2 - 2g \\), so ampleness forces \\( g = 0 \\); conversely every genus 0 curve is \\( \\mathbb{P}^1 \\). Hence the only Fano curve is \\( \\mathbb{P}^1 \\)[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf).\n\n**Surfaces.** Fano surfaces are called del Pezzo surfaces. Writing \\( d = (-K_X)^2 \\) for the degree, Riemann–Roch gives \\( h^0(X, -K_X) = d + 1 \\), and the classical theorem states that \\( 1 \\le d \\le 9 \\) and that for \\( d \\ge 3 \\) the anticanonical linear system is very ample, embedding \\( X \\) in \\( \\mathbb{P}^d \\) as a surface of degree \\( d \\). In particular, del Pezzo surfaces of degree 3 are smooth cubic surfaces in \\( \\mathbb{P}^3 \\), and those of degree 4 are complete intersections of two quadrics in \\( \\mathbb{P}^4 \\). Any del Pezzo surface with \\( \\rho = 1 \\) is isomorphic to \\( \\mathbb{P}^2 \\)[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf).\n\n**Threefolds.** For a Fano threefold of degree \\( d = (-K_X)^3 \\), one has \\( h^0(X, -K_X) = \\tfrac{1}{2}d + 3 \\), and the genus \\( g := \\tfrac{1}{2}d + 1 \\) is the genus of any smooth curve obtained as the complete intersection of two elements of \\( \\lvert -K_X \\rvert \\)[1](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf). The classification of smooth Fano threefolds was completed in stages: Iskovskikh classified those with Picard number one, using the birational double projection technique, and Mori and Mukai finished the classification for higher Picard number[4](https://link.springer.com/article/10.1007/s13163-021-00401-2). The result is 105 deformation families, of which 17 are prime; Mukai reworked the prime classification biregularly using vector bundles, describing most prime Fano varieties as complete intersections in homogeneous or quasi-homogeneous varieties, while the Mori–Mukai treatment of the 88 non-prime families relied on the theory of extremal rays[4](https://link.springer.com/article/10.1007/s13163-021-00401-2). Iskovskikh's approach combined techniques introduced by Grothendieck with Fano's original ideas, taking the index as the main classifying invariant[5](https://link.springer.com/article/10.1007/s10711-023-00882-z).\n\n## Rational curves and boundedness\n\nThe existence of rational curves on Fano varieties is one of the cornerstones of the [minimal model program](https://www.edgepedia.org/minimal-model-program). Kollár, Miyaoka, and Mori proved that two general points on an \\( n \\)-dimensional Fano manifold can be connected by a rational curve \\( C \\) with \\( -K_X \\cdot C \\le n(n+1) \\); that is, Fano manifolds are rationally connected[5](https://link.springer.com/article/10.1007/s10711-023-00882-z). As a consequence they established the inequality \\( -K_X^n \\le (n(n+1))^n \\) and the finiteness of deformation families of \\( n \\)-dimensional Fano manifolds[5](https://link.springer.com/article/10.1007/s10711-023-00882-z). Every Fano variety is in fact uniruled, covered by rational curves \\( C \\) with \\( -C \\cdot K_X \\le 2 \\dim X \\)[9](https://archive.intlpress.com/site/pub/files/_fulltext/journals/sdg/1990/0001/0001/SDG-1990-0001-0001-a003.pdf).\n\n**Singular Fanos and the BAB theorem.** The theory extends to singular varieties once the allowed singularities are controlled. The Fields Medalist Caucher Birkar established the Borisov–Alexeev–Borisov conjecture in 2018: fixing a natural number \\( d \\) and a positive real number \\( \\epsilon \\), the Fano varieties of dimension \\( d \\) having \\( \\epsilon \\)-log canonical singularities make up a bounded family[10](https://arxiv.org/abs/1609.05543). This boundedness implies that birational automorphism groups of rationally connected varieties are Jordan, answering a question of Serre[10](https://arxiv.org/abs/1609.05543). Thanks to this work, there are only finitely many families of Fano varieties of each dimension[11](https://www.lboro.ac.uk/departments/maths/research/research-spotlights/birational-maps-between-varieties/).\n\n## K-stability and Kähler–Einstein metrics\n\nThe Calabi problem asks whether a Fano manifold admits a Kähler–Einstein metric. Building on work of Matsushima, Futaki, and Mabuchi, Yau, Tian, and Donaldson produced an algebraic stability condition, [K-stability](https://www.edgepedia.org/k-stability) (more precisely K-polystability), conjectured to be equivalent to the existence of such metrics; this Yau–Tian–Donaldson conjecture was proved affirmatively about a decade before 2026[6](https://epubs.siam.org/doi/10.1137/25M1804017).\n\n> A Fano manifold admits a Kähler–Einstein metric if and only if it is K-polystable.[6](https://epubs.siam.org/doi/10.1137/25M1804017)\n\nBecause the definition of K-polystability is purely algebro-geometric, the Calabi problem becomes a problem of algebraic geometry, and the condition extends naturally to singular Fano varieties; Odaka showed that a singular K-semistable Fano variety has only klt (Kawamata log terminal) singularities[6](https://epubs.siam.org/doi/10.1137/25M1804017). K-stability has also become the basis for constructing moduli spaces: smooth K-polystable Fano varieties of fixed dimension and volume are parametrized by a quasi-projective moduli space[12](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70259), and explicit K-moduli spaces have been computed for individual threefold families[13](https://arxiv.org/abs/2510.13611).\n\nFor the threefolds, the Calabi problem has been worked through family by family: for each of the 105 irreducible families of smooth Fano threefolds, it is known whether the general member admits a Kähler–Einstein metric[14](https://research.manchester.ac.uk/en/publications/the-calabi-problem-for-fano-threefolds).\n\n## Toric Fano varieties and computation\n\nA toric variety \\( X \\) is Fano when \\( -K_X \\) is ample, with no smoothness condition required[5](https://link.springer.com/article/10.1007/s10711-023-00882-z). Smooth toric Fano varieties correspond to smooth Fano polytopes, lattice polytopes that can be classified efficiently using an algorithm of Øbro. The numbers rise gradually as dimension increases: among the 105 Fano manifolds in three dimensions, 18 are toric, and by the time one reaches dimension eight the count of smooth Fano polytopes stands at just 749,892.[15](https://kasprzyk.work/research/pdf/dialogues.pdf) Reflexive Fano polytopes, corresponding to Gorenstein toric Fano varieties, are important in mirror symmetry, where topologically mirror-symmetric pairs of Calabi–Yau varieties arise as hypersurfaces in them; there are 4,319 reflexive polytopes in dimension three and 473,800,776 in dimension four[15](https://kasprzyk.work/research/pdf/dialogues.pdf).\n\n## Arithmetic of rational points\n\nFano varieties are the class where rational points are expected to be plentiful. Manin conjectured that for a smooth Fano variety with a dense set of rational points, the number of points of anticanonical height bounded above by \\( B \\) grows like \\( C B (\\log B)^{r-1} \\) as \\( B \\to +\\infty \\), where \\( C \\) is a positive constant and \\( r \\) is the rank of the [Picard group](https://www.edgepedia.org/picard-group)[16](https://webspace.science.uu.nl/~piero001/index_research.html). This asymptotic has been verified for some families of varieties and many single cases, but the conjecture remains widely open; even the case of smooth Fano surfaces, the del Pezzo surfaces, is far out of reach[16](https://webspace.science.uu.nl/~piero001/index_research.html) • [17](https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/plms.70203). One standard method of attack is the universal torsor method, which parameterizes points via Cox rings and then counts by lattice-point techniques[16](https://webspace.science.uu.nl/~piero001/index_research.html).\n\n## References\n\n1. [Olivier Debarre, \"How to classify Fano varieties?\" (2013 lecture notes)](https://www.math.ens.psl.eu/~debarre/ExposePoitiers2013.pdf)\n2. [Gino Fano (1871–1952), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk//Biographies/Fano/)\n3. [V. A. Iskovskikh and Yu. G. Prokhorov, *Algebraic Geometry V: Fano Varieties*, Springer (1999)](https://link.springer.com/book/9783540614685)\n4. [L. De Biase, E. Fatighenti, F. Tanturri, \"Fano 3-folds from homogeneous vector bundles over Grassmannians\", Rev. Mat. Complut. 35 (2022)](https://link.springer.com/article/10.1007/s13163-021-00401-2)\n5. [\"Coregularity of Fano varieties\", Geometriae Dedicata (2023)](https://link.springer.com/article/10.1007/s10711-023-00882-z)\n6. [\"Toward Criteria for the K-Stability of Fano Manifolds\", ICM 2026 Proceedings, Vol. 3](https://epubs.siam.org/doi/10.1137/25M1804017)\n7. [\"Fano variety\", Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=Fano_variety)\n8. [Olivier Debarre, \"Introduction to Mori theory\" (lecture notes)](https://citeseerx.ist.psu.edu/document?doi=55d80e22054936de15edab83c35833f268d6bf46&repid=rep1&type=pdf)\n9. [János Kollár, \"Flips and abundance\" (1990), in Several Complex Geometry](https://archive.intlpress.com/site/pub/files/_fulltext/journals/sdg/1990/0001/0001/SDG-1990-0001-0001-a003.pdf)\n10. [Caucher Birkar, \"Singularities of linear systems and boundedness of Fano varieties\", Ann. of Math.](https://arxiv.org/abs/1609.05543)\n11. [\"Birational maps between varieties\", Loughborough University research spotlight](https://www.lboro.ac.uk/departments/maths/research/research-spotlights/birational-maps-between-varieties/)\n12. [\"K-stable Fano threefolds of rank 2 and degree 28\", J. London Math. Soc. (2025)](https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms.70259)\n13. [E. Etxabarri-Alberdi, J. M. Jones, T. S. Papazachariou, \"K-Moduli of Fano Threefolds of Family 3.3\", arXiv:2510.13611](https://arxiv.org/abs/2510.13611)\n14. [C. Araujo et al., \"The Calabi problem for Fano threefolds\", MPIM Preprint 2021-31](https://research.manchester.ac.uk/en/publications/the-calabi-problem-for-fano-threefolds)\n15. [G. Brown, T. Coates, A. Corti, T. Ducat, L. Heuberger, A. Kasprzyk, \"Computation and data in the classification of Fano varieties\"](https://kasprzyk.work/research/pdf/dialogues.pdf)\n16. [Marta Pieropan, research overview: rational points on Fano varieties](https://webspace.science.uu.nl/~piero001/index_research.html)\n17. [\"Manin's conjecture for integral points on toric varieties\", Proc. London Math. Soc. (2026)](https://londmathsoc.onlinelibrary.wiley.com/doi/full/10.1112/plms.70203)\n\n---\n*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra*\n\n*Initially written Oct 15, 2026*\n\n*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*\n\nLicense: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license\n", "same_as": [], "url": "https://www.edgepedia.org/fano-variety", "markdown_url": "https://www.edgepedia.org/fano-variety.md", "license": { "name": "Edgepedia Community License 1.0", "url": "https://www.edgepedia.org/edgepedia/license", "summary": "Free with credit, commercial use included. AI training is open to everyone. For other uses, organizations over USD 100M in revenue or 100M monthly users license separately.", "spdx": "LicenseRef-Edgepedia-Community-1.0" }, "credit": "\"Fano variety\", Edgepedia (EdgeChat), https://www.edgepedia.org/fano-variety. Edgepedia Community License 1.0.", "credit_md": "\"[Fano variety](https://www.edgepedia.org/fano-variety)\", Edgepedia (EdgeChat), [https://www.edgepedia.org/fano-variety](https://www.edgepedia.org/fano-variety). [Edgepedia Community License 1.0](https://www.edgepedia.org/edgepedia/license).", "credit_html": "\"Fano variety\", Edgepedia (EdgeChat), https://www.edgepedia.org/fano-variety. Edgepedia Community License 1.0.", "speakable": "A Fano variety is a smooth projective algebraic variety whose anticanonical divisor is ample, placing it at the positive-curvature end of the classification of algebraic varieties." }