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Let $f: X \to Y$ be a separated morphism between $k$-varieties or more general schemes of finite type. The most common way in standard literature on algebraic geometry to define the sheaf of relative …

I'm reading "Heat kernels and Dirac operators" by Berline, Getzler and Vergne. I have some trouble to understand a identity on the bottom of page 146 which is essential for the proof of the Atiyah-Sin …

I'm reading "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne. The setting is: $E \to M$ is a $\mathbb{Z}_2$-graded vector bundle on a compact Riemannian manifold $M$ and $D : \Gamma(M …

Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) consist …

I have a general question about the motivation behind to definition the smooth morphisms as we know it from algebraic geometry. The most common definition of a smooth morphism $: X \to Y$ between two …

Let $X$ be a Kahler manifold. Using the exponential sequence one obtains a homomorphism $c_1:H^1(X,\mathcal{O}_X^*)\rightarrow H^2(X,\mathbb{Z})$. This is associating to a holomorphic line bundle $L$ …

The Fubini-Study metric $g:=g_{FS}$ is the unique $U(n+1)$-invariant Riemannian metric on the complex projective space $\mathbb{CP}^{n}$ the complex projective space which by $U(n+1)$-invariance can …

supplement/ an "almost" answer: I noticed that OP's of several related questions (1, 2) asked about similar issue. The best explanation I found there was that for a smooth manifold $X$ the tangent spa …

David Mumford remarks in his book Algebraic Geometry I, Complex Projective Varieties on page 109 that the fact that the canonical ring $\oplus_{k=0}^{\infty} \Omega_{k, \mathbb{P}^n}$ of projective sp …

(In following we are working in "classical" complex setting: i.e. all involved schemes are considered to be varieties over $k=\mathbb{C}$) Let $X \subset \mathbb{P}^n$ be irreducible surface and $L $ …

I have a question about a remark on Fubini Study metric on $\mathbb{CP}^n$ from Notes on canonical Kähler metrics on page 8 is remarked (Example 2.12 4.): Fix a Hermitian innerproduct on $\mathbb{C}^ …

Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 …