We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary... more
We present geometric realizations of horospherical two-orbit varieties, by showing that their blow-up along the unique closed invariant orbit is the zero locus of a general section of a homogeneous vector bundle over some auxiliary variety. As an application, we compute the cohomology ring of the G 2 G_2 -variety, including its quantum version. We also consider the S p i n 7 Spin_7 -variety, which deserves a different treatment.
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We study the smooth projective symmetric variety of Picard number one that compactifies the exceptional complex Lie group G2, by describing it in terms of vector bundles on the spinor variety of Spin(14). We call it the double Cayley... more
We study the smooth projective symmetric variety of Picard number one that compactifies the exceptional complex Lie group G2, by describing it in terms of vector bundles on the spinor variety of Spin(14). We call it the double Cayley Grassmannian because quite remarkably, it exhibits very similar properties to those of the Cayley Grassmannian (the other symmetric variety of type G2), but doubled in the certain sense. We deduce among other things that all smooth projective symmetric varieties of Picard number one are infinitesimally rigid.
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We describe a remarkable rank 14 14 matrix factorization of the octic S p i n 14 \mathrm {Spin}_{14} -invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense,... more
We describe a remarkable rank 14 14 matrix factorization of the octic S p i n 14 \mathrm {Spin}_{14} -invariant polynomial on either of its half-spin representations. We observe that this representation can be, in a suitable sense, identified with a tensor product of two octonion algebras. Moreover the matrix factorisation can be deduced from a particular Z \mathbb {Z} -grading of e 8 \mathfrak {e}_8 . Intriguingly, the whole story can in fact be extended to the whole Freudenthal-Tits magic square and yields matrix factorizations on other spin representations, as well as for the degree seven invariant on the space of three-forms in several variables. As an application of our results on S p i n 14 \mathrm {Spin}_{14} , we construct a special rank seven vector bundle on a double-octic threefold, that we conjecture to be spherical.
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Consider the ten-dimensional spinor variety in the projectivization of a half-spin representation of dimension sixteen. The intersection X of two general translates of this variety is a smooth Calabi-Yau fivefold, as well as the... more
Consider the ten-dimensional spinor variety in the projectivization of a half-spin representation of dimension sixteen. The intersection X of two general translates of this variety is a smooth Calabi-Yau fivefold, as well as the intersection Y of their projective duals. We prove that although X and Y are not birationally equivalent, they are derived equivalent and L-equivalent in the sense of Kuznetsov and Shinder.
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We compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov–Witten invariants in the multiplication table of the Schubert... more
We compute the small cohomology ring of the Cayley Grassmannian, that parametrizes four-dimensional subalgebras of the complexified octonions. We show that all the Gromov–Witten invariants in the multiplication table of the Schubert classes are nonnegative and deduce Golyshev’s conjecture [Formula: see text] holds true for this variety. We also check that the quantum cohomology is semisimple and that there exists, as predicted by Dubrovin’s conjecture, an exceptional collection of maximal length in the derived category.
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LetXXbe a general complex Fano threefold of genus88. We prove that the moduli space of rank two semistable sheaves onXXwith Chern numbersc1=1c_1=1,c2=6c_2=6andc3=0c_3=0is isomorphic to the Fano surfaceF(X)F(X)of conics onXX. This surface... more
LetXXbe a general complex Fano threefold of genus88. We prove that the moduli space of rank two semistable sheaves onXXwith Chern numbersc1=1c_1=1,c2=6c_2=6andc3=0c_3=0is isomorphic to the Fano surfaceF(X)F(X)of conics onXX. This surface is smooth and isomorphic to the Fano surface of lines in the orthogonal toXXcubic threefold. InsideF(X)F(X), the nonlocally free sheaves are parameterized by a smooth curve of genus2626isomorphic to the base of the family of lines onX\textrm {X}.
Research Interests: Mathematics, Algebraic Geometry, Pure Mathematics, Algorithm, Annotation, and 4 moreGenus, Pfaffian, F, and J
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Research Interests: Mathematics, Algebraic Geometry, Pure Mathematics, Indexation, M, and 3 moreN, J, and Affine Transformation
It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed (2n - 4)-form on the Fano scheme of lines on a (2n - 2)-dimensional hypersurface Yn of degree n. We... more
It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed (2n - 4)-form on the Fano scheme of lines on a (2n - 2)-dimensional hypersurface Yn of degree n. We provide several definitions of this form — via the Abel–Jacobi map, via Hochschild homology, and via the linkage class — and compute it explicitly for n = 4. In the special case of a Pfaffian hypersurface Yn we show that the Fano scheme is birational to a certain moduli space of sheaves of a (2n - 4)-dimensional Calabi–Yau variety X arising naturally in the context of homological projective duality, and that the constructed form is induced by the holomorphic volume form on X. This remains true for a general non-Pfaffian hypersurface but the dual Calabi–Yau becomes noncommutative.
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This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent role. This selection is largely arbitrary... more
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent role. This selection is largely arbitrary and mainly reflects the interests of the author.
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We prove explicit formulas for Chern classes of tensor products of virtual vector bundles, whose coefficients are given by certain universal polynomials in the ranks of the two bundles.
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ABSTRACT We discuss the problem of determining the possible spectra of a sum of Hermitian matrices each with known spectrum. We explain how the Horn conjecture, which gives a complete answer to this question, is related with algebraic... more
ABSTRACT We discuss the problem of determining the possible spectra of a sum of Hermitian matrices each with known spectrum. We explain how the Horn conjecture, which gives a complete answer to this question, is related with algebraic geometry, symplectic geometry, and representation theory. The first lecture is an introduction to Schubert calculus, from which one direction of Horn's conjecture can be deduced. The reverse direction follows from an application of geometric invariant theory: this is treated in the second lecture. Finally, we explain in the third lecture how a version of Horn's problem for special unitary matrices is related to the quantum cohomology of Grassmannians. Notes for Gael VIII, CIRM, March 2000. 1. Eigenvalues of hermitian matrices and Schubert calculus The problem. Let A; B; C be complex Hermitian n by n matrices. Denote the set of eigenvalues, or spectrum of A by (A) = ( 1 (A) Delta Delta Delta n (A)), and similarly by (B) and (C) the spectra of B and C. The main theme of these notes is the following question: Suppose that A +B = C. What can be their spectra (A), (B), (C) ? There are obvious relations, like trace(C) = trace(A)+trace(B) or 1 (C) 1 (A)+ 1 (B). But a complete answer to this longstanding question was given only recently, and combines works and ideas from representation theory, symplectic and algebraic geometry. Weyl's inequalities. There are various characterizations of the eigenvalues of Hermitian matrices, many of which are variants of the minimax principle. Let ( j ) be the standard Hermitian product on C n . If s = n Gamma r, denote by G r;s the Grassmannian of r-dimensional linear subspaces of C n . Then j+1 (A) = min L2G nGammaj;j max x2L (xjx)=1 (Axjx): The idea is to test the values of (Axjx) on subspaces...
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We show that on a complex flag manifold, a very ample line bundle which is a p p -th power has property N p N_p in the sense of Green and Lazarsfeld. This is a partial answer to a problem raised by Fulton.
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A Steiner bundle over the projective 3-space is the kernel in a trivial bundle of a morphism defined by a matrix of linear forms. We produce various Steiner bundles E of rank n such that E(1) has n-1 sections, the dependency locus of... more
A Steiner bundle over the projective 3-space is the kernel in a trivial bundle of a morphism defined by a matrix of linear forms. We produce various Steiner bundles E of rank n such that E(1) has n-1 sections, the dependency locus of which is a smooth curve.
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On dispose essentiellement de deux resultats importants concernant l'annulation de la cohomologie des puissances exterieures d'un fibre ample E de rang d sur une variete complexe compacte de dimension n. D'apres le premier, du... more
On dispose essentiellement de deux resultats importants concernant l'annulation de la cohomologie des puissances exterieures d'un fibre ample E de rang d sur une variete complexe compacte de dimension n. D'apres le premier, du a Sommese ([19]), les groupes H'(X, /\E) sont nuls lorsque q^d — fc + l.Le second, du Le Potier ([13]), affirme plus generalement que les H-(X, /\E) s'annulent des que p + q^k(d— A:) 4-1. Nous nous proposons, pour notre part, de demontrer l'annulation des groupes precedents sous des hypotheses qui, d'une certaine maniere, f nt le lien entre celles de Sommese et de Le Potier: