Algebraic geometry, as seen by Andrei Tyurin 1. Category of Algebraic geometry:
1.1. Objects:
1.1.1. Affine schemes. 1.1.1.1. Local rings. 1.1.1.2. Resolutions [of modules]. 1.1.1.3. Local homological algebra. 1.1.1.4. Local structure of coherent sheaves. 1.1.1.5. Structure of Artinian sheaves. 1.1.1.6. Local structure of orbifolds. 1.1.1.7. Local rings of super-varieties. 1.1.1.8. Local jet-structure. 1.1.1.9. Local formal schemes. 1.1.1.10. Local semi-algebraic sets. 1.1.1.11. Nilpotent elements. 1.1.2. Singularities. 1.1.2.1. Du Val singularities. 1.1.2.2. Local invariants of a singularity. 1.1.2.3. Canonical class of a singularity. 1.1.2.4. Special resolutions of singularities. 1.1.2.5. Resolution of isolated singularities. 1.1.2.6. General theorem on local desingularisations. 1.1.2.7. Local singularities of curves. 1.1.2.8. Local singularities of surfaces. 1.1.2.9. Local singularities of threefolds. 1.1.2.10. Tangent cone constructions. 1.1.2.11. The tangent cone algorithm. 1.1.2.12. Topology of singularities. 1.1.2.13. Local dualities. 1.1.2.14. Contractions and embeddings. 1.1.2.15. Singularities of super-varieties. 1.1.3. Global objects. 1.1.3.1. Abstract algebraic varieties. 1.1.3.2. Projective algebraic varieties. 1.1.3.3. Orbifolds. 1.1.3.4. Super-varieties. 1.1.3.5. Stacks. 1.1.3.6. Moishezon spaces. 1.1.3.7. Schemes. 1.1.3.8. Semi-algebraic sets. 1.1.3.9. Fields. 1.1.4. Can-classification of algebraic varieties. 1.1.4.1. Canonical class. 1.1.4.2. Canonical ring. 1.1.4.3. Kodaira dimension. 1.1.4.4. Fano variety. 1.1.4.5. Complex orientable varieties. 1.1.4.6. Varieties of general type. 1.1.4.6. Fibrations. 1.1.4.7. Relative version of can-classification.
1.2. Special varieties.
1.2.1. Curves. 1.2.1.1. Plane curves. 1.2.1.2. Coverings of projective line. 1.2.1.3. Complete intersections. 1.2.1.4. Sections of Grassmannians. 1.2.1.5. Hyperelliptic curves. 1.2.2 K3 surfaces. 1.2.2.1. Quartics in projective spaces. 1.2.2.2. Double covers of projective plane. 1.2.2.3. Complete intersections. 1.2.2.4. Sections of Grassmannians
1.2.2.5. Extensions of canonical curves. 1.2.3. Abelian varieties. 1.2.3.1. Plane cubics. 1.2.3.2. Commutative groups laws. 1.2.3.3. Polarizations. 1.2.4. Toric varieties. 1.2.4.1. Projective spaces. 1.2.4.2. Weighted projective spaces. 1.2.4.3. Fans. 1.2.5. Complete intersections in weighted projective spaces. 1.2.5.1. Kodaira type of complete intersections. 1.2.5.2. Fano varieties. 1.2.5.3. Complex oriented varieties.
1.3 Morphisms.
1.3.1. Local structure of algebraic morphisms. 1.3.1.1. Differential of a morphism. 1.3.1.2. Different of a morphism. 1.3.1.3. Local Galois group. 1.3.1.4. Local contraction. 1.3.1.5. Local decomposition. 1.3.1.5. Etale covering. 1.3.1.6. Local blow up. 1.3.1.7. Basic sets. 1.3.1.8. Rational maps. 1.3.1.9. Lefshetz pencils. 1.3.2. Rational maps. 1.3.2.1. Exceptional divisor. 1.3.2.2. Contractions. 1.3.2.3. Basic sets. 1.3.2.4. Coverings. 1.3.2.5. Branch locus. 1.3.2.6. Ramification locus. 1.3.2.7. Hurwitz formula.
1.4. Sub-objects.
1.4.1. Cycles and subschemes. 1.4.1.1. Subschemes. 1.4.1.2. Formal geometry along subscheme. 1.4.1.3. Normal bundles and normal cones. 1.4.1.4. Ideal of a subscheme. 1.4.1.5. Resolutions of ideal sheaves. 1.4.2. Infinitesimal deformations. 1.4.2.1. Formal deformations. 1.4.2.2. Sections of the normal bundle. 1.4.2.3. Obstructions. 1.4.2.4. Kuranishi family. 1.4.2.5. Homological interpretation. 1.4.3. Global deformations. 1.4.3.1. Chow schemes. 1.4.3.2. Hilbert schemes. 1.4.3.3. Linear systems of divisors. 1.4.3.4. 0-cycles. 1.4.4. Equivalence relations. 1.4.4.1. Algebraic equivalence. 1.4.4.2. Lefshetz conjecture. 1.4.4.3. Griffiths's theorem. 1.4.4.4. Rational equivalence. 1.4.4.5. Abel-Jacobi map. 1.4.4.6. rational equivalence of 0-cycles on a
surface.
1.4.4.7. Mumford construction and generalizations.
1.5. Underlying structures. Topology.
1.5.1. Topological invariants. 1.5.1.1. Betti numbers. 1.5.1.2. Euler characteristics. 1.5.1.3. Intersection theory. 1.5.1.4. Signatures. 1.5.1.5. A-genus. 1.5.1.6. Elliptic genus. 1.5.2. Topological structure. 1.5.2.1. Fundamental group. 1.5.2.2. Cell decomposition. 1.5.2.3. Hard Lefshetz theorem. 1.5.2.4. Cohomological operations. 1.5.2.5. Connected sum decompositions. 1.5.3. Topological structure of algebraic curves. 1.5.3.1. Fundamental groups. 1.5.3.2. Homology and cohomology groups. 1.5.3.3. Intersection theory and the symplectic
structure. 1.5.3.4. Cell decomposition. 1.5.3.5. Connected sum decomposition. 1.5.4. Differential topology of algebraic surfaces. 1.5.4.1. Topological invariants of an algebraic surfaces. 1.5.4.2. Elementary surfaces. 1.5.4.3. Topological model of a simple connected algebraic surface. 1.5.4.4. Connected sum decompositions. 1.5.4.5. Deformation type and differential type. 1.5.4.6. Donaldson's un-decomposability theorem. 1.5.5. Homotopy type of Calabi-Yao threefolds. 1.5.5.1. Intersections cubic forms. 1.5.5.2. C.T.C.Wall's decomposition theorem. 1.5.5.3. Euler characteristics. 1.5.5.4. Tables of experiment data. 1.5.5.5. Topological mirror symmetry.
1.6. Underlying structures. Complex analysis.
1.6.1. Transcendental methods. 1.6.1.1. Hodge structures. 1.6.1.2. Kaehler cone. 1.6.1.3. Kodaira theorem. 1.6.1.4. Hard Lefshetz theorem for Kaehler manifolds. 1.6.1.5. Embeddings. 1.6.2. Periods. 1.6.2.1. Period domains. 1.6.2.2. Periods map. 1.6.2.3. Periods of algebraic curves. 1.6.2.4. Periods of algebraic surfaces. 1.6.2.5. Periods of Fano varieties. 1.6.3. Hodge conjecture. 1.6.3.1. Hodge theorem for divisors. 1.6.3.2. Griffiths's approach to Hodge conjecture. 1.6.3.3. Partial cases of Hodge conjecture. 1.6.3.4. Cubic manifolds. 1.6.3.5. Shafarevich conjecture. 1.6.3.6. Mukai approach to Hodge conjecture. 1.6.3.7. Tankeev results on Hodge conjecture for abelian varieties. 2.Motivic category.
2.1. Algebraic correspondences. 2.1.1. Classical correspondence theory. 2.1.1.1. Numerical invariants of correspondences. 2.1.1.2. Homological invariants of correspondences. 2.1.1.3. Mixed Hodge structures. 2.1.1.3. Mixed Hodge structures under correspondences. 2.1.2. Arithmetical correspondence theory.
2.2. Motivic cohomology. 3. Coherent sheaves.
3.1. Local structure. 3.1.1. Local homological algebra. 3.1.1.1. Local resolutions of coherent sheaves. 3.1.1.2. Local resolution for singular points. 3.1.1.3. Torsions. 3.1.1.4. Reflexive hulls. 3.1.1.5. Homological dimension. 3.1.1.6. Local cohomology. 3.1.2. Invertible sheaves. 3.1.2.1. Sections and linear systems. 3.1.2.2. Higher cohomology. 3.1.2.3. Ampleness. 3.1.2.4. Basic points. 3.1.2.5. Maps of linear systems. 3.1.2.6. Vanishing theorems. 3.1.3. Low dimensional cases. 3.1.3.1. Sheaves on curves. 3.1.3.2. Sheaves on surfaces. 3.1.3.3. Sheaves on threefolds. 3.1.4. Vector bundles. 3.1.4.1. Maps to Grassmannians. 3.1.4.2. Stability conditions. 3.1.4.3. Elementary transformations. 3.1.5. Constructions. 3.1.5.1. Local extensions. 3.1.5.2. Global extensions. 3.1.5.3. Long cohomology sequences. 3.1.5.4. Tables. 3.1.5.5. Double-cover constructions.
3.2. Deformations of sheaves. 3.2.1.Infinitesimal deformations. 3.2.1.1. Artamkin - Mukai deformation theory. 3.2.1.2. Geometrical interpretation of infinitesimal deformations. 3.2.1.3. Tangent spaces to spaces of deformations. 3.2.2. Moduli spaces of vector bundles and coherent sheaves. 3.2.2.1. Stability conditions. 3.2.2.2. Constructions of global moduli spaces. 3.2.2.3. Structures on moduli spaces. 3.2.2.4. Universal sheaves. 3.2.2.5. Canonical classes of moduli spaces. 3.2.2.6. Singularity of moduli spaces. 3.2.2.7. Mukai transformations. 3.2.2.8. Geometrical Hecke correspondences. 3.2.3. Moduli spaces of pairs 3.2.3.1. Stability conditions. 3.2.3.2. G_m- actions and polarizations. 3.2.3.3. Geometry of stability conditions. 3.2.3.4. Flips. 3.2.3.5. Chains of flips. 3.2.4. Compactifications of moduli spaces. 3.2.4.1. Gieseker compactifications. 3.2.4.2. Compactifications of moduli spaces of pairs. 3.2.4.3. Local structure of compactifications. 3.2.4.4. Geometrical description of boundary points.
3.3. Transcendental methods in vector bundles theory.
3.3.1. Gauge theory. 3.3.1.1. Structure group of a vector bundle. 3.3.1.2. Hermitian structures. 3.3.1.3. Connections. 3.3.1.4. Gauge groups. 3.3.1.5. Orbit spaces. 3.3.1.6. Cohomology of orbit spaces. 3.3.2. Gauge groups reductions. 3.3.2.1. Symplectic reduction. 3.3.2.2. Moment maps for gauge groups. 3.3.2.3. Kaehler reductions. 3.3.2.4. Hyper-Kaehler reductions. 3.3.2.5. Constructions of moduli spaces.
3.4. Low dimensional base cases.
3.4.1. Moduli spaces of vector bundles on a curve. 3.4.1.1. Jacobians of curves. 3.4.1.2. Classical Brill-Nether theory. 3.4.1.3. Theta-function theory. 3.4.1.4. Stability conditions. 3.4.1.5. Canonical classes. 3.4.1.6. Theta-divisor. 3.4.1.7. Spaces of conformal blocks. 3.4.1.8. Verlinde formula. 3.4.1.9. Fusion rules. 3.4.1.10. Factorization constructions. 3.4.1.11. Oxbury constructions. 3.4.2. Moduli spaces of stable pairs on a curve. 3.4.2.1. Extensions constructions. 3.4.2.2. Secants constructions. 3.4.2.3. Flips. 3.4.2.4. Linear systems of flips. 3.4.2.5. Brill - Noether theory for non linear bundles. 3.4.2.6. Mukai constructions. 3.4.3. Moduli spaces of vector bundles on a surface. 3.4.3.1. Existence theorems. 3.4.3.1. Canonical classes. 3.4.3.2. Compactifications. 3.4.3.3. Polarizations. 3.4.3.4. Wall crossing. 3.4.3.5. Ideal sheaves. 3.4.3.6. Extension constructions. 3.4.3.7. Dividing constructions. 3.4.3.8. Uhlenbeck- Donaldson equivalence theorem. 3.4.3.9. Compactification comparing theorem. 3.4.3.10. Scheme theoretical description of Uhlenbeck compactifications. 3.4.4. Moduli spaces of stable pairs on a surface. 3.4.4.1. Graduation of stable pairs. 3.4.4.2. Torsion pairs and elementary transformations. 3.4.4.3. Bogomolov stability theorem. 3.4.5. Moduli of vector bundles on threefolds. 3.4.5.1. Instantons on 3.4.5.2. Monads and resolutions of instantons. 3.4.5.3. Matrix descriptions of instantons problem. 3.4.5.4. Scandinavian nets of quadrics. 3.4.5.5. Degenerations of instantons. 3.4.5.6. Vector bundles on complex orientable threefolds. 3.4.6. Moduli spaces of vector bundles on hyperkaehler manifolds. 3.4.6.1. Weyl - Peterson metrics. 3.4.6.2. Hyperkaehler structures on moduli spaces. 3.4.6.3. Extension of Weil - Peterson metrics. 3.4.6.4. Twistor spaces of moduli spaces. 3.4.6.5. Special deformations of complex structures. 3.4.7. Vector bundles on higher dimensional projective spaces. 3.4.7.1.The Hartshorne conjecture. 3.4.7.2. The Mumford - Horrocks bundle. 3.4.7.3. Barth construction of MH-bundle. 3.4.7.4. Pencils of special linear complexes. 3.4.7.5. Deformations of MH-bundles. 3.4.7.6. Geometry of jumping lines of MH-bundles. 3.4.7.7. Symmetries of MH-bundles. 3.4.7.8. Commesatti surfaces and Serre constructions. 3.4.8. Higgs bundles. 3.4.8.1. Flat connections. 3.4.8.2. Hitchin constructions. 3.4.8.3. Representations of fundamental groups. 3.4.8.4. Simpson theory. 3.4.8.5. Holomorphic Hamiltonian systems. 3.4.8.6. Partial compactifications of cotangent bundles to moduli spaces of bundles on a curve. 3.4.8.7. Realizations of Teichmuller spaces. 3.4.8.8. Hitchin spaces. 3.4.9. Homogeneous bundles. 3.4.9.1. Cohomology of homogeneous vector bundles and representations theory. 3.4.9.2. Bott's theorem. 3.4.9.3. Vanishing theorems.
3.5. Discrete invariants of sheaves.
3.5.1. Mukai lattice. 3.5.1.1. Characteristic classes of sheaves. 3.5.1.2. Chern characters of sheaves. 3.5.1.3. Todd polynomials. 3.5.1.4. Riemann-Roch theorems. 3.5.1.5. Mukai vectors. 3.5.1.6. Ext - bilinear forms. 3.5.1.7. Elementary reflections. 3.5.2. Modular operations 3.5.2.1. Exceptional bundles. 3.5.2.2. Universal dividing. 3.5.2.3. Universal extension. 3.5.2.4. Mestrano operations. 3.5.2.5. Mukai lattice actions. 3.5.2.6. Arithmetical orbits problem. 3.5.3. Basises in the Mukai lattice. 3.5.3.1. Triangle basis. 3.5.3.2. Transformations of triangle basis. 3.5.3.3. Orbits of transformations. 3.5.3.4. Arithmetical questions. 3.5.4. Helices theory. 3.5.4.1. Resolution of a diagonal. 3.5.4.2. Del Pezzo surfaces. 3.5.4.3. Helices. 3.5.4.4. Braid groups actions. 3.5.4.5. Helices on projective spaces. 3.5.4.6. Markov equations.3 3.5.4.7. Exceptional multiplets (Karpov-Nogin theory). 3.5.4.8. Dracos on K3 surfaces. 3.5.5. Derived categories. 3.5.5.1. Serre functor. 3.5.5.2. Canonical classes. 3.5.5.3. Orlov - Bondal structure theorems. 3.5.5.4. General mirror symmetry.
4. Moduli spaces of algebraic varieties.
4.1. Local formations.
4.1.1. Kodaira- Spencer theory. 4.1.1.1. Infinitesimal deformations. 4.1.1.2. Kodaira-Spencer maps. 4.1.1.3. Kuranishi families. 4.1.1.4. Infinitesimal rigid objects. 4.1.2. Obstructions theory. 4.1.2.1. Obstructions of deformations 4.1.2.2. Unobstructed deformations. 4.1.2.3. Local deformations of complex orientable variety. 4.1.2.4. Local singularities of deformations. 4.1.2.5. Local deformations of singularities. 4.1.2.6. Special geometries. 4.1.2.7. Lowson-Harvey deformations. 4.1.3. Local deformations of pairs and flags. 4.1.3.1. Infinitesimal deformations of pairs. 4.1.3.2. Formal tangent space to deformations of flags. 4.1.4. Local deformations of maps. 4.1.4.1. Local deformations sources and targets. 4.1.4.2. Local deformations of maps of curves. 4.1.4.3. Local deformations of maps of rational curves. 4.1.4.4. Relations to symplectic geometry.
4.2. Global moduli spaces.
4.2.1. Global rigid objects. 4.2.1.1. Projective spaces. 4.2.1.2. Quadrics. 4.2.1.3. Homogeneous varieties. 4.2.1.4. Local homogeneous varieties. 4.2.1.5. Calabi-Visantiti principle. 4.2.2. Stability conditions. 4.2.2.1. Geometrical invariant theory. 4.2.2.2. Stability conditions. 4.2.2.3. Stability conditions for curves. 4.2.2.4. Stability conditions for surfaces. 4.2.2.5. Non-stable objects. 4.2.3. Moduli stacks. 4.2.3.1. Gluing problems for local moduli spaces. 4.2.3.2. Stacks conditions. 4.2.4. Versal and universal families. 4.2.4.1. Versal deformations of local rigid objects. 4.2.4.2. Versal deformations of quadrics. 4.2.4.3. Universal family.
4.3. Deformations of pairs and flags.
4.3.1. Marked objects. 4.3.1.1. Marked curves. 4.3.1.2. Marked surfaces. 4.3.1.3. Marking by singularities. 4.3.2. Global deformations of pairs. 4.3.3. Global deformations of maps. 4.3.3.1. Moduli spaces of maps of curves. 4.3.3.2. Quantum multiplications of cohomology classes. 4.3.3.3. Rigidity conditions for maps. 4.3.3.4. Maps of curves to complex oriented varieties. 4.3.3.5. Generating functions of maps.
4.4. Compactifications problems. 4.4.1. Mumford - Deligne compactification of moduli space of curves. 4.4.1.1. Stable reduction theorem. 4.4.1.2. Stable configurations of curves. 4.4.1.3. Orbifold structures. 4.4.1.4. Manin - Kontzevich compactifications of spaces of
curves maps.
4.5. Low dimensional cases.
4.5.1. Moduli spaces of curves. 4.5.1.1. Chow rings of moduli spaces of curves. 4.5.1.2. Theta-constants embeddings. 4.5.1.3. Stack structure. 4.5.1.4. Genus-towers of moduli spaces. 4.5.1.5. Kaehler cones of moduli spaces of curves. 4.5.1.6. Kodaira dimensions of moduli spaces of curves. 4.5.1.7. Mumford-Harris theorems. 4.5.2. Analytical theory of moduli spaces of curves. 4.5.2.1. Beltrami differentials. 4.5.2.2. Quadratic differentials. 4.5.2.3. Teichmuller operations. 4.5.2.4. Teichmuller metrics. 4.5.2.5. Teichmuller spaces. 4.5.2.6. Kobayashi metrics on Teichmuller spaces. 4.5.2.7. Roiden recognition theorems. 4.5.2.8. Modular groups. 4.5.2.9. Dehn generators of modular groups. 4.5.2.10. Commutants of modular groups and Picard groups of modular stacks. 4.5.2.10. Fricke modules of curves. 4.5.2.11. Analytical torsions. 4.5.2.12. Rodemeister torsions 4.5.2.13. Equivalence theorem. 4.5.2.14. Selberg theorem on zeta-functions. 4.5.3. Moduli spaces of flat structures on curves. 4.5.3.1. Flat structures. 4.5.3.2. Monodromy maps of spaces of flat structures. 4.5.3.3. Eichler cohomology of representations. 4.5.3.4. Differentials of monodromy maps. 4.5.3.5. Images of monodromy maps. 4.5.3.6. Graphtings. 4.5.3.7. Poincare problem. 4.5.3.8. Relations to Hitchin theory. * 4.5.4. Moduli spaces of marked curves. 4.5.4.1. Geometry of moduli spaces of marked rational curves. 4.5.4.2. Chow rings. 4.5.4.3. Fusion rules. 4.5.4.4. Weil - Peterson metrics. 4.5.4.5. Relations to TQFT. 4.5.5. Moduli spaces of K3 surfaces. 4.5.5.1. Moduli spaces of K3 surfaces with Picard lattice conditions. 4.5.5.2. Moduli spaces of K3 with automorphisms structure conditions. 4.5.5.3. Nikulin classifications. 4.5.5.4. Moduli spaces of K3 with real structures. 4.5.5.5. Local invariants of 4-dimensional pseudo-Riemann manifolds. 4.5.5.6. Degeneration loci of surfaces. 4.5.5.7. Relations to Kac-Moody algebras. 4.5.6. Special geometries of moduli spaces. 4.5.6.1. Moduli spaces of complex oriented threefolds. 4.5.6.2. Calabi- Yao metrics. 4.5.6.2.Intermedial Jacobians of Calabi-Yao threefolds. 4.5.6.3. Pre-potentials. 4.5.7. Moduli spaces in QFT. 4.5.7.1. Sigma-models. 4.5.7.2. Moduli spaces of QFT. 4.5.7.3. Zamolodchikov metrics. 4.5.7.4. Moduli spaces of classical solutions. 4.5.7.5. Moduli spaces of quantum vacuums. 4.5.7.6. Moduli spaces of TQFT. 4.5.7.7. Frobenius manifolds.
5. Geometrical functors.
5.1. Classical Torelli theorems. 5.1.1. Geometry of theta divisors of curves. 5.1.1.1. Riemann interpretation theorems. 5.1.1.2. Gauss maps. 5.1.1.3. Special classes. 5.1.1.4. Singular points of theta divisors. 5.1.2. Geometry of canonical embeddings. 5.1.2.1. Canonical embeddings of curves. 5.1.2.2. Resolutions of ideal sheaves. 5.1.2.3. Green conjectures. 5.1.2.4. Curves on K3 surfaces. 5.1.2.5. Wahl maps. 5.1.2.6. Extension problems. 5.1.3. Local Torelli theorems. 5.1.3.1. Differentials of Torelli maps. 5.1.3.2. Hyperelliptic case. 5.1.3.3. Immersion theorems. 5.1.4. Global Torelli theorems. 5.1.4.1. Ramification loci of Gauss maps. 5.1.4.2. Intersections of tangent cones. 5.1.4.3. Weil proof. 5.1.5. Schottki problems. 5.1.1.1. Big and small Schottki loci. 5.1.1.2. Welters theorem. 5.1.1.3. Novikov conjecture. 5.1.1.4. Powers of theta divisors. 5.1.1.5. Andreotti-Mayer approach.
5.2. Moduli spaces of vector bundles on curves.
5.2.1. Conformal blocks vector bundles on moduli spaces of curves. 5.2.1.1. Projective flat connections. 5.2.2.2. Hitchin constructions. 5.2.2. Torelli theorems for vector bundles on curves. 5.2.2.1. Intermediate Jacobians of moduli spaces of vector bundles. 5.2.2.2. Mumford - Newstead proof. 5.2.2.3. Lines and subspaces on moduli spaces. 5.2.2.4. Double induction proof.
5.3. Intermediate Jacobians of Fano varieties.
5.3.1. Local Torelli theorems. 5.3.1.1. Differentials of periods maps. 5.3.1.2. Three dimensional cubic case. 5.3.1.3. Mukai interpretations. 5.3.1.4. Fano threefolds of genus 7. 5.3.2. Global Torelli theorems for Fano varieties. 5.3.2.1. Gauss map for cubics. 5.3.1.2. Degeneration arguments. 5.3.1.3. Mukai arguments.