The K3 category of cubic fourfolds and Gushel-Mukai fourfolds Laura Pertusi Università di Roma Tor Vergata December 20, 2019
Introduction Derived category K3 categories Complex Algebraic Geometry Aim: to study smooth projective varieties over C . Defined as zero loci of systems of homogeneous polynomial equations with coefficients in C . Example (Cubic fourfolds) In P 5 C , given f ∈ C [ x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] homogeneous of degree 3 with ( ∂ f /∂ x i ) � = 0 , define Y := Z ( f ) = { x ∈ P 5 C : f ( x ) = 0 } . For example, consider x 3 0 + x 3 1 + · · · + x 3 5 = 0 in P 5 C . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Complex Algebraic Geometry Aim: to study smooth projective varieties over C . Defined as zero loci of systems of homogeneous polynomial equations with coefficients in C . Example (Cubic fourfolds) In P 5 C , given f ∈ C [ x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] homogeneous of degree 3 with ( ∂ f /∂ x i ) � = 0 , define Y := Z ( f ) = { x ∈ P 5 C : f ( x ) = 0 } . For example, consider x 3 0 + x 3 1 + · · · + x 3 5 = 0 in P 5 C . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Complex Algebraic Geometry Aim: to study smooth projective varieties over C . Defined as zero loci of systems of homogeneous polynomial equations with coefficients in C . Example (Cubic fourfolds) In P 5 C , given f ∈ C [ x 0 , x 1 , x 2 , x 3 , x 4 , x 5 ] homogeneous of degree 3 with ( ∂ f /∂ x i ) � = 0 , define Y := Z ( f ) = { x ∈ P 5 C : f ( x ) = 0 } . For example, consider x 3 0 + x 3 1 + · · · + x 3 5 = 0 in P 5 C . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Complex Algebraic Geometry Our tool: vector bundles ↔ locally free sheaves. Example (Trivial vector bundle) Given a smooth complex projective variety X , consider X × C r → X . If r = 1, this is the structure sheaf O X . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Complex Algebraic Geometry Our tool: vector bundles ↔ locally free sheaves. Example (Trivial vector bundle) Given a smooth complex projective variety X , consider X × C r → X . If r = 1, this is the structure sheaf O X . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Complex Algebraic Geometry Our tool: vector bundles ↔ locally free sheaves. Example (Trivial vector bundle) Given a smooth complex projective variety X , consider X × C r → X . If r = 1, this is the structure sheaf O X . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Complex Algebraic Geometry Our tool: vector bundles ↔ locally free sheaves. Example (Trivial vector bundle) Given a smooth complex projective variety X , consider X × C r → X . If r = 1, this is the structure sheaf O X . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Motivations Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P 1 C vector bundles are direct sums of line bundles. Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required! Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Motivations Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P 1 C vector bundles are direct sums of line bundles. Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required! Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Motivations Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P 1 C vector bundles are direct sums of line bundles. Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required! Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Motivations Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P 1 C vector bundles are direct sums of line bundles. Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required! Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Motivations Vector bundles are classified by using numerical invariants, like the rank or the degree. Theorem (Grothendieck) Over P 1 C vector bundles are direct sums of line bundles. Consider vector bundles with some fixed numerical invariants as points of an algebraic space. Problem Is there a moduli space parametrizing vector bundles on X with fixed numerical invariants having the structure of a smooth projective variety? A notion of stability is required! Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Stability Assume dim ( X ) = 1, i.e. X is a curve. For example, X is the planar curve x 3 0 + x 3 1 + x 3 2 = 0 in P 2 C . Example (Slope stability) Define the function µ ( − ) := deg ( − ) rk ( − ) . A vector bundle E is slope (semi)stable if for every subbundle F ֒ → E , we have µ ( F )( ≤ ) < µ ( E ) . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Stability Assume dim ( X ) = 1, i.e. X is a curve. For example, X is the planar curve x 3 0 + x 3 1 + x 3 2 = 0 in P 2 C . Example (Slope stability) Define the function µ ( − ) := deg ( − ) rk ( − ) . A vector bundle E is slope (semi)stable if for every subbundle F ֒ → E , we have µ ( F )( ≤ ) < µ ( E ) . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Stability Assume dim ( X ) = 1, i.e. X is a curve. For example, X is the planar curve x 3 0 + x 3 1 + x 3 2 = 0 in P 2 C . Example (Slope stability) Define the function µ ( − ) := deg ( − ) rk ( − ) . A vector bundle E is slope (semi)stable if for every subbundle F ֒ → E , we have µ ( F )( ≤ ) < µ ( E ) . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Moduli spaces Theorem (Mumford) If X is a smooth projective curve of genus g ≥ 2 , for every pair ( r , d ) ∈ N ∗ × Z , there is a projective variety M ( r , d ) parametrizing semistable vector bundles of rank r and degree d on X . Moreover, the open subset M s ( r , d ) ⊂ M ( r , d ) of stable bundles is non-empty and smooth. Theorem (Atiyah) Case of vector bundles over curves of genus 1 . Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Coherent sheaves Problem: The category Vect ( X ) of vector bundles over X is not abelian. Solution: consider the category of coherent sheaves Vect ( X ) ⊂ Coh ( X ) . Kernel, image and exact sequence are well-defined notions. 1 Locally of the form 2 O ⊕ k 1 → O ⊕ k 2 → E → 0 . X X Set µ ( E ) = + ∞ if rk ( E ) = 0 � notion of stability for 3 coherent sheaves. Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Coherent sheaves Problem: The category Vect ( X ) of vector bundles over X is not abelian. Solution: consider the category of coherent sheaves Vect ( X ) ⊂ Coh ( X ) . Kernel, image and exact sequence are well-defined notions. 1 Locally of the form 2 O ⊕ k 1 → O ⊕ k 2 → E → 0 . X X Set µ ( E ) = + ∞ if rk ( E ) = 0 � notion of stability for 3 coherent sheaves. Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Coherent sheaves Problem: The category Vect ( X ) of vector bundles over X is not abelian. Solution: consider the category of coherent sheaves Vect ( X ) ⊂ Coh ( X ) . Kernel, image and exact sequence are well-defined notions. 1 Locally of the form 2 O ⊕ k 1 → O ⊕ k 2 → E → 0 . X X Set µ ( E ) = + ∞ if rk ( E ) = 0 � notion of stability for 3 coherent sheaves. Laura Pertusi K3 category of cubic and GM fourfolds
Introduction Derived category K3 categories Pass to complexes Remark : if X is smooth projective of dimension n , then every F ∈ Coh ( X ) has a locally free resolution 0 → E n → · · · → E 1 → E 0 → F → 0 . � introduce the derived category D b ( X ) ! Laura Pertusi K3 category of cubic and GM fourfolds
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